View
18
Download
0
Category
Preview:
Citation preview
Algorithmic Graph Theory 1
A Glimpse at Algorithmic A Glimpse at Algorithmic Graph TheoryGraph Theory
吴耀琨吴耀琨 上海交通大学数学系上海交通大学数学系
(Based on a (Based on a pptppt made by M. made by M. GolumbicGolumbic))
上海交大理科班讨论班, 2009/9/27
2Algorithmic Graph Theory
Martin Charles Golumbic, Landmarks in Algorithmic Graph Theory: Martin Charles Golumbic, Landmarks in Algorithmic Graph Theory: A personal Retrospective, LNCS 5420 (2009) 1A personal Retrospective, LNCS 5420 (2009) 1——14.14.
www.dis.uniroma1.it/~seminf/seminars/www.dis.uniroma1.it/~seminf/seminars/golumbic.pptgolumbic.ppt
3Algorithmic Graph Theory
Defining some termsDefining some terms
graph:graph: a collection of vertices and edgesa collection of vertices and edgescoloring a graph:coloring a graph:assigningassigning a color to every a color to every vertexvertex suchsuch that that adjacentadjacent vertices have different colorsvertices have different colors
4Algorithmic Graph Theory
independent set:independent set: a collection of vertices a collection of vertices NONO two of which are connectedtwo of which are connectedExampleExample: { : { d, e, f d, e, f } or the } or the greengreen setsetcliqueclique (or complete set):(or complete set):
EVERYEVERY two of which are two of which are connectedconnectedExampleExample: { : { a, b, d a, b, d } or { } or { c, e c, e } }
5Algorithmic Graph Theory
complement of a graph:complement of a graph:interchanging the edges and the noninterchanging the edges and the non--edgesedges
The complement G The original graph G__
6Algorithmic Graph Theory
directed graph:directed graph: edges have directionsedges have directions(possibly both directions)(possibly both directions)
orientation:orientation: exactly ONE direction per edgeexactly ONE direction per edge
cyclic orientation acyclic orientation
7Algorithmic Graph Theory
Given a family of sets Given a family of sets S(1),S(1),……,S(n),,S(n), we define its we define its intersection graph to be the graph with vertex intersection graph to be the graph with vertex set set v(1),v(1),……,v(n),v(n) and and v(i)v(j)v(i)v(j) is an edge if and only is an edge if and only if if S(i)S(i) and and S(j)S(j) have nonempty intersection.have nonempty intersection.
IntersectionIntersection Representation of a Representation of a GraphGraph
8Algorithmic Graph Theory
Interval GraphsInterval GraphsThe The intersection graphs of intervals on a lineintersection graphs of intervals on a line::
-- create a vertex for each intervalcreate a vertex for each interval-- connect vertices when their intervals intersectconnect vertices when their intervals intersect
Jan Feb Mar Apr May Jun July Sep Oct Nov Dec
Phase 1Phase 2
Phase 3Task 4
Task 5
1 2 3
4 5The interval graph G
9Algorithmic Graph Theory
applications in computationapplications in computationoperations researchoperations researchmolecular biologymolecular biologyschedulingschedulingdesigning circuitsdesigning circuitsrich mathematical problemsrich mathematical problems
10Algorithmic Graph Theory
History of Interval GraphsHistory of Interval GraphsHajosHajos 1957: 1957: Combinatorics (scheduling)Combinatorics (scheduling)BenzerBenzer 1959: 1959: Biology (genetics)Biology (genetics)Gilmore & Hoffman 1964: CharacterizationGilmore & Hoffman 1964: CharacterizationBooth & Booth & LuekerLueker 1976: First linear time 1976: First linear time
recognition recognition algorithmalgorithmMany other applications:Many other applications:
mobile radio frequency assignmentmobile radio frequency assignmentVLSI designVLSI designtemporal reasoning in AItemporal reasoning in AIcomputer storage allocationcomputer storage allocation
Scheduling ExampleScheduling Example
Lectures need to be assigned classrooms at the Lectures need to be assigned classrooms at the University.University.
Lecture #a: 9:00Lecture #a: 9:00--10:1510:15Lecture #b: 10:00Lecture #b: 10:00--12:0012:00etc.etc.
Conflicting lectures Conflicting lectures →→ Different roomsDifferent roomsHow many rooms?How many rooms?
Scheduling Example (cont.)Scheduling Example (cont.)
Scheduling Example (graphs)Scheduling Example (graphs)
(a) The interval graph (b) Its complement (disjointness)
14Algorithmic Graph Theory
Offline Coloring Interval GraphsOffline Coloring Interval Graphs
Interval graphs have special properties.Interval graphs have special properties.These special structural properties These special structural properties guarantees some efficient algorithms.guarantees some efficient algorithms.The greedy coloring algorithm sweeps The greedy coloring algorithm sweeps across from left to right to assign colors across from left to right to assign colors which causes no conflict and is as small which causes no conflict and is as small as possible (Firstas possible (First--Fit).Fit).
15Algorithmic Graph Theory
Coloring Intervals (greedy)Coloring Intervals (greedy)
16Algorithmic Graph Theory
Coloring Interval GraphsColoring Interval Graphs
17Algorithmic Graph Theory
Is the greedy coloring algorithm optimal in Is the greedy coloring algorithm optimal in minimizing the number of used colors?minimizing the number of used colors?
18Algorithmic Graph Theory
Coloring Intervals (greedy)Coloring Intervals (greedy)P (needs 4 colors)
19Algorithmic Graph Theory
Coloring Interval GraphsColoring Interval Graphs
The clique at point P
20Algorithmic Graph Theory
Is greedy the best we can do in Is greedy the best we can do in general?general?
Can we prove optimality? Can we prove optimality?
21Algorithmic Graph Theory
Is greedy the best we can do?Is greedy the best we can do?Can we prove optimality? Can we prove optimality? Yes: It uses the smallest # colors.Yes: It uses the smallest # colors.
22Algorithmic Graph Theory
Is greedy the best we can do?Is greedy the best we can do?Can we prove optimality? Can we prove optimality? Yes: It uses the smallest # colors.Yes: It uses the smallest # colors.
Proof: Let k be the number of colors used.
Look at the point P, when color k was used first (This is the left endpoint of the first interval with color k).
At P all the colors 1 to k-1 were busy!
We are forced to use k colors at P and the corresponding intervals form a clique of size k in the interval graph.
23Algorithmic Graph Theory
Greedy the best we can do !Greedy the best we can do !
Formally,
(1) at least k colors are required
(because of the clique)
(2) greedy succeeded using k colors.
Therefore,
the solution is optimal. Q.E.D.
24Algorithmic Graph Theory
Perfect GraphsPerfect Graphs
AA graph is perfect if and only if the clique graph is perfect if and only if the clique number and the chromatic number of each number and the chromatic number of each induced induced subgraphsubgraph of it are equal.of it are equal.
Any induced subgraph of a perfect graph is Any induced subgraph of a perfect graph is perfect.perfect.
IntervalInterval graphs are perfect.graphs are perfect.
25Algorithmic Graph Theory
Perfect Perfect Graphs (Contd.)Graphs (Contd.)
AA graph is perfect if and only if the clique graph is perfect if and only if the clique number and the chromatic number of each number and the chromatic number of each induced induced subgraphsubgraph of it are equal.of it are equal.
IsIs the complement of an interval the complement of an interval graphgraph perfect?perfect?
Namely, is the smallest size of a transversal of a Namely, is the smallest size of a transversal of a family of intervals equal to the biggest size of a family of intervals equal to the biggest size of a set of pariwise disjoint intervals from the same set of pariwise disjoint intervals from the same family?family?
26Algorithmic Graph Theory
Perfect Graph Theorem (Perfect Graph Theorem (LovaszLovasz, born March 9, , born March 9, 19481948 ) : ) : A graph is perfect A graph is perfect iffiff its complement is its complement is perfect. perfect.
Laszlo Laszlo LovaszLovasz (1972). "Normal (1972). "Normal hypergraphshypergraphs and the perfect and the perfect graph conjecture". Discrete Mathematics 2: 253graph conjecture". Discrete Mathematics 2: 253––267.267.
Laszlo Laszlo LovaszLovasz won both the Wolf prize and the won both the Wolf prize and the Knuth prize in 1999. He is the president of the Knuth prize in 1999. He is the president of the International Mathematical Union since 2007.International Mathematical Union since 2007.
27Algorithmic Graph Theory
Can you think of a simple graph which is not Can you think of a simple graph which is not perfect?perfect?
28Algorithmic Graph Theory
AnAn induced cycle of odd length at least 5 is induced cycle of odd length at least 5 is called an odd hole. An induced called an odd hole. An induced subgraphsubgraphthat is the complement of an odd hole is that is the complement of an odd hole is called an odd called an odd antiholeantihole. .
AA graph that does not contain any odd graph that does not contain any odd holes or odd holes or odd antiholesantiholes is called a Berge is called a Berge graph. All perfect graphs are Berge graphs.graph. All perfect graphs are Berge graphs.
29Algorithmic Graph Theory
Strong Perfect Graph Theorem. A graph is perfect if and onlyif it is a Berge graph. Namely, the odd holes and odd antiholesare a complete list of obstructions for perfectness of graphs.
Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin (2006), "The strong perfect graph theorem". Annals of Mathematics 164 (1): 51--229.
30Algorithmic Graph Theory
Maria Maria ChudnovskyChudnovsky
Department of Industrial Department of Industrial Engineering and Operations Engineering and Operations ResearchResearch
Department of Mathematics Department of Mathematics
Columbia UniversityColumbia University
31Algorithmic Graph Theory
Interview with Chudnovsky
32Algorithmic Graph Theory
Interview with Chudnovsky
33Algorithmic Graph Theory
Interview with Chudnovsky
34Algorithmic Graph Theory
How good is the greedy algorithm How good is the greedy algorithm for online coloring interval graphs?for online coloring interval graphs?The smallest The smallest number ofnumber of colors needed to color colors needed to color an interval graph is its clique number k.an interval graph is its clique number k.For any order of the arrival of intervals, we use For any order of the arrival of intervals, we use the greedy algorithm (Firstthe greedy algorithm (First--Fit) to color them. Fit) to color them. The number of colors used in the worst case is The number of colors used in the worst case is denoted denoted FF(kFF(k) .) .FF(k) FF(k) ≤≤ 8k8k--3; 3; FF(k) FF(k) ≥≥ 4.99k when k is 4.99k when k is sufficiently large.sufficiently large.Conjecture: As Conjecture: As kk tends to infinity, the ratio tends to infinity, the ratio FFFF((kk)/)/kk tends to 5.tends to 5.
35Algorithmic Graph Theory
Characterizing Interval GraphsCharacterizing Interval Graphs
Properties of interval graphsProperties of interval graphsHow to recognize themHow to recognize themTheir mathematical structureTheir mathematical structure
36Algorithmic Graph Theory
Characterizing Interval GraphsCharacterizing Interval Graphs
Properties of interval graphsProperties of interval graphsHow to recognize themHow to recognize themTheir mathematical structureTheir mathematical structure
Two properties characterize interval graphs:
- The Chordal Graph Property
- The co-TRO Property
37Algorithmic Graph Theory
The coThe co--TRO PropertyTRO Property
The transitive orientation (TRO) of the The transitive orientation (TRO) of the complement complement i.e., the complement must have a TROi.e., the complement must have a TRO
Not transitive ! Transitive !
