(Meta) Kernelization

Hans L. Bodlaender∗ Fedor V. Fomin† Daniel Lokshtanov†Eelko Penninkx∗ Saket Saurabh† Dimitrios M. Thilikos‡

∗Department of Information and Computing Sciences,Utrecht University, TB Utrecht, The Netherlands

Email: {hansb|penninkx}@cs.uu.nl†Department of Informatics,

University of Bergen, Norway.Email: {fedor.fomin|daniello|saket}@ii.uib.no.

‡Department of Mathematics, National & KapodistrianUniversity of Athens, Panepistimioupolis, GR-15784, Athens, Greece.

Email: sedthilk@math.uoa.gr.

Abstract— Polynomial time preprocessing to reduce instancesize is one of the most commonly deployed heuristics to tacklecomputationally hard problems. In a parameterized problem, everyinstance I comes with a positive integer k. The problem is said toadmit a polynomial kernel if, in polynomial time, we can reducethe size of the instance I to a polynomial in k, while preservingthe answer. In this paper, we show that all problems expressiblein Counting Monadic Second Order Logic and satisfying a com-pactness property admit a polynomial kernel on graphs of boundedgenus. Our second result is that all problems that have finite integerindex and satisfy a weaker compactness condition admit a linearkernel on graphs of bounded genus.

The study of kernels on planar graphs was initiated by a seminalpaper of Alber, Fellows, and Niedermeier [J. ACM, 2004 ] whoshowed that PLANAR DOMINATING SET admits a linear kernel.Following this result, a multitude of problems have been shown toadmit linear kernels on planar graphs by combining the ideas ofAlber et al. with problem specific reduction rules. Our theoremsunify and extend all previously known kernelization results forplanar graph problems. Combining our theorems with the Erdos-Posa property we obtain various new results on linear kernels fora number of packing and covering problems.

1. INTRODUCTION

Preprocessing (data reduction or kernelization) as a strat-egy of coping with hard problems is universally used inalmost every implementation. The history of preprocessing,like applying reduction rules to simplify truth functions, canbe traced back to the 1950’s [32]. A natural question inthis regard is how to measure the quality of preprocessingrules proposed for a specific problem. For a long timethe mathematical analysis of polynomial time preprocessingalgorithms was neglected. The basic reason for this anomalywas that if we start with an instance I of an NP-hardproblem and can show that in polynomial time we canreplace this with an equivalent instance I ′ with |I ′| < |I|then that would imply P=NP in classical complexity. The

†Partially supported by the Norwegian Research Council.‡Supported by the project “Kapodistrias” (AΠ 02839/28.07.2008) of theNational and Kapodistrian University of Athens (project code: 70/4/8757).

situation changed drastically with advent of parameterizedcomplexity. Combining tools from parameterized complexityand classical complexity it has become possible to deriveupper and lower bounds on the sizes of reduced instances,or so called kernels.

In parameterized complexity each problem instance comeswith a parameter k and the parameterized problem is said toadmit a polynomial kernel if there is a polynomial time algo-rithm (the degree of polynomial is independent of k), calleda kernelization algorithm, that reduces the input instancedown to an instance with size bounded by a polynomial p(k)in k, while preserving the answer. This reduced instance iscalled a p(k) kernel for the problem. If p(k) = O(k), thenwe call it a linear kernel. Kernelization has been extensivelystudied in parameterized complexity, resulting in polynomialkernels for a variety of problems. Notable examples includea 2k-sized vertex kernel for VERTEX COVER [14], a 355k-vertex kernel for DOMINATING SET on planar graphs [3],which later was improved to a 67k-vertex kernel [12],and an O(k2) kernel for FEEDBACK VERTEX SET [33]parameterized by the solution size.

One of the most important results in the area of kernel-ization is given by Alber et al. [3]. They gave the first linearsized kernel for the DOMINATING SET problem on planargraphs. The work of Alber et al. [3] triggered an explosionof papers on kernelization, and in particular on kernelizationof problems on planar graphs. Combining the ideas of Alberet al. [3] with problem specific data reduction rules, kernelsof linear sizes were obtained for a variety of parameterizedproblems on planar graphs including CONNECTED VERTEX

COVER, MINIMUM EDGE DOMINATING SET, MAXIMUM

TRIANGLE PACKING, EFFICIENT EDGE DOMINATING SET,INDUCED MATCHING, FULL-DEGREE SPANNING TREE,FEEDBACK VERTEX SET, CYCLE PACKING, and CON-NECTED DOMINATING SET [3], [8], [9], [13], [24], [25],[26], [29], [30]. DOMINATING SET has received specialattention from kernelization view point, leading to a linear

2009 50th Annual IEEE Symposium on Foundations of Computer Science

0272-5428/09 $26.00 © 2009 IEEE

DOI 10.1109/FOCS.2009.46

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kernel on graphs of bounded genus [21] and polynomialkernel on graphs excluding a fixed graph H as a minor andon d-degenerated graphs [4], [31]. We refer to the surveyof Guo and Niedermeier [23] for a detailed treatment of thearea of kernelization.

Most of the papers on linear kernels on planar graphshave the following idea in common: find an appropriateregion decomposition of the input planar graph based onthe problem in question, and then perform problem specificrules to reduce the part of the graph inside each region.The first step towards the general abstraction of all thesealgorithms was initiated by Guo and Niedermeier [24], whointroduced the notion of “problems with distance property”.However, to prove that some problem posses a linear kernelon planar graphs, the approach of Guo and Niedermeierstill requires constructing reduction rules which are problemdependent. Thus all previous work on kernelization wasstrongly based on the design of problem specific reductionrules. In this paper we step aside and find properties ofproblems, like expressibility in certain logic, which allowsthese reduction rules to be automated. Our results not onlygive a uniform and natural explanation for a large family ofknown kernelization results but also provide a variety of newresults. In what follows we build up towards our theorems.We first give necessary definitions needed to formulate ourresults.

Let Gg be the family of all graphs that can be embeddedinto a surface Σ of Euler-genus at most g. Given a graphG embedded on a surface Σ of Euler-genus g, and a set S,we define Rr

G(S) to be the set of all vertices of G whoseradial distance from some vertex of S is at most r. The radialdistance between two vertices x, y is the minimum length ofan alternating sequence of vertices and faces starting from xand ending in y, such that every two consecutive elementsof this sequence are incident to each other. We say that aparameterized problem Π ⊆ Gg×N is compact if there existsan integer r such that for all (G = (V,E), k) ∈ Π, there isan embedding of G into a surface Σ of Euler-genus at mostg and a set S ⊆ V such that |S| ≤ r · k and Rr

G(S) = V .Similarly, Π is quasi-compact if there exists an integer rsuch that for every (G, k) ∈ Π, there is an embedding of Ginto a surface Σ of Euler-genus at most g and a set S ⊆ Vsuch that |S| ≤ r · k and tw(G \ Rr

G(S)) ≤ r (by tw(G)we denote the treewidth of G). Notice that if a problem iscompact then it is also quasi-compact.

We use Counting Monadic Second Order Logic(CMSO) [5], [15], [16], an extension of MSO, as a basictool to express properties of vertex/edge sets in graphs. Infact, it is known that every set F of graphs of boundedtreewidth is CMSO-definable if and only if F is finitestate [28]. Our first result concerns a parameterized analogueof graph optimization problems where the objective is to finda maximum or minimum sized vertex or edge set satisfyinga CMSO-expressible property. In particular, the problems

considered are defined as follows. In a p-MIN-CMSO graphproblem Π ⊆ Gg ×N, we are given a graph G = (V,E) andan integer k as input. The objective is to decide whetherthere is a vertex/edge set S of size at most k such thatthe CMSO-expressible predicate PΠ(G,S) is satisfied. Ina p-EQ-CMSO problem the size of S is required to beexactly k and in a p-MAX-CMSO problem the size of Sis required to be at least k. The annotated version Πα ofa p-MIN/EQ/MAX-CMSO problem Π is defined as follows.The input is a triple (G = (V,E), Y, k) where G is a graph,Y ⊆ V is a set of black vertices, and k is a non-negativeinteger. In the annotated version of a p-MIN/EQ-CMSOgraph problem, S is additionally required to be a subsetof Y . For the annotated version of a p-MAX-CMSO graphproblem S is not required to be a subset of Y , but insteadof |S| ≥ k we demand that |S ∩ Y | ≥ k.Our results. For a parameterized problem Π ⊆ Gg × N, letΠ ⊆ Gg × N denotes the set of all no instances of Π. Ourfirst result is the following theorem.

Theorem 1. Let Π ⊆ Gg ×N be a p-MIN/EQ/MAX-CMSOproblem and either Π or Π is compact. Then the annotatedversion Πα admits a cubic kernel if Π is a p-EQ-CMSOproblem and a quadratic kernel if Π is a p-MIN/MAX-CMSO problem.

We remark that a polynomial kernel for an annotatedgraph problem Πα, is a polynomial time algorithm that givenan input (G = (V,E), Y, k) of Πα, computes an equivalentinstance (G′ = (V ′, E′), Y ′, k′) of Πα such that |V ′| andk′ ≤ kO(1). Theorem 1 has the following corollary.

Corollary 1. Let Π ⊆ Gg × N be an NP-complete p-MIN/EQ/MAX-CMSO problem such that either Π or Π iscompact and Πα is in NP . Then Π admits a polynomialkernel.

Theorem 1 and its corollary give polynomial kernels forall graph problems on surfaces for which such kernels areknown. However, many problems in the literature are knownto admit linear kernels on planar graphs. Our next theoremunifies and generalizes all known linear kernels for graphproblems on surfaces. To this end we utilize the notion offinite integer index. This term first appeared in the workby Bodlaender and van Antwerpen-de Fluiter [10], [18] andis similar to the notion of finite state [1], [11], [15] (seeSubsection 2.4 for the formal definitions).

Theorem 2. Let Π ⊆ Gg × N has finite integer index andeither Π or Π is quasi-compact. Then Π admits a linearkernel.

Our two theorems are similar in spirit, yet they have a fewdifferences. In particular, not every p-MIN/EQ/MAX-CMSOgraph problem has finite integer index. For an examplethe INDEPENDENT DOMINATING SET problem is a p-MIN-CMSO problem, but it does not have finite integer index.

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Also, the class of problems that have finite integer indexdoes not have a syntactic characterization and hence it takessome more work to apply Theorem 2 than Theorem 1. Onthe other hand, Theorem 2 yields linear kernels, applies toquasi-compact problems and unifies and generalizes resultspresented in [2], [3], [8], [9], [13], [21], [24], [25], [26],[29], [30] as a corollary.

To demonstrate the applicability of our results we showin Section 6 how our theorems lead to polynomial or linearkernels for a variety of problems. For an ease of referencewe provide in the Section 8, a compendium of parameterizedproblems for which polynomial or linear kernels are derived.

2. PRELIMINARIES

In this section we give various definitions which we makeuse of in the paper. Let G = (V,E) be a graph. For asubset V ′ ⊆ V by G[V ′] we denote the subgraph inducedon V ′. A parameterized problem Π is a subset of Γ∗ × N

for some finite alphabet Γ. An instance of a parameterizedproblem consists of (x, k), where k is called the parameter.The notion of kernelization is formally defined as follows.

Definition 1. [Kernelization] A kernelization algorithm, orin short, a kernel for a parameterized problem Π ⊆ Γ∗ ×N

is an algorithm that given (x, k) ∈ Γ∗ × N outputs in timepolynomial in |x| + k a pair (x′, k′) ∈ Γ∗ × N such that(a) (x, k) ∈ Π if and only if (x′, k′) ∈ Π and (b) |x′|, k′ ≤g(k), where g is some computable function. The function gis referred to as the size of the kernel. If g(k) = kO(1) org(k) = O(k) then we say that Π admits a polynomial kerneland linear kernel respectively.

We will assume that the parameter k is given in unaryand hence in the pair (x, k), k ≤ |x|O(1).

2.1. Surfaces and Distances

In this paper we consider graphs embeddable into sur-faces. Let Gg be the class of all graphs that can be embeddedinto a surface Σ of Euler-genus at most g. We say thata graph G is Σ-embedded if it is accompanied with anembedding of the graph into Σ.

We define the normal distance between two vertices x andy to be the shortest path distance between them. The radialdistance between x and y is defined to be one less than theminimum length of a sequence starting from x and endingat y such that vertices and faces alternate in the sequence.Given an Σ-embedded graph G = (V,E) and a set S ⊆ V ,we denote by Rr

G(S) and BrG(S) the set of all vertices that

are in radial distance at most r and normal distance at most rfrom some vertex in S respectively. Notice that for every setS ⊆ V and every r ≥ 0, it holds that Br

G(S) ⊆ R2r+1G (S)

for any embedding of G into a surface Σ. An alternative wayof viewing radial distance is to consider the radial graph,RG: an embedded multigraph whose vertices are the verticesand the faces of G (each face f of G is represented by a

point vf in it). An edge between a vertex v and a vertexvf is drawn if and only if v is incident to f . Thus RG isa bipartite multigraph, embedded in the same surface as G.Hence, if G ∈ Gg then RG ∈ Gg . Also the radial distance ofa pair of vertices in G corresponds to the normal distancein RG.

Let G = (V,E) be a graph. A tree decomposition ofG is a pair (T = (VT , ET )),X = {Xt}t∈VT

) such that:∪u∈VT

= V , ∀e = (u, v) ∈ E,∃t ∈ VT : {u, v} ⊆ Xt and∀v ∈ V , T [{t | v ∈ Xt}] is connected. The width of a treedecomposition is maxt∈VT

|Xt|−1 and the treewidth of G =(V,E) is the minimum width over all tree decompositions ofG. A tree decomposition is called a nice tree decompositionif the following conditions are satisfied: Every node of thetree T has at most two children; if a node t has two childrent1 and t2, then Xt = Xt1 = Xt2 ; and if a node t has onechild t1, then either |Xt| = |Xt1 | + 1 and Xt1 ⊂ Xt or|Xt| = |Xt1 | − 1 and Xt ⊂ Xt1 . It is possible to transforma given tree decomposition into a nice tree decompositionin time O(|V | + |E|) [6].

Given an edge e = (u, v) of a graph G = (V,E), bycontracting an edge (u, v) we mean identifying the verticesu and v and removing all the loops and duplicate edges.A minor of a graph G is a graph H that can be obtainedfrom a subgraph of G by contracting edges. A graph classC is minor closed if any minor of any graph in C is also anelement of C. A minor closed graph class C is H-minor-freeor simply H-free if H /∈ C.

2.2. Protrusions

Given a graph G = (V,E) and S ⊆ V , we define ∂G(S)as the set of vertices in S that have a neighbor in V \S. For aset S ⊆ V the neighbourhood of S is NG(S) = ∂G(V \S).When it is clear from the context, we omit the subscripts.We now define the notion of a protrusion.

Definition 1. [r-protrusion] Given a graph G = (V,E), wesay that a set X ′ ⊆ V is an r-protrusion of G if |N(X ′)| ≤ rand tw(G[X ′ ∪ N(X ′)]) ≤ r.

For an r-protrusion X ′, the vertex set X = X ′ ∪ N(X ′)is an extended r-protrusion. The set X is the extendedprotrusion of X ′ and X ′ is the protrusion of X .

2.3. Counting Monadic Second Order Logic

The syntax of MSO of graphs includes the logical con-nectives ∨, ∧, ¬, ⇔, ⇒, variables for vertices, edges, setof vertices and set of edges, the quantifiers ∀, ∃ that canbe applied to these variables, and the following five binaryrelations: (1) u ∈ U where u is a vertex variable andU is a vertex set variable; (2) d ∈ D where d is anedge variable and D is an edge set variable; (3) inc(d, u),where d is an edge variable, u is a vertex variable, and theinterpretation is that the edge d is incident on the vertexu; (4) adj(u, v), where u and v are vertex variables u,

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and the interpretation is that u and v are adjacent; (5)equality of variables representing vertices, edges, set ofvertices and set of edges. In addition to the usual featuresof monadic second-order logic, if we have atomic formulastesting whether the cardinality of a set is equal to n modulop, where n and p are integers such that 0 ≤ n < p andp ≥ 2, then this extension of the MSO is called the countingmonadic second-order logic. So essentially CMSO is MSOwith following atomic formula: If U denotes a set X , thencardn,p(U) = true if and only if |X| is n mod p. We referto [5], [15], [16] for a detailed introduction on CMSO.

2.4. t-Boundaried Graphs and Finite Integer Index

We first define the notion of t-boundaried graphs andvarious operations on them.

Definition 2. [t-Boundaried Graphs] A t-boundariedgraph is a graph G = (V,E) with t distinguished vertices,uniquely labelled from 1 to t. The set ∂(G) of labelledvertices is called the boundary of G. The vertices in ∂(G)are referred to as boundary vertices or terminals.

For a graph G = (V,E) and a vertex set S ⊆ V , we willsometimes consider the graph G[S] as the |∂(S)|-boundariedgraph with ∂(S) being the boundary.

Definition 3. [Gluing by ⊕] Let G1 and G2 be two t-boundaried graphs. We denote by G1⊕G2 the t-boundariedgraph obtained by taking the disjoint union of G1 and G2

and identifying each vertex of ∂(G1) with the vertex of∂(G2) with the same label; that is, we glue them togetheron the boundaries. In G1⊕G2 there is an edge between twolabelled vertices if there is an edge between them in G1 orin G2.

Definition 4. [Legality] Let G be a graph class, G1 andG2 be two t-boundaried graphs, and G1, G2 ∈ G. We saythat G1 ⊕G2 is legal with respect to G if the unified graphG1 ⊕G2 ∈ G. If the class G is clear from the context we donot say with respect to which graph class the operation islegal.

Definition 5. [Replacement] Let G = (V,E) be a graphcontaining an extended r-protrusion X . Let X ′ be therestricted protrusion of X and let G1 be an r-boundariedgraph. The act of replacing X ′ with G1 corresponds tochanging G into G[V \ X ′] ⊕ G1. Replacing G[X] withG1 corresponds to replacing X ′ with G1.

Definition 6. For a parameterized problem, Π on a graphclass G and two t-boundaried graphs G1 and G2, we saythat G1 ≡Π G2 if there exists a constant c such that for allt-boundaried graphs G3 and for all k: (a) G1 ⊕G3 is legalif and only if G2 ⊕G3 is legal; (b) (G1 ⊕G3, k) ∈ Π if andonly if (G2 ⊕ G3, k + c) ∈ Π.

Definition 7. [Finite Integer Index] Π has finite integerindex in G if for every t there exists a finite set S of

t-boundaried graphs such that S ⊆ G and for any t-boundaried graph G1 there exists a G2 ∈ S such thatG2 ≡Π G1. Such a set S is called a set of representativesfor (Π, t).

Note that for every t, the relation ≡Π on t-boundariedgraphs is an equivalence relation. A problem Π is finiteinteger index, if and only if for every t, ≡Π is of finiteindex, that is, has a finite number of equivalence classes.

3. REDUCTION RULES

In this section we give reduction rules for compact anno-tated p-MIN/EQ/MAX-CMSO problems and quasi-compactparameterized problems having finite integer index. Ourreduction rules have the following form:

“If there is a constant size separator such that afterits removal we obtain a connected component ofunbounded size and of constant treewidth, then wereplace this component with something of constantsize.”

The implementation of this rule depends on whether weare dealing with an annotated p-MIN-CMSO, p-EQ-CMSOor p-MAX-CMSO problems, or whether the problem inquestion has finite integer index. In the following discussionwe only treat annotated p-MIN/EQ/MAX-CMSO problemswhere the set S being searched for is a set of vertices. Thecase where S is a set of edges can be dealt in an identicalmanner.

3.1. Reduction for Annotated p-MIN-CMSO Problems

We now describe the reduction rule that we apply for an-notated p-MIN-CMSO problems. The technique employedin this section will act as a template for how we handle theannotated p-EQ/MAX-CMSO problems. Recall that in anannotated p-MIN-CMSO problem Πα we are given a graphG = (V,E) where a subset Y ⊆ V is colored black and aninteger k. The objective is to find a set S ⊆ Y of size atmost k such that a fixed CMSO-definable property PΠ(G,S)holds. For our reduction rule, we are also given a sufficientlylarge r-protrusion X ′. In the first step of the reduction, weshow that the set Y ∩ X ′ can be reduced to O(k) verticeswithout changing whether (G, k) is a yes instance to Πα ornot. In the second step we show that the r-protrusion X ′

can be covered by O(k) r′-protrusions such that each r′-protrusion contains at most a constant number of verticesfrom Y . In the third and final step of the reduction rule,we replace the largest r′-protrusion with an equivalent, butsmaller r′-boundaried graph. We now provide the reductionrule for annotated p-MIN-CMSO problems.

Lemma 1. Let Πα be an annotated p-MIN-CMSO problem.Let G = (V,E) be a graph, Y ⊆ V be the set of blackvertices and k be an integer. Let X be an extended r-protrusion of G. Then there is an integer c, and an O(|X|)time algorithm, that computes a set of vertices Z ⊆ X ∩ Y

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with |Z| ≤ ck, such that if there exists a W ⊆ Y withPΠ(G,W ) and |W | ≤ k, then there exists a W ′ ⊆ Y withPΠ(G,W ′), |W ′| ≤ k′, and W ′ ∩ X ⊆ Z.

Proof: The algorithm starts by making a tree decom-position of G[X] of width at most r, using the linear timealgorithm to compute treewidth by Bodlaender [6]. Now weadd ∂(X) to each bag, and add one bag containing only thevertices in ∂(X). The tree decomposition has width at most2r as the bag size is at most r+1+|∂(X)| ≤ 2r+1. Considerthe following equivalence relation on subsets Q ⊆ X ∩ Y .We say that Q ∼B Q′, if and only if for all R ⊆ V − Xwe have PΠ(G,Q ∪ R) ⇔ PΠ(G,Q′ ∪ R). The numberof equivalence classes of the relation ∼B is bounded by afunction of the treewidth [11], [17] of G[X], and thus forfixed r, can be assumed to be bounded by a constant, sayc. We would like to find a minimum sized representative ofeach of the equivalence classes. We describe an algorithmrunning in time O(|X|) to find the desired set in eachequivalence class. Let us consider an algorithm that solvesan optimization version of the problem min{|W | | W ⊆Y ∧ PΠ(G,W )} on graphs of bounded treewidth, usinga dynamic programming approach. For an example, wecan use the algorithm described by Borie et al. [11]. Thealgorithm of Borie et al. [11] computes for each equivalenceclass in relation ∼B (or, possibly, a refinement of the relation∼B) the minimum size of a set in the class. This is donein a dynamic programming fashion, computing each valuegiven a table of these values for the children of the bag inthe tree decomposition. The running time is linear for fixedc. It is not hard to observe that we can also compute foreach equivalence class a minimum size set Q ⊆ X ∩Y thatbelongs to the class. This can be done in linear time. If thereare more than one minimum size sets in a class, then thealgorithm just picks one.

Let Q denote that set of equivalence classes of rela-tion ∼B and suppose that for each class q ∈ Q that isnon-empty, we have a minimum size representative Qq.By the above argument we can find a minimum size rep-resentative Qq for each class in O(|X|) time. Now, setZ =

⋃q∈Q,|Qq|≤k Qq .

Suppose now that there exists W ⊆ Y with PΠ(G,W )and |W | ≤ k. Consider the equivalence class q that containsX ∩ W . Let Qq be the selected minimum size represen-tative of q. Consider the set W ′ = (W \ X) ∪ Qq. AsPΠ(G, (W \ X) ∪ (X ∩ W )), we have that PΠ(G, (W \X) ∪ Qc) = PΠ(G,W ′). Since, Qq is a minimum sizerepresentative from q, we have that |Qq| ≤ |W \ X|, andthat |W ′| ≤ |W |. Finally, since the number of equivalenceclasses in Q is a function of r and each representative Qq

has size at most k, we have that |Z| = O(k). This provesthe lemma.

Using Lemma 1, we change the set Y to (Y \ X) ∪ Z.We now show how to exploit the fact that Z contains O(k)

vertices.

Partitioning Protrusions: In the second step of the re-duction rule we partition the extended r-protrusion X intosmaller r′-protrusions.

Lemma 2. Let G = (V,E) be a graph, Y ⊆ V be the setof black vertices of G and k be an integer. Furthermore,let X be an extended r-protrusion and Z = X ∩ Y . Thereis a O(|X|) time algorithm that finds O(|Z|) extended r′-protrusions X1,X2, . . . , X� such that X = X1 ∪X2 ∪ . . .∪X� and for every i ≤ �, Z ∩ Xi ⊆ ∂(Xi).

Proof: We start by making a nice tree decompositionof G[X], and adding ∂(X) to all bags. Now, we mark anumber of bags. First, we mark the root bag and each forgetnode where a vertex in Z is forgotten. As each vertex isforgotten at most once in a nice tree decomposition, so farwe have O(|Z|) marked bags. Now, mark each bag that isthe lowest common ancestor of two marked bags, until wecannot marks in this fashion. Standard counting argumentsfor trees show that this operation at most doubles the numberof marks. Hence, there are at most O(|Z|) marked bags.

We now split X into X1,X2, . . . , X� as follows: we takeparts of the tree decomposition, with internally no markedbags, plus the marked bags at the border. Note that each suchpart has at most two marked bags, each of size at most 2rand thus the size of ∂(Xi) is at most 4r for every i ≤ �. Notethat by the way we constructed the sets Xi, Z∩Xi ⊆ ∂(Xi)for every i.

Reducing Protrusions: In the third phase of our reductionrule, we find a protrusion to replace, and perform thereplacement. Notice that this part of the reduction worksboth for annotated p-MIN-CMSO and for annotated p-EQ-CMSO problems.

Lemma 3. Let Πα be an annotated p-MIN-CMSO or p-EQ-CMSO problem. There is a fixed constant c dependingonly on Πα such that there is an algorithm that given agraph G = (V,E) ∈ Gg , a set Y ⊆ V of black vertices, aninteger k and an extended r-protrusion X with |X| > c suchthat Y ∩ X ⊆ ∂(X), runs in time O(|X|), and produces agraph G∗ = (V ∗, E∗) ∈ Gg such that |V ∗| < |V | and(G∗, k) ∈ Πα if and only if (G, k) ∈ Πα.

Proof: For two t-boundaried graphs G1 and G2, wesay that they are equivalent with respect to a subset Sof ∂(G1) = ∂(G2) if for every G3 = (V3, E3) and setS′ ⊆ V3 we have that PΠ(G1 ⊕ G3, S ∪ S′) if and onlyif PΠ(G2 ⊕ G3, S ∪ S′). If G1 and G2 are equivalentwith respect to S we say that G1 ∼S G2. The canonicalequivalence relation for CMSO properties with free setvariables has finite index [11], [17] and hence the numberof equivalence classes of ∼S depends only on t for everyfixed S ⊆ ∂(G1). We make a new equivalence relationdefined on the set of t-boundaried graphs belonging to a

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graph class G. Two t-boundaried graphs G1, G2 ∈ Gg areequivalent if (a) for every t-boundaried graph G3, G1 ⊕G3

is legal if and only if G2 ⊕ G3 is legal; and (b) for everyS ⊆ ∂(G1) = ∂(G2), G1 ∼S G2. Now, ∼S has a finitenumber of equivalence classes for every S ⊆ ∂(G1) and theclass Gg is characterized by a finite set of forbidden minors.Hence the number of equivalence classes in the equivalencerelation defined above is a function of t. Let S be a set of r-boundaried graphs containing one smallest representative foreach equivalence class of the relation above. Let c the sizeof the largest graph in S. We have that X is a r-boundariedgraph with boundary ∂(X). Let G1 be a graph in S suchthat G1 and G[X] are equivalent.

Now we replace G[X] with G1 = (V1, E1) in the graphG, and let the resulting graph be G∗ = (V ∗, E∗). Let Y1

be the set of black vertices in ∂(G1). We let Y ∗ = Y \X ∪ Y1 be the set of black vertices in G∗. Since |X| > c,we have that |V1| < |X| and hence |V ∗| < |V |. It remainsto prove that (G, k) ∈ Πα if and only if (G∗, k) ∈ Πα.In one direction, suppose there is a set S ⊆ Y such thatPΠ(G,S) holds. Then S ∩ X ⊆ ∂(G[X]) and since G[X]and G1 are equivalent with respect to the relation above wehave that PΠ(G∗, S) holds. In the other direction, supposePΠ(G∗, S) holds. Since Y ∗∩V1 ⊆ ∂(G1) and G[X] and G1

are equivalent with respect to the relation above, we havethat PΠ(G,S) holds. This concludes the proof.

Lemmata 1, 2 and 3 together yield a reduction rule forall annotated p-MIN-CMSO problems.

Lemma 4. [�]1 Let Πα be an annotated p-MIN-CMSOproblem. There is a fixed constant c depending only onΠα such that there is an algorithm that given a graphG = (V,E) ∈ Gg , a set Y ⊆ V of black vertices, an integerk and an extended r-protrusion X with |X| > ck, runs intime O(|X|), and produces a graph G∗ = (V ∗, E∗) ∈ Gg

such that |V ∗| < |V | and (G∗, k) ∈ Πα if and only if(G, k) ∈ Πα.

3.2. Reduction for Annotated p-EQ-CMSO Problems

In this section we give a reduction rule for annotated p-EQ-CMSO problems. The rule is very similar to the onefor the p-MIN-CMSO problems described in the previoussection. Therefore we only highlight the differences betweenthe two rules in our arguments. The main difference betweenthe two problem variants is that we need to keep track ofsolutions of every fixed size between 0 and k, instead of justthe smallest one in each class. Because of this we requirethe protrusion to contain at least ck2 vertices instead of ckvertices, in order to be able to reduce it.

Lemma 5. [�] Let Πα be an annotated p-EQ-CMSOproblem. There is a fixed constant c depending only on

1Proofs for results marked with [�] can be found in the extended versionof the paper at http://arxiv.org/abs/0904.0727.

Πα such that there is an algorithm that given a graphG = (V,E) ∈ Gg , a set Y ⊆ V of black vertices, an integerk and an extended r-protrusion X with |X| > ck2, runs intime O(k|X|), and produces a graph G∗ = (V ∗, E∗) ∈ Gg

such that |V ∗| < |V | and (G∗, k) ∈ Πα if and only if(G, k) ∈ Πα.

3.3. Reduction for Annotated p-MAX-CMSO Problems

We now give a reduction rule for annotated p-MAX-CMSO problems. The rule is still similar to the onesdescribed in the two previous sections, but differs more fromthe p-MIN-CMSO problems than p-EQ-CMSO did.

Lemma 6. [�] Let Πα be an annotated p-MAX-CMSOproblem. There is a fixed constant c depending only onΠα such that there is an algorithm that given a graphG = (V,E) ∈ Gg , a set Y ⊆ V of black vertices, an integerk and an extended r-protrusion X with |X| > ck, runs intime O(|X|), and produces a graph G∗ = (V ∗, E∗) ∈ Gg

such that |V ∗| < |V | and (G∗, k) ∈ Πα if and only if(G, k) ∈ Πα.

3.4. Reductions Based on Finite Integer Index

In the previous sections we gave reduction rules for an-notated p-MIN/EQ/MAX-CMSO problems. These reductionrules, together with the results proved later in this article willgive quadratic or cubic kernels for the problems in question.However, for many problems we can in fact show that theyadmit a linear kernel. In this section we provide reductionrules for graph problems that have finite integer index. Thesereduction rules will yield linear kernels for the problems theyapply to. We are now ready to prove the reduction lemmafor problems that have finite integer index.

Lemma 7. Let Π ⊆ Gg × N has finite integer index in Gg .Then there exists a constant c and an algorithm that givena graph G = (V,E) ∈ G, an integer k and an extendedr-protrusion X in G with |X| > c, runs in time O(|X|) andreturns a graph G∗ = (V ∗, E∗) ∈ Gg and an integer k∗

such that |V ∗| < |V |, k∗ ