Some classes of perfectly orderable graphs

Some Classes of Perfectly Orderable Graphs

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C.T. Hoang* DEPARTMENT OF COMPUTER SClENCE

RUTGERS UNlVERSlTY NEW BRUNSWICK, NEW JERSEY

B.A. Reedt DlSCRETE MATHEMATICS GROUP

BELL COMMUNICATIONS RESEARCH MORRISTOWN, NEW JERSEY

ABSTRACT

In 1981, Chvatal defined the class of perfectly orderable graphs. This class of perfect graphs contains the comparability graphs and the triangulated graphs. In this paper, we introduce four classes of perfectly orderable graphs, including natural generalizations of the comparability and triangulated graphs. We provide recognition algorithms for these four classes. We also discuss how to solve the clique, clique cover, coloring, and stable set problems for these classes.

1. INTRODUCTION

A natural way to color the vertices of a graph is to put them in order u, < u2 < - - - < u, and then assign colors in the following manner:

(1) Consider the vertices sequentially (in the sequence given by the order). (2) Assign to each ui the smallest color (positive integer) not used on any

We shall call this the greedy coloring algorithm. An order < of a graph G is perfect if for each subgraph H of G, the greedy algorithm applied to H , gives an optimal coloring of H . AR obstruction in an ordered graph (G, <) is a set of

neighbor uj of ui with j < i.

*Present address: Research Institute for Discrete Mathematics, University of Bonn, Bonn, Federal Republic of Germany. 'Present address: Department of Combinatorics and Optimization, University of Waterloo, Waterloo. Ontario, Canada.

Journal of Graph Theory, Vol. 13, No. 4, 445-463 (1989) 0 1989 by John Wiley & Sons, Inc. CCC 0364-90241'891040445-1 9$04.00

446 JOURNAL OF GRAPH THEORY

four vertices {a, b , c, d } with edges ab, bc, cd (and no other edges) and a < b, d < c. Clearly, if < is a perfect order on G , then (G, <) contains no obstruction (because the chromatic number of an obstruction is two, but the greedy coloring algorithm uses three colors).

In 1981, V. Chvital [2] introduced perfect order and perfectly orderable graphs. A graph is perfectly orderable if it admits a perfect order. He proved that

(3) an order is perfect if and only if it contains no obstruction, and (4) all perfectly orderable graphs are perfect.

In this paper, we discuss four classes of perfectly orderable graphs. To do this, it will be convenient to work with orientations instead of orders. An orientation U of a graph G is an antisymmetric directed graph obtained from G by assigning a direction to each edge of G. To an ordzed graph (G, <), there corresponds an orientation D(G, <) of G such that ah E D(G, <) if and only if ab E E(G) and a < b. Similarly, given an acyclic directed graph F we can construct an ordered graph (G, <) such that D ( G , <) = F. Thus, we can restate (3) as follows: a graph is perfectly orderable if and only if it admits an acyclic orientation that does not contain an induced subgraph isomorphic to the graph F of Figure 1.

Equivalently: a graph is perfectly orderable if and only if it admits an acyclic orientation in which each induced path of length three is one of the three types in Figure 2.

We obtain six classes of graphs by permitting any nonempty proper subset of these three types. In this paper, we discuss four of these six classes of perfectly orderable graphs. In Section 3, we give characterizations of these four classes. In Section 4, we provide polynomial-time recognition algorithms for these four

FIGURE 1

FIGURE 2.

PERFECTLY ORDERABLE GRAPHS 447

classes. In Section 5, we show how to solve the clique, coloring, clique cover, and stable set problems for graphs in these four classes in polynomial time. These results are interesting because there is no known polynomial-time algorithm to recognize perfectly orderable graphs. Also, two of the classes gener- alize the classes of comparability and triangulated graphs, respectively. In Section 2, we discuss comparability graphs, triangulated graphs, and some other classes of perfectly orderable graphs. This section also contains basic definitions and notations.

2. BACKGROUND

2.1. Definitions

For a graph G, denotes the complement of G. In this paper, “subgraphs” shall always mean “induced subgraphs.” Given a graph G = ( V , E ) , we define G * = (V, X ) to be a directed graph such that xy E E if and only if xy, yx E X . Let C, (respectively Pk) be the chordless cycle (respectively path) with k vertices. We are especially interested in the P4. If G is a graph, then EP4(G) denotes the set of edges of G that are contained in some P4 of G. By the P4 abcd we shall always mean a P4 with vertices a , b , c , d and edges ab, bc, cd; the edge bc is called a rib, the edges ab, cd are called wings, the vertices b , c are called joints, and the vertices a , d are called tips.

For vertices x , y of a graph G , if xy E E(G) then we say that x sees y; otherwise, we say that x misses y. A set of vertices H of a graph G is homogeneous if 2 5 IH I < IG 1, and each vertex outside H sees either all vertices of H or no vertex of H .

+ +

2.2 Classical Perfect Orders

A graph (G = V , E ) is a comparability graph if G satisfies the following three equivalent conditions:

(i) There is a partially ordered set corresponding to G such that two vertices are adjacent in G if and only if the corresponding elements are comparable in the partially ordered set.

(ii) G permits an order < on its vertices so that no induced P3 with vertices a , b, c and edges ab, bc has a < b, b < c (a transitive order).

(iii) G admits an acyclic orientation that contains no induced subgraph isomorphic to the graph F shown in Figure 3 (a transitive orientation).

FIGURE 3.


Observation 2.1. Every comparability graph is perfectly orderable.

Proof. Cledrly, in a transitive orientation, every P., is of type 3. I

The following theorem of Ghouila-Houri [3] is the key to a polynomial-time algorithm to recognize comparability graphs.

Theorem 2.2 A graph is a comparability graph if and only if it admits an orientation that contains no induced subgraph isomorphic to the graph F (a semitransitive orientation). I

Ghouila-Houri used the following two lemmas to prove Theorem 2.2:

Lemma 2.3. then every semitransitive orientation of G is acyciic (and thus transitive).

If a comparability graph G does not contain a homogeneous set, I

Lemma 2.4. set. I

No minimal noncomparable graph contains a homogeneous

A graph G is triangulated (see [4]) if it satisfies the following four equivalent conditions:

(i) G contains no induced subgraph isomorphic to C, for k 2 4. (ii) Every induced subgraph of G contains a vertex whose neighborhood in-

(iii) G admits an order <, such that u, is simplicial in the subgraph induced

(iv) There is an acyclic orientation U of G that contains no induced subgraph

duces a clique (a simplicial vertex).

by H = {x 1 u, < x} + u, (a simplicial order).

isomorphic to the graph shown in Figure 4 (a simplicial orientation).

Observation 2.5. Every triangulated graph is perfectly orderable.

Proof. type 2. I

Clearly, in a simplicial orientation every Ps is either of type 1 or

By properties (ii) and (iii), we can determine if a graph is triangulated using the following simple-minded algorithm:

Algorithm SIMPLICIAL (G)

Input. A graph G . Output. A simplicial order of G or a message “failure.” -

FIGURE 4.


Step 0 . Set <= 0, set H = G. Step I . Locate a simplicial vertex x of H. If there is no simplicial vertex,

Step 2. If H is empty, then return the order <, and stop. Else go to Step 1. then return “failure,” and stop. Set H = H - x. Set y < x for each y in H.

It is well known that triangulated graphs can be recognized in linear time

A graph G is an indifference graph (see [4]) if it satisfies the following (references can be found in [4]).

equivalent conditions:

We can associate the vertices of G with unit intervals on the real line such that two vertices are adjacent if and only if the corresponding intervals intersect. G is triangulated, is a comparability graph, and G contains no induced subgraph isomorphic to the graph F of Figure 5 . G permits an order < such that for any P3 abc (with ab, bc E E(G)) , a < b and b < c , or c < b and b < a (an indifference order). There is an acyclic orientation U of G in which every P3 is directed as shown in Figure 6.

For a polynomial-time algorithm recognition for indifference graph, see [4]. Chvhtal defined the class of brittle graphs. A graph G is brittle if the follow-

(i) Every induced subgraph H of G contains a vertex that is either not a tip

(ii) There is an order < on G such that for every induced subgraph H or G,

ing two equivalent conditions hold:

of any P., of H or not a joint of any P4 of H.

either

(a) the smallest vertex of H is not the tip of any P4 in H, or (b) the largest vertex of H is not the joint of any P4 of H (a brittle order).

The following algorithm will recognize a brittle graph in O(n5) time:

FIGURE 5.

FIGURE 6.


Algorithm BRITTLE

Input. A graph G. Output. Step 0 . Step 1 .

Step 2 .

Step 3 . Step 4 .

A brittle order < for G, or a message “failure, G is not brittle.” Set <= 0, H = G . If H contains a vertex x that is not the tip of any P,, then set H =

If H contains a vertex x that is not a joint of any P4, then set H =

Return “failure, G is not brittle,” and stop. If H = 0, then return the brittle order < and stop; else go to

H - x, set x < y for each y in H , and go to Step 4.

H - x, set y < x for each y in H , and go to Step 4.

Step 1 .

3. CHARACTERIZATIONS

3.1. PJndifference Graphs

A graph G is P,-indifferent if the following three equivalent conditions hold:

(1) There is an order on the vertices of G that is an indifference order when restricted to any P4 of G (a P,-indifference ordering).

( 2 ) There is an acyclic orientation of G that is an indifference orientation when restricted to any P4 of C (a P,-indifference orientation).

(3) There is an acyclic orientation of G in which every P4 is of type 2 .

Note that in a P4-indifference Orientation, the direction of any one edge of a P4 uniquely determines the directions of the other two edges. We are thus moti- vated to make the following definitions:

G i v z a graph G, we-define a relation R-on the arcs of G * as follows. First, ab R ah for all arcs ab of G*. Second, ab R 2 if either +

(i) b = c, a # d, and acdw or wacd is a P4 for some vertex w of G. (ii) a = d, b # c, and wcab or cabw is a P4 for some vertex w of G.

C o 5 d e r any P,-indifference orientation U of G. Clearly 2 E U and & R cd implieshat cd E U (otherwise there will be an improperly directed P4). Now, let S(ab) be the equivalence class of directed edges under the transitive closure R* of R, which c o n t 2 s &. &t T(z) be the e l z e n t s of S ( 2 ) but with noAirection-Note that cd E S(ab) if and only if dc E S(z) and therefore T(ab) = T(ba).

Clearly E ( G ) is partitioned into disjoint equivalence classes T , = T(a ,b , ) , T , = T(a,b2), . . . , Tk = T(a,h,). By the precedinEemark- P4-indifference orientation restricted to some T, must be eithe_rS(a,b,) or S(b,a,) . Clearly U z - stricted to TI must be acyclic. Trivially, S(a,b,) is acyclic if and o n l y ~ ( b , a , ) is. Thus, if a graph permits a P,-indifference orientation, then each S( a,b,) must


be acyclic. In fact, this necessary condition is generally sufficient. (We shall refer to the graph in Figure 7 as the “pyramid.”)

Theorem 3.1. A graph G is P,-indifferent if and only if G does not obtain a pyramid, and for all arcs 3 of G*, S ( 3 ) - the equivalence class of 3 under the transitive closure R * -is acyclic.

Proof The “only if” part is easy to prove: it suffices to see that the pyramid is not P4-indifferent. Now to prove the “if” part, assume the theorem is false. Let G be a smallest counter example to the theorem. By the minimality of G , all proper subgraphs of G are P,-indifferent.

Claim 3.2. G contains no homogeneous set.

Assume G contains a homogeneous set H . Let x be an arbitrary vertex of H . Both G - H + x and H are P,-indifferent and thus permit P4-indifference orientations U , and U2, respectively. To obtain a P4-indifference orientation U o f G, orient G - H as in UI, orient H as &U,, and for y &H, z in G - H , yz (respectively zy ) is in U if and only if xz (respectively zx) is in U1. It is easy to verify that U is a P,-indifference orientation, contradicting the fact that G is not a P4-indifference graph.

Claim 3.3. G contains no induced cycle of length greater than 5.

Let C = vo, v I , . . . , v, be a cycle of length at least five in G. Trivially, vov, R vlvz R . . . R v z R v , u I . But then C induces a cycle in S(v,v, ), contradicting our assumptions about G.

- - -3

Claim 3.4. G contains none of the graphs Fl through F6 depicted in Figure 8.

It is a simple matter to verify that, in each of these graphs, some S ( 2 ) contains a cycle.

FIGURE 7.


Claim 3.5. tains one of the graphs F, - F3 depicted in Figure 8.

If a graph H contains a C4 and no homogeneous set, then it con-

Let uouI~2u3 be a C4 in a graph H, with no homogeneous set, where uiuitl is an edge (addition taken modulo 4). Consider the subgraph A of H induced by vertices that see both u, and u3. Let A , be the component in 7 containing uo and u2. Since H has no homogeneous set, there exists a vertex x that is not an element of A , that disagrees (in H ) on two elements of A , . Since A , is connected (in the complement of H), x must disagree on two vertices w l , w2 that are adjacent in A, and hence miss each other in H. We can assume that x is adjacent to w, and not adjacent to w2. We know x is not an element of A because every vertex of A - A, is adjacent to all the vertices of A , . Thus x sees at most one of u, and us. Now {w,, u , , w2, u3} induces a C, in H. If x saw precisely one of u1 and u3, then {x, w , , u , , w2,u,} would induce an F, in G. Thus x sees neither u, nor

F3

T

FIGURE 8.


u,. Now, consider the graph B induced by those vertices that see both w, and w,, but do not see x . Let B , be the component in containing u, and uj. Clearly there exists in H a vertex y that disagrees on B , (otherwise B , is a homogeneous set) and therefore on two nonadjacent (in H) vertices u , and u, of B , . We may assume that y sees u , and misses u,. We know y is not an element of B as B , is homogeneous in B . If y saw both w, and w, then, since y is not an element of B , y would see x . But then {x , y, wl, w,, u,) would induce an F, in H . Thus y sees at most one of w , , w,. If y saw precisely one of w , , w, then {y, u l , u,, w,, w,} would induce an F, in H. Thus y sees neither w, nor w,. Now {x , y, w l r w,, u I , u2} induces an F, or an F,, depending on whether or not x sees y. Claim 3.5 is justified.

Claims 3.2, 3.4, and 3.5 taken together imply that G contains no C,. Combining this with Claim 3.3, we see that G must be triangulated. We obtain an orientation U of the edges of EP, by arbitrarily selecting one of the two possible orientations for each nonsingleton equivalence class T,, thereby ensuring that each P4 is properly directed. If U does not contain a cycle, it can be extended into a P,-indifference orientation of G , a contradiction. Let K be a directed cycle of minimum cardinality in U . Clearly, K contains edges from two distinct equivalence classes as the orientation on each equivalence class in G is acyclic. We shall show that K must be a triangle. We shall also show that if K is a triangle then K is contained in a pyramid. These two statements imply Theorem 3. I . (We shall prove them in reverse order.)

Step 1. If K is a triangle then G contains a pyramid.

Label the vertices of K as uou,u,. Assume uoul is the wing of a P4 with tip u,,. Label the P4 uou,ba. If b is not adjacent to u:, then, since G is triangulated, a is not adjacent to u2. But then u,u,ba is an incorrectly directed P4, which cannot occur by our choice of the orientation. Therefore b sees u2. If a did not see u2, then abu,uo would be an incorrectly directed P4, which is impossible. Using symmetric arguments, we see that if an edge of K is the wing of a P,, then the other vertex of K sees every vertex of this P,.

Case I. 1 . Every edge of K is the wing of a P,.

In this case, by the above remark, we have three vertices uo, u , , u2 such that ui sees precisely ui in K. Moreover, these three vertices form a stable set, as otherwise G would contain a C,. But then the six vertices u , , u2, u3 , u, , u2, u3 form an F, in G, which by Claim 3 we know cannot occur.

Case 1.2. Two edges of K are the ribs of a P4 in G.

Without loss of generality we may assume that the two edges are uou, and qv,. Then uoul extends to a P4 au,,u,b, and uIu2 extends to a P4 cv,u,d. If b did not see u, then (b , u,, u,, d } would induce either a C, or an improperly directed


P,. Thus b sees u2 , and similarly, c sees uo. If b was adjacent to c then {b, c ,uo ,u2} would induce a C, in G. Thus b is not adjacent to c. We know a is adjacent to u2, as otherwise auou,b would form an improperly directed P,. Also, a does not see c or {a , c , u I , u2} would be a C, in G. But then {a, b , c} + K induces a pyramid, as required.

Case I .3. Two edges of K are wings of P4s

We have already remarked that if an edge of K is a wing of a P, then the other vertex of K sees every vertex of the P, (we will use this fact without stat- ing it from now on). We can assume that the two edges that are wings of P4s are uoul and uIu2. There are four subcases to consider.

Case 1.3.1. Both P4s have u, as a joint.

In this case, we can label the P,s as uou,ab and u,u,cd. We know that a , b , c , d are distinct vertices as they all have differing neighborhoods on K . There is no edge between d and b in G or {d, uo, u2, b} would induce a C, in G. For precisely the same reason, ca, cb, da are not in E(G) . Now, by considering the subgraph of G induced by { a , b , c , d , uo, u I , u2}, we may see that all the edges of K are in the same equivalence class, a contradiction.

Case 1.3.2. uI is a joint of one of the P4s; u2 is the joint of the other.

In this case, we can labcl the P,s as u,,v,uh and ulu2cd. We know that a , b, c, d are distinct vertices as they all have differing neighborhoods on K . As before, since G is triangulated, ca, cb, d a , db E E(G) . Now, by considering the subgraph of G induced by { a , b , c ,d ,uo ,u l ,u2} , it is easy to see that all the edges of K are in the same equivalence class, a contradiction.

Case 1.3.3. u, is the joint of one of the P4s; u, is the joint the other.

This case is symmetric to Case 1.3.2.

Case 1.3.4. u, is the tip of both P ~ S .

In this case, we can label the P4s as u,u,ab and u,u2cd. It is possible that a and c are the same vertex. Regardless, we know that d does not sees b or duou2b would be a C, in G. Now, if d saw a then {b, a , d , uo, u , , u2} would be a pyramid. Similarly, if b saw c then { b , c , d , u , , u , , u , } would be a pyramid. Since G contains no pyramid, it follows that a misses d , b misses c, and therefore a and c are distinct vertices. Now { a , b , c , d , ~ ~ , ~ ~ , ~ ~ } induces either an F5 or an F6, depending on whether or not a sees c. But we know that G contains neither of these graphs, and so this contradiction completes the proof for this case.


Step 2. K is a triangle.

We shall assume that IK I b 4 and derive a contradiction. Since K is not contained in one equivalence class, clearly there exist two incident edges in K from different equivalence classes. Let vovl and vlv2 be these edges.

Case 2 . I . vo is not adjacent to v2.

Assume there exists a vertex u in G that is adjacent to vo but not v,. Clearly, u is not adjacent to v2 or uvovlv2 would be a C, in G . But then uvov,v2 is a P4 in G , contradicting the fact that uovl and v1v2 are in different equivalence classes. Thus no such vertex u exists. Now, since vovl is in a P,, it clearly must be a wing with joint v , of some P4 vov,ab. By an analogous argument, vzul is in a P4 u2vlcd. Now d does not see v,, for then it would see u,, and not v,, which we have just shown to be impossible. However c must see v,, or uou,cd would be an improperly directed P4. Similarly, a sees v2 but b does not. If a does not see c then neither does b as otherwise {a , b, c, v,} would be a C,. But then bav,c is a P4 with edges from two equivalence classes, which is impossible. Thus a is adjacent to c . If b saw c then {a, 6 , c, vo,vI,vz} would induce a pyramid in G , and we are done. Thus b misses c. Similarly, d misses a. Now, b is not adjacent to d or {a ,b ,d ,c} would be a C, in G . However, in this case, bacd is a P4 with edges in two equivalence classes, a contradiction.

Case 2.2. vo is adjacent to v2.

We note that the edge vov2 is in no P4 for otherwise one of K - u, or vovIv2 is a directed cycle, contradicting the minimality of K .

Claim 2.2.1. There do not exist vertices x and y such that x sees vo and misses v2, and y sees v2 and misses vo.

Proof. Clearly x must be adjacent to y, for otherwise xvov,y would be a P4. But then {x,vo, v2, y} is a C4 in G , a contradiction. I

Claim 2.2.2. If an edge of triangle vovlvz is the wing of a P4, then the other vertex of the triangle sees all the vertices of this P.,. I

Claim 2.2.2 is a restatement of a remark in Case 1.3.

Case 2.2.1. vov, and vIv2 are both wing of P4s.

By Claim 2.2.2, the two tips of these P4s that are not in the triangle contra- dict Claim 2.2.1.


Case 2 .2 .2 . uou, and uIuz are both ribs of P4s.

In this case, we can label the P,s as au,u,b and cu,u,d. If b misses u, then bu,u,d is either a P, with edges from two equivalence classes or a C,. Both these possibilities lead to a contradiction. It follows that b sees u2 and, by sym- metry, c sees uo. Now c sees u, but misses u, and b sees u2 but misses u,. This contradicts Claim 2.2.1.

Case 2.2.3. uoul is a rib of a P4 and uluz is the wing of a P4 with joint u2.

In this case, we can label the P4s as uv,u,b and u,u2cd. If b misses u2 then bv,u2c is either a P4 with edges from two equivalence classes or a C,. Both these possibilities lead to a contradiction. It follows that b sees u2. This contradicts Claim 2.2. I , with b and d the offending vertices.

Case 2.2.4. uovl is a rib of a P4 and ulu2 is the wing of a P, with joint u , .

In this case, we can label the P,s as au,u,b and u,u,cd. By Claim 2.2.2, uo sees c and d. Now b cannot see c or bcu,u, would be a P, containing uouz or a C,. Similarly, b cannot see d. Now dcu,b is a P, with edges from two equivalence classes, a contradiction.

Case 2.2.5. u2uI is the rib of a P4 and uluo is the wing ofthe f, with joint uo.

This case is symmetrical with Case 2.2.3.

Case 2.2.6. uzu, is the rib of a P, and uluo is the wing of a P, with joint u, .

This case is symmetrical with Case 2.2.4.

3.2. P,-Comparability Graphs

A graph G is P,-comparability if it satisfies the following three equivalent conditions:

(i) It permits an ordering that is transitive when restricted to any P, of G (a

(ii) It permits an acyclic orientation that is transitive when restricted to any

(iii) It permits an acyclic orientation in which each P, is of type 3.

Just as in a f,-indifference orientation, the direction of the edge of a f., in a f,-transitive orientation determines the direction of the other two edges of that P,. However, the interrelationship is slightly different. GivLn a g y h G, we define a relation L on the arcs of G* as follows: First, a b L a b . Secondly, & L 2 if either

P,-transitive ordering).

P, of G (a P,-transitive Orientation).


(i) c = a , b # d , and wbcd or bcdw is a P4 for some vertex w of H , or (ii) b = d, a # c , and wadc or adcw is a P4 for some vertex w of G.

Consider any P4-transitive orientation U of G. Clearly & E U and z L 2 implieLcd E U (otherwise, there would be an improperly directed P4). Now, let M(ab) be an equivalence class of directed edges under the transitive closure L* of L thatcontains z. Let N ( z ) be the elements of M ( 3 ) but with no direct&n. Note that cd E M ( z ) if and only if 2 E M ( 2 ) and therefore N ( J ) = N*).

Clearly, E(G) is partitioned by L* into disjoint equivalence class N , = N(a,b,) , N(&), . . . , Nk = (a&). By the preceding remarks,aP,-transitive orientation U restricted to some N , must be either M ( x ) or M ( b,a,). Clearly, U restricted to N , must be acyclic. Trivially, M ( x ) is acyclic if and only-( bp, ) is. Thus if a graph permits a P4-transitive orientation, each M ( a , b , ) must be acyclic. In fact, this necessary condition is also sufficient.

+

Theorem 3.6. ab of G * , M(&) (the equivalence class of z under L ) is acyclic.

A graph G is a P,-comparability graph if and only if for each

Proof of Theorem 3.6. In proving the analogous statement for P4-indifference graphs, we arbitrarily directed each of the nonsingleton equivalence classes in E(G), then we showed that either the resulting orientation could be extended into a P4-indifference orientation, or the graph had a homogeneous set or contained a pyramid. Precisely the same statement holds for P4-comparability graphs. We shall say a P4 is a proper under some orientation if the orientation is transitive when restricted to that P4. We define a proper orientation of EP4(G) to be one in which every P4 is proper. A graph G is interesting if M(&) is acyclic for every pair of adjacent vertices a and b in G. We shall show that if U is a proper Orientation of the P4-edge set of an interesting graph G, then one of the following holds:

(i) G has a homogeneous set. (ii) G contains a pyramid.

(iii) U is acyclic and thus can be extended into P,-transitive orientation of G.

Unfortunately, although the pyramid is not a P4-indifference graph, it is a P4-comparability graph. Therefore, we need to examine in more detail the graphs for which (ii) holds. To do this, we will need the following definition:

Definition 3.7. sets C, S, P, Q, R good if the following properties hold.

Let G be a graph. We call a partitioning of G into five disjoint

(1) C is a clique with at least two vertices; S is a stable set. (2) x E P .$ x is adjacent to all of C U S. (3) x E R j x is adjacent to all of C and none of S. (4) x E Q 3 x is nonadjacent to all of C U S U R . (5) P U Q U R is nonempty.


We actually show that if (ii) holds while (i) and (iii) do not, then the pyramid is directed as in Figure 9, and furthermore, that this implies that G permits a good partition.

Thus, we actually prove the following claim:

Claim 3.8. one of the following four conditions holds:

Let U be a proper orientation of the P4-edge set of a graph G; then

(i) G has a homogeneous set. (ii) G has a good partition.

(iii) G is not interesting. (iv) U is acyclic and thus can be extended into a P,-transitive orientation of G .

If Theorem 3.6 fails, then there exists a minimal counterexample of G . With- out loss of generality, we may assume that G is minimally P,-incomparable. Also, G is interesting since the orientation of each equivalence class is antisymmetric and acyclic. We know that neither (iii) nor (iv) holds on G . We show that neither (i) nor (ii) can hold any minimally P4-incomparable graph. But this contradicts Claim 3.8 and thus Theorem 3.6 must hold. We shall only give a sketch of the proof of Claim 3.8, since the entire proof is very involved; we shall present it in [7].

Claim 3.9. set. I

No minimally P,-incomparable graph contains a homogeneous

The proof of Claim 3.9 is similar to 3.2 and is left to the reader.

Claim 3.10. tion. I

No minimally P,-incomparable graph contains a good parti-

To complete the proof of Theorem 3.6, we need the following two claims:

Claim 3.11. contains a directed triangle. I

If a proper orientation of an interesting graph is cyclic, then it

FIGURE 9


Claim 3.12. a directed triangle, then G contains a homogeneous set or a good partition.

If an interesting graph G admits a proper orientation that contains I

3.3. P,-simplicia1 Graphs

A graph G is P,-simplicial if the following three equivalent conditions hold:

(i) G permits an ordering that is simplicial when restricted to any P4 of G (a P4-simplicia1 ordering).

(ii) G permits an orientation that is simplicia1 when restricted to any P4 of G (a P4-simplicia1 orientation).

(iii) Every P4 of G is of type 1 or 2.

We shall show that P4-simplicia1 graphs are a special class of brittle graphs. Recall that a graph G is brittle if each subgraph H of G contains a vertex x such that x is either not the joint of a P4 in H or not the tip of a P4 in H. A graph G is strongly brittle if each induced subgraph H of G contains a vertex x such that x is not the joint of a P4 in G, abxc with b, c E H (clearly every strongly brittle graph is brittle).

Theorem 3.13. A graph is P,-simplicial if and only if it is strongly brittle.

Proof of Theorem 3.13.

Claim 3.14. Every P,-simplicial graph is strongly brittle

Proof. Let G be a P,-simplicial graph and let H be a subgraph of G. Con- sider an arbitrary P,-simplicia1 ordering of the vertices of G and let x be the largest vertex of H in this ordering. Then x is not the joint of a P4 abxc in G , with b , c E H . I

Claim 3.15. Every strongly brittle graph is P,-simplicial.

Proof. The following algorithm, given a strongly brittle graph G, produces a P4-simplicial ordering of G .

Algorithm

Input. Output. Step 0 . Step I . Step 2 . Step 3 .

A strongly brittle graph G. A P4-simplicia1 order < on G. Set H = G. Set <= 0. Choose x in H such that x is in no P4 abxc (of G ) with b, c E H. Set H = H - x. Set y < x for all y in H. If H # 0 then go to step 1; else stop.


3.4. Raspail Graph

A graph is Raspail if it permits an orientation in which every P4 is of type 1. In 1984, Reed discovered a minimal forbidden subgraph characterization of Ras- pail graphs. In order to describe this characterization, we need to define clique wheels. For n at least two, a n-clique wheel is the graph formed from a clique of order n as follows: First, label the vertices of the clique uo, u , , . . . , u,,-,. Then, introduce vertices po. p , , . . . , p n - ] and qo, q , , . . . , qn-l such that

(a) for i = 0, 1 , . . . , n - I , p , misses u, , , but sees u, for j f i + I ; (b) for i = 0, I , . . . , n - 1, q, misses u,, u,,, but sees u, for j # i , j # i + 1; (c) P = { po, p , , . . . , pn- ,} and Q = {qo, q , , . . . , qn-l) are stable sets; and (d) p , sees q! but misses q, f o r j # i.

Theorem 3.16. isomorphic to a clique wheel or any of the graphs F, - F, depicted in Figure 10.

A graph G is Raspail if and only if G contains no subgraph

This theorem was proved independently by Hertz and deWerra 161.

FIGURE 10


4. RECOGNITIONS

Each of the four classes of graphs described in the previous section can be recognized in polynomial time. In this section, we shall describe polynomial-time algorithms to recognize these four classes.

4.1. P,-lndifference Graphs

In Section 3.1, we described how to partition the symmetric graph G * of a graph G into equivalence classes S,, . . . ,S, by considering the P4s in G . We then proved that a graph is P4-indifferent if and only if

(i) for each equivalence class Si, the directed subgraph of G * formed by

(ii) G contains no pyramid.

Thus, to determine if a graph is P4-indifferent we need only construct the classes Si and then test that (i) and (ii) hold. Clearly, we can verify (ii) in O(n6) time. Furthermore, Zf=, lE(Si)l 5 2n2, so having constructed the classes Si, we verify (i) in O(n2) time. See [l] for an algorithm that determines if a directed graph G is acyclic in O(IE(G)l). We shall now describe an O(n4) algorithm to construct the equivalence classes Si.

Recall that we partition G into the classes Si by considering the transitive closure of the relation RAefined as follows:

We say that & R cd if either

(1) b = c, a # d , and acdw or wacd is a P4 for some vertex w of G , or ( 2 ) a = d , b # c, and wcab or cabw is a P4 for some vertex w of G.

To construct the classes Si we only need construct the transitive closure R * of R. To construct R we only need scan through all P4s of G, adding pairs of arcs to R as appropriate. This takes O(n4) time. Using standard transitive closure algorithm (see [ l]) we can then construct R * in O(n4) time.

Since we can construct the classes Si in O(n4) time, and then verify (i) and (ii) in O(n4) and O(n6) time, respectively, we can recognize P4-indifference graphs in o(n6) time.

the arcs of Si is acyclic, and

4.2. P,-Cornparability Graphs

Our algorithm to recognize P4 comparability graphs is similar to, but much sim- pler than, the recognition algorithm for P4-indifference graphs. In Section 3.2, we described how to partition the arcs of the symmetric graph G* of a graph G into equivalence classes M,, M2, . . . , Mk such that G is P,-comparability graph if and only if the directed graph formed by the arcs of each Mi is acyclic. We can construct the M i s and verify that the corresponding subgraphs are acyclic in O(n4) time. The method we use to do this was described in the last section.


4.3. P,-Simplicia1 Graphs

In Section 3.3, we described an algorithm that, given a P4-simplicia1 graph G, constructs a P,-simplicial ordering of the vertices of G. We can trivially modify this algorithm to provide a recognition algorithm for P4-simplicial graphs that runs in O(n5) time (the algorithm will halt with failure if we cannot find a vertex x of the appropriate type in step I ) .

4.4. Raspail Graphs

In a Raspail orientation, every wing must be directed toward its tip. The following algorithm will determine if a graph is Raspail in O(n4) time:

Algorithm RASPAIL

Input. A graph G Output. Step 0. Step 1 .

Step 2.

A message “G is Raspail” or “G is not Raspail.” Set H = 0. For each P4 ubcd in G, set H = H + {g) + {a> (takes O(n4)

If H is acyclic, return “G is Raspail”; else return “G is not Raspail.” time).

5. OPTIMIZATION

Grotschel, LovBsz, and Schrijver [S] developed a polynomial-time algorithm to solve the following four optimization problems for perfect graphs: maximum clique, maximum stable set, minimum coloring, and minimum clique-cover. This algorithm is based on the ellipsoid method and provides no insight into the structure of perfect graphs. Therefore, it is of interest to find “combinatorial” algorithms to solve the four optimization problems for special classes of graphs. Chvatal [2] described an algorithm that, given a graph G and a perfect order on the vertices of G, finds, in polynomial time, a largest clique and an optimal coloring of G. We shall use his algorithm to solve these two problems for our classes of graphs in polynomial time. Clearly, Raspail, P4-indifference, and P,-simplicia1 graphs are brittle. We can construct a perfect order of any brittle graphs in polynomial time (see Algorithm Brittle). Thus for these three classes of graphs, the maximum clique problem and minimum coloring problems can be solved in polynomial time. Since complements of brittle graphs are brittle, we can also solve the other two optimization problems in polynomial time for these three classes.

We have developed an algorithm that, given a P,-comparability graph G , constructs a P4-transitive order for C . The details of the algorithm is given in


[7]. We shall informally describe the algorithm here. Given a P,-comparability graph G, the algorithm performs one of the following two actions:

(1) If G has a homogeneous set H then the algorithm

(i) selects a vertex x of H, (ii) finds a P,-transitive order { u , , u2, . . . , uI , x , uIt ,, . . . , uk} of G -

(iii) combines them into a P,-transitive order { u , , u2, . . . , u,, u , , . . . ,

(2) If G has a good partition C, S , P, Q , R and no homogeneous set, then the

(H - x) and a P4-transitive order {ul , uz, . . . , u,} of H , and

u,, u,+i , . . . 9 uk} of G .

algorithm

(i) selects s E S and c E C such that sc 65 E(G), (ii) finds a P,-transitive order V, = { u , , u,, . . . , u,} of C + S and a P4-

transitive order V, = {ul, u2, . . . , u,} of G - (C - c) - ( S - s), (iii) transforms V, into a new P4-transitive order V, = {u , , u,, . . . ,

u,, c, s, uIt3, . . . , urn) or V3 = {ul, u2, . . . , u,, s, c, uIt3, . . . , urn) in which there is no z with s < z < c or c < z < s,

(iv) combines V, and V, into a P,-transitive order {u l , u,, . . . , u , , u , , . . . , u,, u,+,, . . . , u,} of G.

If G has no good partition and no homogeneous set, then any interesting orientation of G is acyclic. Thus, any interesting orientation corresponds to a par- tial order all of whose linear extensions are P4-transitive orders. In this case the algorithm constructs an interesting orientation of G and then extends the corresponding order to a linear order of G. The algorithm runs in O(n5) time.

References

[ l ] A. V. Aho, J. E. Hopcroft, and J. D. Ullman, The Design and Analysis of Computer Algorithms. Addison-Wesley, Menlo Park, CA (1974).

121 V. Chvatal, Perfectly ordered graphs. Topics on Perfect Graphs. North- Holland, Amsterdam (1984) 63-65.

[3] A. Ghouila-Houri, CaractCrisation des graphes non orient& dont on peut orienter les aretes de manibre h obtenir le graphe d’une relation d’ordre. C.R. Acad. Sci. Paris 254 (1962) 1370-1371.

[4] M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs. Aca- demic Press, New York (1980).

I51 M. Grotschel, L. LovBsz, and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1 (198 1) 169-197.

[6] A. Hertz and D. de Werra, Les graphes bipolarisables. Report ORWP, Swiss Federal Institute of Technology in Lausanne (1986).

[7] C. T. Hohng and B. A. Reed. Recognizing P4-comparability graphs. Ann. Discrete Math., to appear.

Documents

Some classes of perfectly orderable graphs