38Algorithmic Graph Theory
Interval Graphs are coInterval Graphs are co--TROTRO
The complement of an Interval graph has a The complement of an Interval graph has a transitive orientation!transitive orientation!
-- Why?Why?
The complement is the disjointness graph.
So, orient from the earlier interval
to the later interval.
39Algorithmic Graph Theory
The Chordal Graph PropertyThe Chordal Graph Propertychordal graph: chordal graph:
every cycle of length every cycle of length >> 4 has a chord4 has a chord(connecting two vertices that are not consecutive)(connecting two vertices that are not consecutive)
i.e., they may not contain chordless cycles!i.e., they may not contain chordless cycles!
40Algorithmic Graph Theory
Interval Graphs are ChordalInterval Graphs are Chordal
Interval graphs may not contain chordless cycles!Interval graphs may not contain chordless cycles!
-- i.e., they are chordal. Why?i.e., they are chordal. Why?
41Algorithmic Graph Theory
Interval Graphs are ChordalInterval Graphs are Chordal
Interval graphs may not contain chordless cycles!Interval graphs may not contain chordless cycles!
-- i.e., they are chordal. Why?i.e., they are chordal. Why?
42Algorithmic Graph Theory
Interval Graphs are ChordalInterval Graphs are Chordal
Let [v(1)v(2)Let [v(1)v(2)……v(n)] be a cycle corresponding to v(n)] be a cycle corresponding to intervals I(1),intervals I(1),……,I(n). Without loss of generality, ,I(n). Without loss of generality, suppose that the right endpoint of I(1) is the suppose that the right endpoint of I(1) is the smallest among all right endpoints of intervals smallest among all right endpoints of intervals I(1),I(1),……,I(n). Then we see that v(2) and v(n) are ,I(n). Then we see that v(2) and v(n) are joined by an edge which is a chord of the cycle if joined by an edge which is a chord of the cycle if nn≥≥4.4.
43Algorithmic Graph Theory
There are tons of different There are tons of different characterizations (definitions, characterizations (definitions, representations) of representations) of chordalchordal graphs graphs (interval (interval graphs). This fact is really graphs). This fact is really AMAZINGAMAZING!!ChordalChordal graphs are exactly the graphs are exactly the intersection graphs of a family of intersection graphs of a family of subtreessubtreesof a common host tree.of a common host tree.When the host tree is a path, the When the host tree is a path, the chordalchordalgraph is an interval graph.graph is an interval graph.
44Algorithmic Graph Theory
Gilmore and Hoffman (1964)Gilmore and Hoffman (1964)
Theorem:Theorem:A graph A graph G G is an interval graph is an interval graph ifif and only if and only if G G Is chordal and Is chordal and its complement its complement G G is transitively is transitively orientableorientable. .
__
This provides the basis for the first set of recognition algorithms in the early 1970’s.
45Algorithmic Graph Theory
A Mystery in the LibraryA Mystery in the LibraryThe Berge Mystery Story:The Berge Mystery Story:
Six professors had been to the library on the Six professors had been to the library on the day that the rare tractate was stolen. day that the rare tractate was stolen.
Each had entered once, stayed for some time Each had entered once, stayed for some time and then left. and then left.
If two were in the library at the same time, then If two were in the library at the same time, then at least one of them saw the other. at least one of them saw the other.
Detectives questioned the professors and Detectives questioned the professors and gathered the following testimony: gathered the following testimony:
Abe said that he saw Burt and Eddie Abe said that he saw Burt and Eddie Burt reported that he saw Abe and Ida Burt reported that he saw Abe and Ida Charlotte claimed to have seen Charlotte claimed to have seen DesmondDesmond and Idaand IdaDesmond said that he saw Abe and IdaDesmond said that he saw Abe and IdaEddie testified to seeing Burt and CharlotteEddie testified to seeing Burt and CharlotteIda said that she saw Charlotte and EddieIda said that she saw Charlotte and Eddie
One of the Professor LIED One of the Professor LIED by asserting to see by asserting to see somebody whom he did not see!!somebody whom he did not see!! Who was it?Who was it?
The Facts:The Facts:
Solving the MysterySolving the Mystery
The Testimony Graph
Clue #1:
Double arrows imply TRUTH
Solving the MysterySolving the Mystery
Undirected Testimony Graph
We know there is a lie, since {A, B, I, D} is a chordless 4-cycle.
cycle
Intersecting Intervals Intersecting Intervals cannotcannotform Chordless Cyclesform Chordless Cycles
Burt Desmond
Abe
No place for Ida’s interval:It must hit both B and D but cannot hit A.
Impossible!
Solving the MysterySolving the Mystery
There are three chordless 4-cycles:{A, B, I, D} {A, D, I, E} {A, E, C, D}
The liar must be a common member of all the above 4-cycles and hence is either Abe or Desmond.
WHO IS THE LIAR? Abe or Desmond ?
One professor from the chordless 4One professor from the chordless 4--cycle must be a liar.cycle must be a liar.
Solving the Mystery (cont.)Solving the Mystery (cont.)
WHO IS THE LIAR? Abe or Desmond ?
If Abe were the liar and Desmond truthful, then {A, B, I, D} would remain a chordless 4-cycle in the interval graph with the time periodof each professor staying in the library as an intersection model.
Therefore:
Desmond is the liar.
53Algorithmic Graph Theory
Was Desmond Stupid or Was Desmond Stupid or Just Ignorant?Just Ignorant?
If Desmond had studied algorithmic graph theory, he If Desmond had studied algorithmic graph theory, he would have known that his testimony to the police would have known that his testimony to the police would not hold up.would not hold up.
Can you formulate any general results on those kind Can you formulate any general results on those kind of lies which will be detected? Namely, how to of lies which will be detected? Namely, how to recognize the testimony graph from which the liar can recognize the testimony graph from which the liar can be detected? Can you try to develop an efficient be detected? Can you try to develop an efficient algorithm to detect such a liar provided it is possible algorithm to detect such a liar provided it is possible to be detected?to be detected?
54Algorithmic Graph Theory
Many other Families of Many other Families of Intersection GraphsIntersection Graphs
Victor Klee, in a paper in 1969:Victor Klee, in a paper in 1969:
``What are the intersection graphs of arcs ``What are the intersection graphs of arcs in a circle?in a circle?’’’’
55Algorithmic Graph Theory
Many other Families of Many other Families of Intersection GraphsIntersection Graphs
Victor Klee, in a paper in 1969:Victor Klee, in a paper in 1969:
``What are the intersection graphs of arcs in a ``What are the intersection graphs of arcs in a circle?circle?““
56Algorithmic Graph Theory
Many other Families of Many other Families of Intersection GraphsIntersection Graphs
Victor Klee, in a paper in 1969:Victor Klee, in a paper in 1969:
``What are the intersection graphs of arcs in a circle?``What are the intersection graphs of arcs in a circle?““
KleeKlee’’s paper was an implicit challenges paper was an implicit challenge-- consider a whole variety of problems consider a whole variety of problems -- on many kinds of intersection graphs. on many kinds of intersection graphs.
57Algorithmic Graph Theory
The problem should be of intrinsic interest in even a The problem should be of intrinsic interest in even a very special form, but should admit of interesting very special form, but should admit of interesting extensions. In my opinion, a good problem is extensions. In my opinion, a good problem is sufficiently specific so that even the specific form is of sufficiently specific so that even the specific form is of interest to someone, but of course itinterest to someone, but of course it’’s best if a specific s best if a specific solution inspires further questions and generalizations. I solution inspires further questions and generalizations. I deal with specific case, if a meaningful (i.e, not obvious deal with specific case, if a meaningful (i.e, not obvious but not impossible) one can be found. Then but not impossible) one can be found. Then ““brainstormbrainstorm””, looking for natural generalizations and, if , looking for natural generalizations and, if possible, applications. possible, applications. –– Victor L. Klee Victor L. Klee
58Algorithmic Graph Theory
59Algorithmic Graph Theory
Families of Intersection Families of Intersection GraphsGraphsArising from ApplicationsArising from Applications
boxes in the planeboxes in the planepaths in a treepaths in a treechords of a circlechords of a circlespheres in 3spheres in 3--spacespacetrapezoids, parallelograms, curves of functionstrapezoids, parallelograms, curves of functionsmany other geometrical and topological bodiesmany other geometrical and topological bodies
60Algorithmic Graph Theory
Families of Intersection Families of Intersection Graphs Graphs Arising from ApplicationsArising from Applications
boxes in the planeboxes in the planepaths in a treepaths in a treechords of a circlechords of a circlespheres in 3spheres in 3--spacespacetrapezoids, parallelograms, curves of functionstrapezoids, parallelograms, curves of functionsmany other geometrical and topological bodiesmany other geometrical and topological bodies
The Algorithmic Problems:– recognize them– color them– find maximum cliques – find maximum independent sets
61Algorithmic Graph Theory
The Interval Graph Sandwich The Interval Graph Sandwich ProblemProblem
BenzerBenzer’’ss original problem original problem partial intersection datapartial intersection dataIs it consistent ?Is it consistent ?
For complete data this is the For complete data this is the recognition problem for interval graphs recognition problem for interval graphs (polynomial)(polynomial)For partial data we arrive at a different For partial data we arrive at a different model model thatthat is NPis NP--complete.complete.
62Algorithmic Graph Theory
Interval Graph Sandwich ProblemInterval Graph Sandwich Problem
given a partially specified graph given a partially specified graph EE11 required edgesrequired edgesEE22 optional edgesoptional edgesEE33 forbidden edgesforbidden edges
Can you fillCan you fill--in some of the optional in some of the optional edges, soedges, so that the that the result will be an interval result will be an interval graph?graph?Namely, is there a set E such that Namely, is there a set E such that EE11⊆⊆EE⊆⊆ EE11 ∪∪ EE3 3
and that (V,E) is an interval graph?and that (V,E) is an interval graph?
Golumbic & Shamir (1993): Golumbic & Shamir (1993): NPNP--CompleteComplete
63Algorithmic Graph Theory
Interval Probe GraphsInterval Probe Graphs
A special tractable case of interval A special tractable case of interval graph graph sandwich problemsandwich problemComputational biology motivatedComputational biology motivated
Interval probe graph: vertices are partitionedInterval probe graph: vertices are partitionedP P probesprobes & N & N nonnon--probesprobes (independent set)(independent set)
can fillcan fill--in some of the N x N edges,in some of the N x N edges,
so that the result will be an interval graphso that the result will be an interval graph
64Algorithmic Graph Theory
Example: Interval Probe GraphsExample: Interval Probe GraphsNon-Probes are white
Probe graph NOT a Probe graph no matter how you partition vertices!
65Algorithmic Graph Theory
(Golumbic, Maffray, Morel, Annals of Operations Research, (Golumbic, Maffray, Morel, Annals of Operations Research, DOI10.1007/s10479DOI10.1007/s10479--009009--05840584--66 ) A bipartite graph is an ) A bipartite graph is an interval probe graph if and only if it contains none of the six interval probe graph if and only if it contains none of the six forbidden graphs in the figure below.forbidden graphs in the figure below.
66Algorithmic Graph Theory
A A ChordalChordal Graph Sandwich Graph Sandwich ProblemProblem
The The kthkth power of a graph G is the graph power of a graph G is the graph on the same vertex set and two different on the same vertex set and two different vertices are joined by an edge if and only if vertices are joined by an edge if and only if their distance in G is at most k.their distance in G is at most k.A Conjecture of A Conjecture of GavoilleGavoille : Suppose G is : Suppose G is not a not a chordalchordal graph and the longest graph and the longest chordlesschordless cycle in G has length k. Then, we cycle in G has length k. Then, we can find a can find a chordalchordal graph graph inbetweeninbetween G and G and the the ┌┌k/3k/3┐┐thth power of G.power of G.
67Algorithmic Graph Theory
Tolerance GraphsTolerance GraphsWhat if you only have 3 classrooms?What if you only have 3 classrooms?Cancel a Lecture? or show Tolerance?Cancel a Lecture? or show Tolerance?
68Algorithmic Graph Theory
Tolerance GraphsTolerance Graphs
Measured intersection: small, or ``tolerable’’ amount of overlap, may be ignored does NOT produce an edge
at least one of them has to be ``bothered’’
69Algorithmic Graph Theory
Tolerance GraphsTolerance Graphs
Assignment of positive numbers Assignment of positive numbers {{ttvv} (} (v v ∈∈ VV)) such thatsuch that
vwvw ∈∈ E if and only if E if and only if | | IIv v ∩∩ IIww | | ≥≥ min min {{ttvv ,, ttww}}
Measured intersection: small, or ``tolerable’’ amount of overlap, may be ignored does NOT produce an edge
at least one of them has to be ``bothered’’
70Algorithmic Graph Theory
Tolerance Graphs: ExampleTolerance Graphs: Example
c and f will no longer conflict
| Ic ∩ If | < 60 = min {tc , tf}
71Algorithmic Graph Theory
Rather than the arduous and systematic study of every new Rather than the arduous and systematic study of every new concept definable with a graph, the main task for the concept definable with a graph, the main task for the mathematician is to eliminate the often arbitrary and mathematician is to eliminate the often arbitrary and cubersom definitions, keeping only the cubersom definitions, keeping only the ““deepdeep””mathematical problems.mathematical problems.Of course, the deep problems may well be elusive; indeed, Of course, the deep problems may well be elusive; indeed, there have been many definitions (from Dieudonne, among there have been many definitions (from Dieudonne, among others) of what a deep problem is. In graph theory, it others) of what a deep problem is. In graph theory, it should relate to a variety of other combinatorial structures should relate to a variety of other combinatorial structures and must therefore be connected with many difficult and must therefore be connected with many difficult practical problems. Among these will be problems that practical problems. Among these will be problems that classical algebra is not able to solve completely or that the classical algebra is not able to solve completely or that the computer scientists would not attack by himself. computer scientists would not attack by himself. –– Claude Claude BergeBerge
72Algorithmic Graph Theory
Some BooksSome Books
Andreas Andreas BrandstBrandstäädtdt, Van Bang Le, Jeremy P. , Van Bang Le, Jeremy P. SpinradSpinrad, Graph , Graph Classes: A Survey, Classes: A Survey, SIAM,SIAM, 1999.1999.Jeremy P. Jeremy P. SpinradSpinrad, Efficient Graph Representations, AMS, 2003., Efficient Graph Representations, AMS, 2003.MartinMartin Charles Charles GolumbicGolumbic, Algorithmic Graph Theory and , Algorithmic Graph Theory and Perfect Graphs, North Holland, 2004. Perfect Graphs, North Holland, 2004. Martin Charles Martin Charles GolumbicGolumbic, Tolerance Graphs, Cambridge , Tolerance Graphs, Cambridge University Press, 2004.University Press, 2004.Terry A. McKee, F.R. Terry A. McKee, F.R. McMorrisMcMorris, Topics in Intersection Graph , Topics in Intersection Graph Theory, Theory, SIAM,SIAM, 1999. 1999. P.C. P.C. FishburnFishburn, Interval , Interval OrdersOrders and and IntervalInterval GraphsGraphs, Wiley, 1985. , Wiley, 1985. N.V.R. Mahadev, U.N. Peled, Threshold Graphs and Related N.V.R. Mahadev, U.N. Peled, Threshold Graphs and Related Topics, Elsevier, 1995. Topics, Elsevier, 1995.
Some tree-likeness parameters: Chordality,hyperbolicity and tree-length
�n
Shanghai Jiao Tong University
þ°��n��?Ø�, 2009/9/27.
Outline
Introduction
Hyperbolicity
Tree-length
Chordality
Hyperbolicity vs chordality
Conclusion
Tree-likeness
I Trees are graphs with some very distinctive andfundamental properties and it is legitimate to ask to whatdegree those properties can be transferred to moregeneral structures that are tree-like in some sense. – R.Diestel [32, p. 253]
I Roughly speaking, tree-likeness stands for somethingrelated to low dimensionality, low complexity, efficientinformation deduction (from local to global),information-lossless decomposition (from global intosimple pieces) and nice shape for efficient implementationof divide-and-conquer strategy.
Tree-likeness
I Trees are graphs with some very distinctive andfundamental properties and it is legitimate to ask to whatdegree those properties can be transferred to moregeneral structures that are tree-like in some sense. – R.Diestel [32, p. 253]
I Roughly speaking, tree-likeness stands for somethingrelated to low dimensionality, low complexity, efficientinformation deduction (from local to global),information-lossless decomposition (from global intosimple pieces) and nice shape for efficient implementationof divide-and-conquer strategy.
Tree-likeness, Contd.
I Researchers in different fields develop many differentmeasures of tree-likeness and it deserves to investigatethe relationship among them.
I In vast applications, one finds that the borderline betweentractable and intractable cases may be the tree-like degreeof the structure in consideration.
I Many practical structures we face with are very tree-like.
Tree-likeness, Contd.
I Researchers in different fields develop many differentmeasures of tree-likeness and it deserves to investigatethe relationship among them.
I In vast applications, one finds that the borderline betweentractable and intractable cases may be the tree-like degreeof the structure in consideration.
I Many practical structures we face with are very tree-like.
Tree-likeness, Contd.
I Researchers in different fields develop many differentmeasures of tree-likeness and it deserves to investigatethe relationship among them.
I In vast applications, one finds that the borderline betweentractable and intractable cases may be the tree-like degreeof the structure in consideration.
I Many practical structures we face with are very tree-like.
Internet and hyperbolicity
I Hyperbolicity is a measure of tree-likeness and it isreasonable to say that lower hyperbolicity stands for beingcloser to a tree.
I A network with low hyperbolicity allows many goodnetworking algorithms.
I Many experiments say that some large practicalcommunication networks, including the internet, havesurprisingly low hyperbolicity.
I It is interesting to understand why the internet has lowhyperbolicity.
Internet and hyperbolicity
I Hyperbolicity is a measure of tree-likeness and it isreasonable to say that lower hyperbolicity stands for beingcloser to a tree.
I A network with low hyperbolicity allows many goodnetworking algorithms.
I Many experiments say that some large practicalcommunication networks, including the internet, havesurprisingly low hyperbolicity.
I It is interesting to understand why the internet has lowhyperbolicity.
Internet and hyperbolicity
I Hyperbolicity is a measure of tree-likeness and it isreasonable to say that lower hyperbolicity stands for beingcloser to a tree.
I A network with low hyperbolicity allows many goodnetworking algorithms.
I Many experiments say that some large practicalcommunication networks, including the internet, havesurprisingly low hyperbolicity.
I It is interesting to understand why the internet has lowhyperbolicity.
Internet and hyperbolicity
I Hyperbolicity is a measure of tree-likeness and it isreasonable to say that lower hyperbolicity stands for beingcloser to a tree.
I A network with low hyperbolicity allows many goodnetworking algorithms.
I Many experiments say that some large practicalcommunication networks, including the internet, havesurprisingly low hyperbolicity.
I It is interesting to understand why the internet has lowhyperbolicity.
Internet and hyperbolicity, Contd.
Some data from M. Soto in the CAIDA project:
Graph of the routing machines:Tree-width ≥ 234Tree-length ≤ 10Diameter = 9Gromov hyperbolicity = 3, but for for 96% of the vertices itsvalue is 1.
Autonomus system internet topology:Tree-width ≥ 82Tree-length ≤ 6Diameter = 10Gromov hyperbolicity = 2, but for 98% of the vertices its valueis 1
Internet and hyperbolicity, Contd.
Some data from M. Soto in the CAIDA project:
Graph of the routing machines:Tree-width ≥ 234Tree-length ≤ 10Diameter = 9Gromov hyperbolicity = 3, but for for 96% of the vertices itsvalue is 1.
Autonomus system internet topology:Tree-width ≥ 82Tree-length ≤ 6Diameter = 10Gromov hyperbolicity = 2, but for 98% of the vertices its valueis 1
Internet and hyperbolicity, Contd.
Some data from M. Soto in the CAIDA project:
Graph of the routing machines:Tree-width ≥ 234Tree-length ≤ 10Diameter = 9Gromov hyperbolicity = 3, but for for 96% of the vertices itsvalue is 1.
Autonomus system internet topology:Tree-width ≥ 82Tree-length ≤ 6Diameter = 10Gromov hyperbolicity = 2, but for 98% of the vertices its valueis 1
A problem from metric graph theory: characterizinglow hyperbolicity graphs
I Hyperbolicity takes nonnegative half-integer value.I 0-Hyperbolic graphs are just block graphs and are
well-understood.I Bandelt and Chepoi [7] obtained a characterization of all
12 -hyperbolic graphs in terms of a convexity condition. Butno corresponding finite forbidden isometric subgraphcharacterization is known till now.
I Bandelt and Chepoi [8]: “a characterization of all1-hyperbolic graphs by forbidden isometric subgraphs isnot in sight, in as much as isometric cycles of lengths up to7 may occur, thus complicating the picture”.
A problem from metric graph theory: characterizinglow hyperbolicity graphs
I Hyperbolicity takes nonnegative half-integer value.I 0-Hyperbolic graphs are just block graphs and are
well-understood.I Bandelt and Chepoi [7] obtained a characterization of all
12 -hyperbolic graphs in terms of a convexity condition. Butno corresponding finite forbidden isometric subgraphcharacterization is known till now.
I Bandelt and Chepoi [8]: “a characterization of all1-hyperbolic graphs by forbidden isometric subgraphs isnot in sight, in as much as isometric cycles of lengths up to7 may occur, thus complicating the picture”.
A problem from metric graph theory: characterizinglow hyperbolicity graphs
I Hyperbolicity takes nonnegative half-integer value.I 0-Hyperbolic graphs are just block graphs and are
well-understood.I Bandelt and Chepoi [7] obtained a characterization of all
12 -hyperbolic graphs in terms of a convexity condition. Butno corresponding finite forbidden isometric subgraphcharacterization is known till now.
I Bandelt and Chepoi [8]: “a characterization of all1-hyperbolic graphs by forbidden isometric subgraphs isnot in sight, in as much as isometric cycles of lengths up to7 may occur, thus complicating the picture”.
A problem from metric graph theory: characterizinglow hyperbolicity graphs
I Hyperbolicity takes nonnegative half-integer value.I 0-Hyperbolic graphs are just block graphs and are
well-understood.I Bandelt and Chepoi [7] obtained a characterization of all
12 -hyperbolic graphs in terms of a convexity condition. Butno corresponding finite forbidden isometric subgraphcharacterization is known till now.
I Bandelt and Chepoi [8]: “a characterization of all1-hyperbolic graphs by forbidden isometric subgraphs isnot in sight, in as much as isometric cycles of lengths up to7 may occur, thus complicating the picture”.
A problem from metric graph theory: characterizinglow hyperbolicity graphs
I Hyperbolicity takes nonnegative half-integer value.I 0-Hyperbolic graphs are just block graphs and are
well-understood.I Bandelt and Chepoi [7] obtained a characterization of all
12 -hyperbolic graphs in terms of a convexity condition. Butno corresponding finite forbidden isometric subgraphcharacterization is known till now.
I Bandelt and Chepoi [8]: “a characterization of all1-hyperbolic graphs by forbidden isometric subgraphs isnot in sight, in as much as isometric cycles of lengths up to7 may occur, thus complicating the picture”.
A problem from metric graph theory: characterizinglow hyperbolicity graphs
I Hyperbolicity takes nonnegative half-integer value.I 0-Hyperbolic graphs are just block graphs and are
well-understood.I Bandelt and Chepoi [7] obtained a characterization of all
12 -hyperbolic graphs in terms of a convexity condition. Butno corresponding finite forbidden isometric subgraphcharacterization is known till now.
I Bandelt and Chepoi [8]: “a characterization of all1-hyperbolic graphs by forbidden isometric subgraphs isnot in sight, in as much as isometric cycles of lengths up to7 may occur, thus complicating the picture”.
Finite excluded isometric subgraph characterization
Koolen and Moulton [72, p. 696]: In the proof of this propositionwe showed that G was 5-hyperbolic, although we suspect thatthe bound of 5 can be improved upon. In fact, we believe thatthe sum xy + uv in the proof of Proposition 4 can be boundedabove by 5. This would imply that only finitely many graphswould have to be excluded as isometric subgraphs – inaddition to assuming the breadth and short-cut properties– to assume that G would be 1
2 -hyperbolic. However,perhaps more importantly, this proposition indicates that theconcept of short-cuts together with the implicitly well-knownconcept of breadth could be useful for both determining thestructure and finding good bounds on the hyperbolicity ofhyperbolic graphs.
Our work
I As a small step in pursuing further understanding oftree-likeness, Chengpeng Zhang and I recently take up themodest task of comparing two parameters of tree-likeness,namely (Gromov) hyperbolicity and chordality of a graph.
I For any k ≥ 4, we show that k -chordal graphs must beb k
2 c2 -hyperbolic and there does exist a k -chordal graph with
hyperbolicity achieving this asserted bound.I We determine a complete set of 6 unavoidable isometric
subgraphs of 5-chordal graphs attaining hyperbolicity 1, asa minor attempt to respond to the general question that“what is the structure of graphs with relative smallhyperbolicity” [17, p. 62] and the even more generalquestion of “what is the structure of a very tree-like graph”.
Our work
I As a small step in pursuing further understanding oftree-likeness, Chengpeng Zhang and I recently take up themodest task of comparing two parameters of tree-likeness,namely (Gromov) hyperbolicity and chordality of a graph.
I For any k ≥ 4, we show that k -chordal graphs must beb k
2 c2 -hyperbolic and there does exist a k -chordal graph with
hyperbolicity achieving this asserted bound.I We determine a complete set of 6 unavoidable isometric
subgraphs of 5-chordal graphs attaining hyperbolicity 1, asa minor attempt to respond to the general question that“what is the structure of graphs with relative smallhyperbolicity” [17, p. 62] and the even more generalquestion of “what is the structure of a very tree-like graph”.
Our work
I As a small step in pursuing further understanding oftree-likeness, Chengpeng Zhang and I recently take up themodest task of comparing two parameters of tree-likeness,namely (Gromov) hyperbolicity and chordality of a graph.
I For any k ≥ 4, we show that k -chordal graphs must beb k
2 c2 -hyperbolic and there does exist a k -chordal graph with
hyperbolicity achieving this asserted bound.I We determine a complete set of 6 unavoidable isometric
subgraphs of 5-chordal graphs attaining hyperbolicity 1, asa minor attempt to respond to the general question that“what is the structure of graphs with relative smallhyperbolicity” [17, p. 62] and the even more generalquestion of “what is the structure of a very tree-like graph”.
Our work
I As a small step in pursuing further understanding oftree-likeness, Chengpeng Zhang and I recently take up themodest task of comparing two parameters of tree-likeness,namely (Gromov) hyperbolicity and chordality of a graph.
I For any k ≥ 4, we show that k -chordal graphs must beb k
2 c2 -hyperbolic and there does exist a k -chordal graph with
hyperbolicity achieving this asserted bound.I We determine a complete set of 6 unavoidable isometric
subgraphs of 5-chordal graphs attaining hyperbolicity 1, asa minor attempt to respond to the general question that“what is the structure of graphs with relative smallhyperbolicity” [17, p. 62] and the even more generalquestion of “what is the structure of a very tree-like graph”.
Our work
I As a small step in pursuing further understanding oftree-likeness, Chengpeng Zhang and I recently take up themodest task of comparing two parameters of tree-likeness,namely (Gromov) hyperbolicity and chordality of a graph.
I For any k ≥ 4, we show that k -chordal graphs must beb k
2 c2 -hyperbolic and there does exist a k -chordal graph with
hyperbolicity achieving this asserted bound.I We determine a complete set of 6 unavoidable isometric
subgraphs of 5-chordal graphs attaining hyperbolicity 1, asa minor attempt to respond to the general question that“what is the structure of graphs with relative smallhyperbolicity” [17, p. 62] and the even more generalquestion of “what is the structure of a very tree-like graph”.
Our work
I As a small step in pursuing further understanding oftree-likeness, Chengpeng Zhang and I recently take up themodest task of comparing two parameters of tree-likeness,namely (Gromov) hyperbolicity and chordality of a graph.
I For any k ≥ 4, we show that k -chordal graphs must beb k
2 c2 -hyperbolic and there does exist a k -chordal graph with
hyperbolicity achieving this asserted bound.I We determine a complete set of 6 unavoidable isometric
subgraphs of 5-chordal graphs attaining hyperbolicity 1, asa minor attempt to respond to the general question that“what is the structure of graphs with relative smallhyperbolicity” [17, p. 62] and the even more generalquestion of “what is the structure of a very tree-like graph”.
Overview of the talk
I We only consider simple, unweighted, connected, butnot necessarily finite graphs.
I This talk will introduce three graph parameters measuringtree-likeness, namely chordality, hyperbolicity andtree-length.
I We will present some current knowledge/problems on thegeneral relationship among these parameters.
Overview of the talk
I We only consider simple, unweighted, connected, butnot necessarily finite graphs.
I This talk will introduce three graph parameters measuringtree-likeness, namely chordality, hyperbolicity andtree-length.
I We will present some current knowledge/problems on thegeneral relationship among these parameters.
Overview of the talk
I We only consider simple, unweighted, connected, butnot necessarily finite graphs.
I This talk will introduce three graph parameters measuringtree-likeness, namely chordality, hyperbolicity andtree-length.
I We will present some current knowledge/problems on thegeneral relationship among these parameters.
Outline
Introduction
Hyperbolicity
Tree-length
Chordality
Hyperbolicity vs chordality
Conclusion
Graph metric space
Any graph G together with the usual shortest-path metric on it,dG : V (G)× V (G) 7→ {0, 1, 2, . . .}, gives rise to a metric space.We often use the shorthand xy for d(x , y). Note that a pair ofvertices x and y forms an edge if and only if xy = 1.
Hyperbolicity
I For any vertices x , y , u, v of a graph G, put δG(x , y , u, v) tobe the absolute value of the difference between the largestand the second largest of the three sums
uv + xy2
,ux + vy
2, and
uy + vx2
.
I Clearly, δ(x , y , u, v) = 0 if x , y , u, v are not four differentvertices.
I A graph G, viewed as a metric space, is δ-hyperbolic (ortree-like with defect at most δ) provided for any verticesx , y , u, v in G it holds δ(x , y , u, v) ≤ δ and the (Gromov)hyperbolicity of G, denoted δ∗(G), is the minimum halfinteger δ such that G is δ-hyperbolic.
I Note that it may happen δ∗(G) = ∞ when G is an infinitegraph. But for a finite graph G, δ∗(G) is clearly polynomialtime computable.
Hyperbolicity
I For any vertices x , y , u, v of a graph G, put δG(x , y , u, v) tobe the absolute value of the difference between the largestand the second largest of the three sums
uv + xy2
,ux + vy
2, and
uy + vx2
.
I Clearly, δ(x , y , u, v) = 0 if x , y , u, v are not four differentvertices.
I A graph G, viewed as a metric space, is δ-hyperbolic (ortree-like with defect at most δ) provided for any verticesx , y , u, v in G it holds δ(x , y , u, v) ≤ δ and the (Gromov)hyperbolicity of G, denoted δ∗(G), is the minimum halfinteger δ such that G is δ-hyperbolic.
I Note that it may happen δ∗(G) = ∞ when G is an infinitegraph. But for a finite graph G, δ∗(G) is clearly polynomialtime computable.
Hyperbolicity
I For any vertices x , y , u, v of a graph G, put δG(x , y , u, v) tobe the absolute value of the difference between the largestand the second largest of the three sums
uv + xy2
,ux + vy
2, and
uy + vx2
.
I Clearly, δ(x , y , u, v) = 0 if x , y , u, v are not four differentvertices.
I A graph G, viewed as a metric space, is δ-hyperbolic (ortree-like with defect at most δ) provided for any verticesx , y , u, v in G it holds δ(x , y , u, v) ≤ δ and the (Gromov)hyperbolicity of G, denoted δ∗(G), is the minimum halfinteger δ such that G is δ-hyperbolic.
I Note that it may happen δ∗(G) = ∞ when G is an infinitegraph. But for a finite graph G, δ∗(G) is clearly polynomialtime computable.
Hyperbolicity
I For any vertices x , y , u, v of a graph G, put δG(x , y , u, v) tobe the absolute value of the difference between the largestand the second largest of the three sums
uv + xy2
,ux + vy
2, and
uy + vx2
.
I Clearly, δ(x , y , u, v) = 0 if x , y , u, v are not four differentvertices.
I A graph G, viewed as a metric space, is δ-hyperbolic (ortree-like with defect at most δ) provided for any verticesx , y , u, v in G it holds δ(x , y , u, v) ≤ δ and the (Gromov)hyperbolicity of G, denoted δ∗(G), is the minimum halfinteger δ such that G is δ-hyperbolic.
I Note that it may happen δ∗(G) = ∞ when G is an infinitegraph. But for a finite graph G, δ∗(G) is clearly polynomialtime computable.
Examples
I The hyperbolicity of a tree is 0.I The hyperbolicity of the n-cycle is bn
4c −12 if n is congruent
to 1 modulo 4 and is bn4c else.
I The hyperbolicity of a graph with diameter D is at most D2 .
Examples
I The hyperbolicity of a tree is 0.I The hyperbolicity of the n-cycle is bn
4c −12 if n is congruent
to 1 modulo 4 and is bn4c else.
I The hyperbolicity of a graph with diameter D is at most D2 .
Examples
I The hyperbolicity of a tree is 0.I The hyperbolicity of the n-cycle is bn
4c −12 if n is congruent
to 1 modulo 4 and is bn4c else.
I The hyperbolicity of a graph with diameter D is at most D2 .
Hyperbolicity is a tree-likeness measure
I The hyperbolicity of a graph is a way to measure theadditive distortion with which every four-points sub-metricof the given graph metric embeds into a tree metric [1].The condition that the hyperbolicity is zero is known as thefour-point condition (4PC) and is a characterization ofgeneral tree-like metric spaces, so-called real tree[41, 45, 67].
I Moreover, the fact that hyperbolicity is a tree-likenessparameter is reflected in the easy fact that the hyperbolicityof a graph is the maximum hyperbolicity of its 2-connectedcomponents – This observation implies the classical resultthat 0-hyperbolic graphs are exactly block graphs, namelythose graphs in which every 2-connected subgraph iscomplete, which are also known to be exactly thosediamond-free chordal graphs.
Hyperbolicity is a tree-likeness measure
I The hyperbolicity of a graph is a way to measure theadditive distortion with which every four-points sub-metricof the given graph metric embeds into a tree metric [1].The condition that the hyperbolicity is zero is known as thefour-point condition (4PC) and is a characterization ofgeneral tree-like metric spaces, so-called real tree[41, 45, 67].
I Moreover, the fact that hyperbolicity is a tree-likenessparameter is reflected in the easy fact that the hyperbolicityof a graph is the maximum hyperbolicity of its 2-connectedcomponents – This observation implies the classical resultthat 0-hyperbolic graphs are exactly block graphs, namelythose graphs in which every 2-connected subgraph iscomplete, which are also known to be exactly thosediamond-free chordal graphs.
Outline
Introduction
Hyperbolicity
Tree-length
Chordality
Hyperbolicity vs chordality
Conclusion
Tree-length
I A chordal graph is a graph without chordless cycle oflength greater than 3.
I The tree-length [34, 35, 78, 89] of a graph G, denotedtl(G), was introduced by Dourisboure and Gavoille in 2007and is the minimum integer k such that there is a chordalgraph G′ satisfying V (G) = V (G′), E(G) ⊆ E(G′) andmax(dG(u, v) : dG′(u, v) = 1) = k . We use the conventionthat the tree-length of a graph without any edge is 1.
I It is straightforward from the definition that chordal graphsare exactly the graphs of tree-length 1. It is also knownthat AT-free graphs, permutation graphs anddistance-hereditary graphs have tree-length at most 2 [34,p. 367].
Tree-length
I A chordal graph is a graph without chordless cycle oflength greater than 3.
I The tree-length [34, 35, 78, 89] of a graph G, denotedtl(G), was introduced by Dourisboure and Gavoille in 2007and is the minimum integer k such that there is a chordalgraph G′ satisfying V (G) = V (G′), E(G) ⊆ E(G′) andmax(dG(u, v) : dG′(u, v) = 1) = k . We use the conventionthat the tree-length of a graph without any edge is 1.
I It is straightforward from the definition that chordal graphsare exactly the graphs of tree-length 1. It is also knownthat AT-free graphs, permutation graphs anddistance-hereditary graphs have tree-length at most 2 [34,p. 367].
Tree-length
I A chordal graph is a graph without chordless cycle oflength greater than 3.
I The tree-length [34, 35, 78, 89] of a graph G, denotedtl(G), was introduced by Dourisboure and Gavoille in 2007and is the minimum integer k such that there is a chordalgraph G′ satisfying V (G) = V (G′), E(G) ⊆ E(G′) andmax(dG(u, v) : dG′(u, v) = 1) = k . We use the conventionthat the tree-length of a graph without any edge is 1.
I It is straightforward from the definition that chordal graphsare exactly the graphs of tree-length 1. It is also knownthat AT-free graphs, permutation graphs anddistance-hereditary graphs have tree-length at most 2 [34,p. 367].
Tree decomposition
The concept of tree decompositions was introduced byRobertson and Seymour in 1984 [83] and has since beenextensively studied in both mathematics and lots of appliedfields.A tree decomposition of a graph G is a tree T such that eachvertex v of T corresponds to a bag Sv ⊆ V (G) and thefollowing conditions are satisfied:
I (Vertex Covering) ∪v∈V (T )Sv = V (G).I (Edge Covering) For any edge {u, w} ∈ E(G) there exists
v ∈ V (T ) such that u, w ∈ Sv .
I (Running Intersection Property) For any u ∈ V (G),{v ∈ V (T ) : u ∈ Sv} induces a subtree of T . In otherwords, for any v , w ∈ V (T ), Sv ∩ Sw can be seen in everybag along the path connecting v and w in T .
Tree decomposition
The concept of tree decompositions was introduced byRobertson and Seymour in 1984 [83] and has since beenextensively studied in both mathematics and lots of appliedfields.A tree decomposition of a graph G is a tree T such that eachvertex v of T corresponds to a bag Sv ⊆ V (G) and thefollowing conditions are satisfied:
I (Vertex Covering) ∪v∈V (T )Sv = V (G).I (Edge Covering) For any edge {u, w} ∈ E(G) there exists
v ∈ V (T ) such that u, w ∈ Sv .
I (Running Intersection Property) For any u ∈ V (G),{v ∈ V (T ) : u ∈ Sv} induces a subtree of T . In otherwords, for any v , w ∈ V (T ), Sv ∩ Sw can be seen in everybag along the path connecting v and w in T .
Tree decomposition
The concept of tree decompositions was introduced byRobertson and Seymour in 1984 [83] and has since beenextensively studied in both mathematics and lots of appliedfields.A tree decomposition of a graph G is a tree T such that eachvertex v of T corresponds to a bag Sv ⊆ V (G) and thefollowing conditions are satisfied:
I (Vertex Covering) ∪v∈V (T )Sv = V (G).I (Edge Covering) For any edge {u, w} ∈ E(G) there exists
v ∈ V (T ) such that u, w ∈ Sv .
I (Running Intersection Property) For any u ∈ V (G),{v ∈ V (T ) : u ∈ Sv} induces a subtree of T . In otherwords, for any v , w ∈ V (T ), Sv ∩ Sw can be seen in everybag along the path connecting v and w in T .
Tree decomposition
The concept of tree decompositions was introduced byRobertson and Seymour in 1984 [83] and has since beenextensively studied in both mathematics and lots of appliedfields.A tree decomposition of a graph G is a tree T such that eachvertex v of T corresponds to a bag Sv ⊆ V (G) and thefollowing conditions are satisfied:
I (Vertex Covering) ∪v∈V (T )Sv = V (G).I (Edge Covering) For any edge {u, w} ∈ E(G) there exists
v ∈ V (T ) such that u, w ∈ Sv .
I (Running Intersection Property) For any u ∈ V (G),{v ∈ V (T ) : u ∈ Sv} induces a subtree of T . In otherwords, for any v , w ∈ V (T ), Sv ∩ Sw can be seen in everybag along the path connecting v and w in T .
Tree decomposition
The concept of tree decompositions was introduced byRobertson and Seymour in 1984 [83] and has since beenextensively studied in both mathematics and lots of appliedfields.A tree decomposition of a graph G is a tree T such that eachvertex v of T corresponds to a bag Sv ⊆ V (G) and thefollowing conditions are satisfied:
I (Vertex Covering) ∪v∈V (T )Sv = V (G).I (Edge Covering) For any edge {u, w} ∈ E(G) there exists
v ∈ V (T ) such that u, w ∈ Sv .
I (Running Intersection Property) For any u ∈ V (G),{v ∈ V (T ) : u ∈ Sv} induces a subtree of T . In otherwords, for any v , w ∈ V (T ), Sv ∩ Sw can be seen in everybag along the path connecting v and w in T .
Tree-length: another definition
The length of a tree decomposition of a graph G is themaximum distance in G between two vertices in the same bagof the decomposition.
The tree-length of a graph G is just the shortest length of alltree decompositions of G.
Example (Dourisboure, Gavoille)The tree-length of an n-cycle is dn
3e.
Tree-length and hyperbolicity are comparable
The following are two results from Chepoi, Dragan, Estellon,Habib, Vaxés [25].
Theorem[25, Proposition 13] A graph G is k-hyperbolic provided itstree-length is no greater than k .
The proof is a generalization of the easy proof that every treehas hyperbolicity 0. We just take a look at a tree-decompositionwith maximum bag diameter no greater than k .
Theorem[25, Proposition 14] The inequality tl(G) ≤ 12k + 8k log2 n + 17holds for any k-hyperbolic graph G with n vertices.
Tree-length and hyperbolicity are comparable
The following are two results from Chepoi, Dragan, Estellon,Habib, Vaxés [25].
Theorem[25, Proposition 13] A graph G is k-hyperbolic provided itstree-length is no greater than k .
The proof is a generalization of the easy proof that every treehas hyperbolicity 0. We just take a look at a tree-decompositionwith maximum bag diameter no greater than k .
Theorem[25, Proposition 14] The inequality tl(G) ≤ 12k + 8k log2 n + 17holds for any k-hyperbolic graph G with n vertices.
Tree-length and hyperbolicity are comparable
The following are two results from Chepoi, Dragan, Estellon,Habib, Vaxés [25].
Theorem[25, Proposition 13] A graph G is k-hyperbolic provided itstree-length is no greater than k .
The proof is a generalization of the easy proof that every treehas hyperbolicity 0. We just take a look at a tree-decompositionwith maximum bag diameter no greater than k .
Theorem[25, Proposition 14] The inequality tl(G) ≤ 12k + 8k log2 n + 17holds for any k-hyperbolic graph G with n vertices.
Grid graph
ExampleFor any two natural numbers m and n, the grid graph Gm,n isthe graph with vertex set {1, 2, . . . , m} × {1, 2, . . . , n} and (i1, i2)and (j1, j2) are adjacent in Gm,n if any only if(i1 − j1)2 + (j1 − j2)2 = 1. Dourisboure and Gavoile showed thatthe tree-length of Gn,m is min(n, m) if n 6= m or n = m is evenand is n − 1 if n = m is odd [35, Theorem 3]. Based on thisresult it is not hard to further show thatδ∗(Gm,n) ∈ {min(m, n), min(m, n)− 1} andδ∗(Gm,n) = min(m, n)− 1 if n = m is odd. This says thatTheorem 2 is quite tight.
Is it true that the equality δ∗(Gm,n) = min(m, n)− 1 holds for allpositive integers m, n?
Grid graph
ExampleFor any two natural numbers m and n, the grid graph Gm,n isthe graph with vertex set {1, 2, . . . , m} × {1, 2, . . . , n} and (i1, i2)and (j1, j2) are adjacent in Gm,n if any only if(i1 − j1)2 + (j1 − j2)2 = 1. Dourisboure and Gavoile showed thatthe tree-length of Gn,m is min(n, m) if n 6= m or n = m is evenand is n − 1 if n = m is odd [35, Theorem 3]. Based on thisresult it is not hard to further show thatδ∗(Gm,n) ∈ {min(m, n), min(m, n)− 1} andδ∗(Gm,n) = min(m, n)− 1 if n = m is odd. This says thatTheorem 2 is quite tight.
Is it true that the equality δ∗(Gm,n) = min(m, n)− 1 holds for allpositive integers m, n?
Grid graph
ExampleFor any two natural numbers m and n, the grid graph Gm,n isthe graph with vertex set {1, 2, . . . , m} × {1, 2, . . . , n} and (i1, i2)and (j1, j2) are adjacent in Gm,n if any only if(i1 − j1)2 + (j1 − j2)2 = 1. Dourisboure and Gavoile showed thatthe tree-length of Gn,m is min(n, m) if n 6= m or n = m is evenand is n − 1 if n = m is odd [35, Theorem 3]. Based on thisresult it is not hard to further show thatδ∗(Gm,n) ∈ {min(m, n), min(m, n)− 1} andδ∗(Gm,n) = min(m, n)− 1 if n = m is odd. This says thatTheorem 2 is quite tight.
Is it true that the equality δ∗(Gm,n) = min(m, n)− 1 holds for allpositive integers m, n?
Grid graph
ExampleFor any two natural numbers m and n, the grid graph Gm,n isthe graph with vertex set {1, 2, . . . , m} × {1, 2, . . . , n} and (i1, i2)and (j1, j2) are adjacent in Gm,n if any only if(i1 − j1)2 + (j1 − j2)2 = 1. Dourisboure and Gavoile showed thatthe tree-length of Gn,m is min(n, m) if n 6= m or n = m is evenand is n − 1 if n = m is odd [35, Theorem 3]. Based on thisresult it is not hard to further show thatδ∗(Gm,n) ∈ {min(m, n), min(m, n)− 1} andδ∗(Gm,n) = min(m, n)− 1 if n = m is odd. This says thatTheorem 2 is quite tight.
Is it true that the equality δ∗(Gm,n) = min(m, n)− 1 holds for allpositive integers m, n?
Grid graph
ExampleFor any two natural numbers m and n, the grid graph Gm,n isthe graph with vertex set {1, 2, . . . , m} × {1, 2, . . . , n} and (i1, i2)and (j1, j2) are adjacent in Gm,n if any only if(i1 − j1)2 + (j1 − j2)2 = 1. Dourisboure and Gavoile showed thatthe tree-length of Gn,m is min(n, m) if n 6= m or n = m is evenand is n − 1 if n = m is odd [35, Theorem 3]. Based on thisresult it is not hard to further show thatδ∗(Gm,n) ∈ {min(m, n), min(m, n)− 1} andδ∗(Gm,n) = min(m, n)− 1 if n = m is odd. This says thatTheorem 2 is quite tight.
Is it true that the equality δ∗(Gm,n) = min(m, n)− 1 holds for allpositive integers m, n?
Outline
Introduction
Hyperbolicity
Tree-length
Chordality
Hyperbolicity vs chordality
Conclusion
Chordality
I We say that a graph is k-chordal if it does not contain anyinduced n-cycle for n > k . Clearly, trees are nothing but2-chordal graphs.
I A 4-chordal graph is also called a hole-free graph and a3-chordal graph is nothing but a chordal graph.
I The chordality of a graph G is the smallest integer k suchthat G is k -chordal [11]. Following [11], we use the notationlc(G) for this parameter as it is merely the length of thelongest chordless cycle in G when G is not a tree.
Chordality
I We say that a graph is k-chordal if it does not contain anyinduced n-cycle for n > k . Clearly, trees are nothing but2-chordal graphs.
I A 4-chordal graph is also called a hole-free graph and a3-chordal graph is nothing but a chordal graph.
I The chordality of a graph G is the smallest integer k suchthat G is k -chordal [11]. Following [11], we use the notationlc(G) for this parameter as it is merely the length of thelongest chordless cycle in G when G is not a tree.
Chordality
I We say that a graph is k-chordal if it does not contain anyinduced n-cycle for n > k . Clearly, trees are nothing but2-chordal graphs.
I A 4-chordal graph is also called a hole-free graph and a3-chordal graph is nothing but a chordal graph.
I The chordality of a graph G is the smallest integer k suchthat G is k -chordal [11]. Following [11], we use the notationlc(G) for this parameter as it is merely the length of thelongest chordless cycle in G when G is not a tree.
Chordality, Contd.
The recognition of k -chordal graphs is coNP-complete fork = Θ(nε) for any constant ε > 0 [88]. Especially, to determinethe chordality of the hypercube is attracting much attentionunder the name of the snake-in-the-box problem due to itsconnection with some error-checking codes problem [71].
Nevertheless, just like many other tree-likeness parameters,quite a few natural graph classes are known to have smallchordality [15].
Chordality, Contd.
The recognition of k -chordal graphs is coNP-complete fork = Θ(nε) for any constant ε > 0 [88]. Especially, to determinethe chordality of the hypercube is attracting much attentionunder the name of the snake-in-the-box problem due to itsconnection with some error-checking codes problem [71].
Nevertheless, just like many other tree-likeness parameters,quite a few natural graph classes are known to have smallchordality [15].
Chordality and tree-length
What follows is a result of Gavoille, Katz, Katz, Paul, Peleg [51,Lemma 6].
Theorem[51, Lemma 6] [52, Theorem 3.3] If G is a k-chordal graph, thentl(G) ≤ bk
2c.
Proof.To obtain a minimal triangulation of G, it suffices to select amaximal set of pairwise parallel minimal separators of G andadd edges to make each of them a clique [81, Theorem 4.6]. Itis easy to check that each such new edge connects two pointsof distance at most b k
2c apart in G.
Chordality and tree-length
What follows is a result of Gavoille, Katz, Katz, Paul, Peleg [51,Lemma 6].
Theorem[51, Lemma 6] [52, Theorem 3.3] If G is a k-chordal graph, thentl(G) ≤ bk
2c.
Proof.To obtain a minimal triangulation of G, it suffices to select amaximal set of pairwise parallel minimal separators of G andadd edges to make each of them a clique [81, Theorem 4.6]. Itis easy to check that each such new edge connects two pointsof distance at most b k
2c apart in G.
Chordality and tree-length
What follows is a result of Gavoille, Katz, Katz, Paul, Peleg [51,Lemma 6].
Theorem[51, Lemma 6] [52, Theorem 3.3] If G is a k-chordal graph, thentl(G) ≤ bk
2c.
Proof.To obtain a minimal triangulation of G, it suffices to select amaximal set of pairwise parallel minimal separators of G andadd edges to make each of them a clique [81, Theorem 4.6]. Itis easy to check that each such new edge connects two pointsof distance at most b k
2c apart in G.
Outline
Introduction
Hyperbolicity
Tree-length
Chordality
Hyperbolicity vs chordality
Conclusion
Impossibility of bounding hyperbolicity from below interms of chordality
Firstly, we point out that a graph with low hyperbolicity mayhave arbitrarily large chordality.Indeed, take any graph G and form the new graph G′ by addingan additional vertex and connecting this new vertex with everyvertex of G. It is obvious that δ∗(G′) ≤ 1 while lc(G′) = lc(G) ifG is not a tree. Moreover, it is equally easy to see that G′ iseven 1
2 -hyperbolic if G does not have any induced 4-cycle [72,p. 695].Surely, this example does not preclude the possibility that formany important graph classes we can bound their chordality interms of their hyperbolicity.
Impossibility of bounding hyperbolicity from below interms of chordality
Firstly, we point out that a graph with low hyperbolicity mayhave arbitrarily large chordality.Indeed, take any graph G and form the new graph G′ by addingan additional vertex and connecting this new vertex with everyvertex of G. It is obvious that δ∗(G′) ≤ 1 while lc(G′) = lc(G) ifG is not a tree. Moreover, it is equally easy to see that G′ iseven 1
2 -hyperbolic if G does not have any induced 4-cycle [72,p. 695].Surely, this example does not preclude the possibility that formany important graph classes we can bound their chordality interms of their hyperbolicity.
Impossibility of bounding hyperbolicity from below interms of chordality
Firstly, we point out that a graph with low hyperbolicity mayhave arbitrarily large chordality.Indeed, take any graph G and form the new graph G′ by addingan additional vertex and connecting this new vertex with everyvertex of G. It is obvious that δ∗(G′) ≤ 1 while lc(G′) = lc(G) ifG is not a tree. Moreover, it is equally easy to see that G′ iseven 1
2 -hyperbolic if G does not have any induced 4-cycle [72,p. 695].Surely, this example does not preclude the possibility that formany important graph classes we can bound their chordality interms of their hyperbolicity.
Bound hyperbolicity from above
We now turn to show that it is possible to bound thehyperbolicity from above in terms of chordality.The following result is notified to us by Dragan [38] and ispresumably in the folklore.
TheoremEvery k-chordal graph is b k
2c-hyperbolic.
Proof.It follows directly from Theorems 2 and 5.
Bound hyperbolicity from above
We now turn to show that it is possible to bound thehyperbolicity from above in terms of chordality.The following result is notified to us by Dragan [38] and ispresumably in the folklore.
TheoremEvery k-chordal graph is b k
2c-hyperbolic.
Proof.It follows directly from Theorems 2 and 5.
A side remark
A graph is bridged [3, 75] if it does not contain any finiteisometric cycles of length at least four, or equivalently, if it iscop-win and has no chordless cycle of length 4 or 5. In contrastto Theorem 6, it is interesting to note that the hyperbolicity ofbridged graphs can be arbitrarily high [72, p. 684].
I In view of Example 4, to get better estimate than Theorem6 along the same approach one may try to beef upTheorem 5.
I We point out Dourisboure and Gavoille posed the openproblem that whether or not
tl(G) ≤ d lc(G)
3e (1)
is true [35, Question 1].I If (1) can be established, it will be the best we can expect
in the sense that for every outerplanar graph G, it holdstl(G) = d lc(G)
3 e [35, Theorem 1].
I In view of Example 4, to get better estimate than Theorem6 along the same approach one may try to beef upTheorem 5.
I We point out Dourisboure and Gavoille posed the openproblem that whether or not
tl(G) ≤ d lc(G)
3e (1)
is true [35, Question 1].I If (1) can be established, it will be the best we can expect
in the sense that for every outerplanar graph G, it holdstl(G) = d lc(G)
3 e [35, Theorem 1].
I In view of Example 4, to get better estimate than Theorem6 along the same approach one may try to beef upTheorem 5.
I We point out Dourisboure and Gavoille posed the openproblem that whether or not
tl(G) ≤ d lc(G)
3e (1)
is true [35, Question 1].I If (1) can be established, it will be the best we can expect
in the sense that for every outerplanar graph G, it holdstl(G) = d lc(G)
3 e [35, Theorem 1].
A reformulation of the open problem of Dourisboureand Gavoille
The kth-power of a graph G, denoted Gk , is the graph withV (G) as vertex set and there is an edge connecting twovertices u and v if and only if dG(u, v) ≤ k .
A Chordal Graph Sandwich Problem:For any graph G, is there always a graph inbetween G andGd lc(G)
3 e which is a chordal graph?
A reformulation of the open problem of Dourisboureand Gavoille
The kth-power of a graph G, denoted Gk , is the graph withV (G) as vertex set and there is an edge connecting twovertices u and v if and only if dG(u, v) ≤ k .
A Chordal Graph Sandwich Problem:For any graph G, is there always a graph inbetween G andGd lc(G)
3 e which is a chordal graph?
ExampleThe chordality of the n-cycle is n while the hyperbolicity of then-cycle is bn
4c −12 if n is congruent to 1 modulo 4 and is bn
4celse. It is also known that the tree-length of the n-cycle is dn
3e[35, Lemma 4]. Note that
δ∗(Cn) =
{ b n2 c2 , if n ≡ 0 (mod 4);
b n2 c2 + 1
2 , else.
ExampleFor any k ≥ 2, let Fk be the graph obtained from the 4k -cycle[v1v2 · · · v4k ] by adding the two edges {v1, v3} and{v2k+1, v2k+3}. Clearly, δ(v2, vk+2, v2k+2, v3k+2) = k − 1
2 .Furthermore, we have lc(Fk ) = 4k − 2 andδ∗(Fk ) = k − 1
2 = δ(v2, vk+2, v2k+2, v3k+2) = lc(Fk )4 .
For the graph in Fig. 1 (it is just F2 in Example 8), we haveδ(x , y , u, v) = 3
2 as it happens xy + uv = 3 + 4 = 7 andxu + yv = xv + yu = 2 + 2 = 4.
rxra
ru
rc
rb rvrd
ry
Figure: A graph with hyperbolicity 32 , tree-length 2 and chordality 6.
TheoremFor each k ≥ 4, all k-chordal graphs are b k
2 c2 -hyperbolic.
It is clear that if Theorem 9 is tight for k = 4t (k = 4t + 2) then itis tight for k = 4t + 1 (k = 4t + 3). Consequently, Examples 7and 8 indeed mean that the bound reported in Theorem 9 istight for every k ≥ 4. Surely, the next natural step may be tocharacterize all those extremal k -hyperbolic graphs whose
hyperbolicity attain b k2 c2 – but there seems to be still a long haul
ahead in this direction.
Let C4, H1, H2, H3, H4 and H5 be the graphs displayed in nextslide. It is simple to check that each of them has hyperbolicity 1and is 5-chordal.
73Algorithmic Graph Theory
We can characterize the structure of those 5-chordal graphsachieving maximum possible hyperbolicity.
TheoremA 5-chordal graph has hyperbolicity one if and only if one ofC4, H1, H2, H3, H4, H5 appears as an isometric subgraph of it.
We can characterize the structure of those 5-chordal graphsachieving maximum possible hyperbolicity.
TheoremA 5-chordal graph has hyperbolicity one if and only if one ofC4, H1, H2, H3, H4, H5 appears as an isometric subgraph of it.
ConjectureA 6-chordal graph is 1
2 -hyperbolic if and only if it does notcontain any of a list of eleven special graphsG1, G2, G3, F2, C4, C6, Hi , i = 1, . . . , 5, as an isometricsubgraph.
74Algorithmic Graph Theory
Note thatlc(C4) = 4, lc(H1) = lc(H2) = 3, lc(H3) = lc(H4) = lc(H5) = 5.The next two results follow immediately from Theorem 10.
CorollaryA 4-chordal graph is 1-hyperbolic and has hyperbolicity one ifand only if it contains one of C4, H1 and H2 as an isometricsubgraph.
Corollary (Brinkmann, Koolen, Moulton)[17, Theorem 1.1] A chordal graph is 1-hyperbolic and hashyperbolicity one if and only if it contains either H1 or H2 as anisometric subgraph.
We remark that as long as every 4-chordal graph is1-hyperbolic is known, Corollary 12 also immediately followsfrom Corollary 13. We also mention that the first part ofCorollary 13, namely every chordal graph is 1-hyperbolic isimmediate from Theorem 2 as chordal graphs have tree-length1.
Corollary (Bandelt, Chepoi)[8, p. 16] A distance-hereditary graph is always 1-hyperbolicand is 1
2 -hyperbolic exactly when it is chordal, or equivalently,when it contains no induced 4-cycle.
Proof.It is easy to see that distance-hereditary graphs must be4-chordal and can contain neither H1 nor H2 as an isometricsubgraph. The result now follows from Corollary 12.
CorollaryEvery AT -free graph is 1-hyperbolic and has hyperbolicity one ifand only if it contains C4 as an isometric subgraph.
Proof.First observe that an AT -free graph must be 5-chordal. Furthernotice that the triple u, y , v is an AT in any of the graphsH1, . . . , H5. Now, an application of Theorem 10 concludes theproof.
Outline
Introduction
Hyperbolicity
Tree-length
Chordality
Hyperbolicity vs chordality
Conclusion
I The proof of our main results consists of a vastapplications of the triangle inequality. These triangleinequalities are obtained by examining several localcombinatorial objects, especially their behavior undercertain extremality assumptions.
I There exist several ways to connect chordality andhyperbolicity via some other tree-likeness parameters.
I There are lots of other tree-likeness parameters and theirstudy are very active in mathematics and applied fields.See the references for a small selection of some suchpapers.
I Thank you for your attention!
I The proof of our main results consists of a vastapplications of the triangle inequality. These triangleinequalities are obtained by examining several localcombinatorial objects, especially their behavior undercertain extremality assumptions.
I There exist several ways to connect chordality andhyperbolicity via some other tree-likeness parameters.
I There are lots of other tree-likeness parameters and theirstudy are very active in mathematics and applied fields.See the references for a small selection of some suchpapers.
I Thank you for your attention!
I The proof of our main results consists of a vastapplications of the triangle inequality. These triangleinequalities are obtained by examining several localcombinatorial objects, especially their behavior undercertain extremality assumptions.
I There exist several ways to connect chordality andhyperbolicity via some other tree-likeness parameters.
I There are lots of other tree-likeness parameters and theirstudy are very active in mathematics and applied fields.See the references for a small selection of some suchpapers.
I Thank you for your attention!
I The proof of our main results consists of a vastapplications of the triangle inequality. These triangleinequalities are obtained by examining several localcombinatorial objects, especially their behavior undercertain extremality assumptions.
I There exist several ways to connect chordality andhyperbolicity via some other tree-likeness parameters.
I There are lots of other tree-likeness parameters and theirstudy are very active in mathematics and applied fields.See the references for a small selection of some suchpapers.
I Thank you for your attention!
I The proof of our main results consists of a vastapplications of the triangle inequality. These triangleinequalities are obtained by examining several localcombinatorial objects, especially their behavior undercertain extremality assumptions.
I There exist several ways to connect chordality andhyperbolicity via some other tree-likeness parameters.
I There are lots of other tree-likeness parameters and theirstudy are very active in mathematics and applied fields.See the references for a small selection of some suchpapers.
I Thank you for your attention!
I The proof of our main results consists of a vastapplications of the triangle inequality. These triangleinequalities are obtained by examining several localcombinatorial objects, especially their behavior undercertain extremality assumptions.
I There exist several ways to connect chordality andhyperbolicity via some other tree-likeness parameters.
I There are lots of other tree-likeness parameters and theirstudy are very active in mathematics and applied fields.See the references for a small selection of some suchpapers.
I Thank you for your attention!
I. Abraham, M. Balakrishnan, F. Kuhn, D. Malkhi, V.Ramasubramanian, K. Talwar, Reconstructing approximatetree metrics, PODC 2007, 43-52.
J.M. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M.Mihalik, M. Shapiro, H. Short, Notes on word hyperbolicgroups, Group Theory from a Geometrical Viewpoint, ICTPTrieste 1990 (E. Ghys, A. Haefliger, A. Verjovsky, eds.),World Scientific, 1991, pp. 3–63. Available at:http://homeweb1.unifr.ch/ciobanul/pub/Teaching/kggt/MSRInotes2004.pdf
R.P. Anstee, M. Farber, On bridged graphs and cop-wingraphs, J. Combin. Theory 44 (1988), 22–28.
S. Arnborg, J. Lagergren, D. Seese, Easy problems fortree-decomposable graphs, Journal of Algorithms 12(1991), 308–340.
P. Bahls, Asymptotic connectivity of infinite graphs, Disc.Math. 309 (2009), 2250–2259.
V. Bálint, A distance approximating trees, Disc. Appl. Math.156 (2008), 2740–2752.
H-J. Bandelt, V. Chepoi, 1-Hyperbolic graphs, SIAM J. Disc.Math. 16 (2003), 323–334.
H-J. Bandelt, V. Chepoi, Metric graph theory and geometry:A survey, In: Surveys on Discrete and ComputationalGeometry: Twenty Years Later, J.E. Goodman, J. Pach, R.Pollack (eds.), AMS, pp. 49–86, 2008.
H-J. Bandelt, H.M. Mulder, Distance-hereditary graphs, J.Combin. Theory Ser. B 41 (1986), 182–208.
H.L. Bodlaender, J. Engelfriet, Domino treewidth, J.Algorithms 24 (1997), 94–123.
H.L. Bodlaender, D.M. Thilikos, Treewidth for graphs withsmall chordality, Discrete Applied Mathematics 79 (1997),45–61.
H.L. Bodlaender, A partial k -arboretum of graphs withbounded treewidth, Theoret. Comput. Sci. 209 (1998),1–45.
B. Bowditch, Notes on Gromov’s hyperbolicity criterion forpath-metric spaces, in: Group Theory from a GeometricalViewpoint, E. Ghys, A. Haefliger, A. Verjovsky (eds.), WorldScientific, Singapore, pp. 64–167, 1991.
A. Brandstädt, V. Chepoi, F. Dragan, Distanceapproximating trees for chordal and dually chordal graphs,J. Algorithms 30 (1999), 166–184.
A. Brandstädt, V.B. Le, J.P. Spinrad, Graph Classes: ASurvey, SIAM, 1999.
M. Bridson, A. Haefliger, Metric Spaces of Non-PositiveCurvature, Springer, Berlin, 1999.
G. Brinkmann, J.H. Koolen, V. Moulton, On the hyperbolicityof chordal graphs, Annals of Combinatorics 5 (2001),61–69.
L.S. Chandran, V.V. Lozin, C.R. Subramanian, Graphs oflow chordality, Discrete Mathematics and TheoreticalComputer Science 7 (2005), 25–36.
L.S. Chandran, L.S. Ram, On the number of minimum cutsin a graph, SIAM J. Disc. Math. 18 (2004), 177–194.
M-S. Chang, H. Müller, On the tree-degree of graphs,Lecture Notes in Computer Science 2204 (2001), 44–54.
B. Chen, S.-T. Yau, Y.-N. Yeh, Graph homotopy andGraham homotopy, Discrete Math. 241 (2001), 153–170.
H. Chen, V. Dalmau, Beyond hypertree width:Decomposition methods without decompositions, LectureNotes in Computer Science 3709 (2005), 167–181.
V. Chepoi, F. Dragan, A note on discrete approximatingtrees in graphs, Europ. J. Combinatorics 21 (2000),761–766.
V. Chepoi, F. Dragan, B. Estellon, M. Habib, Y. Vaxés,Notes on diameters, centers, and approximating trees of
δ-hyperbolic geodesic spaces and graphs, Electronic Notesin Discrete Mathematics 31 (2008), 231–234.
V. Chepoi, F. Dragan, B. Estellon, M. Habib, Y. Vaxés,Diameters, centers, and approximating trees of δ-hyperbolicgeodesic spaces and graphs, Annual Symposium onComputational Geometry pp. 59–68, 2008.
V. Chepoi, F.F. Dragan, B. Estellon, M. Habib, Y. Vaxés, Y.Xiang, Additive spanners and distance and routing labelingschemes for hyperbolic graphs, submitted. Available at:http://pageperso.lif.univ-mrs.fr/~victor.chepoi/HypRoutJournal.pdf
V. Chepoi, B. Estellon, Packing and covering δ-hyperbolicspaces by balls, Lecture Notes in Computer Science 4627(2007), 59–73.
D. Cohen, P. Jeavons, M. Gyssens, A unified theory ofstructural tractability for constraint satisfaction problems,Journal of Computer and System Sciences 74 (2008),721–743.
E. Dahlhaus, P.L. Hammer, F. Maffray, S. Olariu, Ondomination elimination orderings and domination graphs,Lecture Notes in Computer Science 903 (1994), 81–92.
N.D. Dendris, L.M. Kirousis, D.M. Thilikos, Fugitive-searchgames on graphs and related parameters, TheoreticalComputer Science 172 (1997), 233–254.
M.M. Deza, E. Deza, Encyclopedia of Distances, Springer,2009.
R. Diestel, Graph Theory, Springer, 1997.
M. Dinitz, Online, dynamic, and distributed embeddings ofapproximate ultrametrics, Lecture Notes in ComputerScience 5218 (2008), 152–166.
Y. Dourisboure, Compact routing schemes for boundedtree-length graphs and for k -chordal graphs, Lecture Notesin Computer Science 3274 (2004), 365–378.
Y. Dourisboure, C. Gavoille, Tree-decompositions with bagsof small diameter, Discrete Math. 307 (2007), 2008–2029.
R.G. Downey, C. McCartin, Bounded persistencepathwidth, In: M. Atkinson and F. Dehne, editors, EleventhComputing: The Australasian Theory Symposium(CATS2005), volume 41 of CRPIT, pages 51–56,Newcastle, Australia, 2005. ACS (Australian ComputerSociety).
R.G. Downey, C. McCartin, Online promise problems withonline width metrics, Journal of Computer and SystemSciences 73 (2007), 57–72.
F.F. Dragan, Private communication to Yaokun Wu, June12, 2009.
F.F. Dragan, Y. Xiang, How to use spanning trees tonavigate in graphs, Lecture Notes in Computer Science5734 (2009), 282–294.
F.F. Dragan, C. Yan, Distance approximating trees:Complexity and algorithms, Lecture Notes in ComputerScience 3998 (2006), 260–271.
A. Dress, Trees, tight extensions of metric spaces, and thecohomological dimension of certain groups: A note oncombinatorial properties of metric spaces, Adv. Math. 53(1984), 321–402.
A. Dress, B. Holland, K.T. Huber, J.H. Koolen, V. Moulton, J.Weyer-Menkhoff, ∆ additive and ∆ ultra-additive maps,Gromov’s trees, and the Farris transform, Discrete AppliedMathematics 146 (2005), 51–73.
A. Dress, K.T. Huber, V. Moulton, Some uses of the Farristransform in mathematics and phylogenetics – A Review,Annals of Combinatorics 11 (2007), 1–37.
A. Dress, V. Moulton, M. Steel, Trees, taxonomy, andstrongly compatible multi-state characters, Adv. Appl. Math.19 (1997), 1–30.
A. Dress, V. Moulton, W. Terhalle, T-theory: An overview,Europ. J. Combin. 17 (1996), 161–175.
R. Duke, Types of cycles in hypergraphs, Annals ofDiscrete Mathematics 27 (1985), 399–418.
R. Fagin, Degrees of acyclicity for hypergraphs andrelational database schemes, Journal of the Association forComputing Machinery 30 (1983), 514–550.
M. Farber, R.E. Jamison, On local convexities in graphs,Discrete Math. 66 (1987), 231–247.
J.S. Farris, A.G. Kluge, M.J. Eckardt, A numerical approachto phylogenetic systematics, Systematic Zoology(continued by Syst. Biol.) 19 (1970), 172–189.
T. Gallai, Transitiv orientierbare Graphen, Acta Math. Acad.Sci. Hungary 18 (1967), 25–66.
C. Gavoille, M. Katz, N.A. Katz, C. Paul, D. Peleg,Approximate distance labeling schemes, Lecture Notes inComputer Science 2161 (2001), 476–487.
C. Gavoille, M. Katz, N.A. Katz, C. Paul, D. Peleg,Approximate distance labeling schemes, LaBRI ResearchReport, RR-1250-00, University of Bordeaux, December2000. Available at: http://dept-info.labri.fr/~gavoille/gavoille_publis.html#GKKPP00
C. Gavoille, O. Ly, Distance labeling in hyperbolic graphs,Lecture Notes in Computer Science 3827 (2005),1071–1079.
Cyril Gavoille, Private communication to Yaokun Wu,September 3, 2009.
F. Gavril, Algorithms for maximum weight induced paths,Information Processing Letters 81 (2002), 203–208.
M.C. Golumbic, C.L. Monma, W.T. Trotter, Tolerancegraphs, Disc. Appl. Math. 9 (1984), 157–170.
M.C. Golumbic, A.N. Trenk, Tolerance Graphs, CambridgeUniversity Press, Cambridge, 2004.
G. Gottlob, N. Leone, F. Scarcello, A comparison ofstructural CSP decomposition methods, ArtificialIntelligence 124 (2000), 243–282.
G. Gottlob, N. Leone, F. Scarcello, Hypertreedecompositions: A survey, Lecture Notes in ComputerScience 2136 (2001), 37–57.
G. Gottlob, N. Leone, F. Scarcello, Hypertreedecompositions and tractable queries, Journal of Computerand System Sciences 64 (2002), 579–627.
M. Gromov, Hyperbolic groups, in: Essays in GroupTheory, S. Gersten (ed.), MSRI Series, 8, Springer-Verlag,pp. 75–263, 1987.
A. Gupta, R. Krauthgamer, J.R. Lee, Bounded geometries,fractals, and low-distortion embeddings, 44th Annual IEEE
Symposium on Foundations of Computer Science(FOCS’03), pp. 534–543, 2003.
R.B. Hayward, Weakly triangulated graphs, Journal ofCombinatorial Theory Ser. B 39 (1985), 200–208.
P. Hlinený, S-I. Oum, D. Seese, G. Gottlob, Widthparameters beyond tree-width and their applications, TheComputer Journal 51 (2008), 326–362.
E. Howorka, On metric properties of certain clique graphs,J. Combin. Theory Ser. B 27 (1979), 67–74.
P. Hunter, S. Kreutzer, Digraph measures: Kellydecompositions, games, and orderings, TheoreticalComputer Science 399 (2008), 206–219.
W. Imrich, On metric properties of tree-like spaces,Contribution to graph theory and its applications (Internat.Colloq. Oberhof, 1977) pp. 129–156, Tech. HochschuleIlmenau, Ilmenau, 1977.
E. Jonckheere, P. Lohsoonthorn, F. Bonahon, ScaledGromov hyperbolic graphs, J. Graph Theory 57 (2008),157–180.
E. Jonckheere, M. Lou, J. Hespanha, P. Barooah, Effectiveresistance of Gromov-hyperbolic graphs: Application toasymptotic sensor network problems, 46th IEEEConference on Decision and Control (2007), 1453–1458.
L.M. Kirousis, D.M. Thilikos, The linkage of a graph, SIAMJ. Comput. 25 (1996), 626–647.
V. Klee, What is the maximum length of a d-dimensionalsnake? The American Mathematical Monthly 77 (1970),63–65.
J.H. Koolen, V. Moulton, Hyperbolic bridged graphs,European J. Combin. 23 (2002), 683–699.
F.R. Kschischang, B.J. Frey, H-A. Loeliger, Factor graphsand the sum-product algorithm, IEEE Transactions onInformation Theory 47 (2001), 498–519.
M. Laurent, On the sparsity order of a graph and itsdeficiency in chordality, Combinatorica 21 (2001), 543–570.
V.B. Le, J. Spinrad, Consequence of an algorithm forbridged graphs, Dis. Appl. Math. 280 (2004), 271–274.
T.T. Lee, The Euler formula of cyclomatic numbers ofhypergraphs, Southeast Asian Bulletin of Mathematics 21(1997), 113–137.
D.R. Lick, A.T. White, k -Degenerate graphs, Canadian J. ofMathematics 22 (1970), 1082–1096.
D. Lokshtanov, On the complexity of computing treelength,Lecture Notes in Computer Science 4708 (2007), 276–287.
T.A. McKee, E.R. Scheinerman, On the chordality of agraph, Journal of Graph Theory 17 (1993), 221–232.
V. Moulton, M. Steel, Retractions of finite distance functionsonto tree metrics, Discrete Applied Mathematics 91 (1999),215–233.
A. Parra, P. Scheffler, Characterizations and algorithmicapplications of chordal graph embeddings, Disc. Appl.Math. 79 (1997), 171–188.
F.S. Roberts, On the boxicity and cubicity of a graph, in:Recent Progresses in Combinatorics, Academic Press,New York, 1969, pp. 301–310.
N. Robertson, P.D. Seymour, Graph minors III. Planartree-width, J. Combin. Theory Ser. B 36 (1984), 49–63.
I. Rusu, J. Spinrad, Domination graphs: Examples andcounterexamples, Disc. Appl. Math. 110 (2001), 289–300.
Y. Shavitt, T. Tankel, Hyperbolic embedding of internetgraph for distance estimation and overlay construction,IEEE/ACM Transactions on Networking 16 (2008), 25–36.
V.P. Soltan, V.D. Chepoi, Conditions for invariance of setdiameters under d-convexification in a graph, Cybernetics19 (1983), 750–756.
M. Thorup, All structured programs have small tree-widthand good register allocation, Information and Computation142 (1998), 159–181.
R. Uehara, Tractable and intractable problems ongeneralized chordal graphs, IEICE Technical Report,COMP98-83, pages 1–8, 1999. Available at: http://www.jaist.ac.jp/~uehara/pub/tech.html
K. Umezawa, K. Yamazaki, Tree-length equalsbranch-length, Discrete Mathematics 309 (2009),4656–4660.
J. Väisälä, Gromov hyperbolic spaces, ExpositionesMathematicae 23 (2005), 187–231.
D.R. Wood, On tree-partition-width, European Journal ofCombinatorics 30 (2009), 1245–1253.
Recommended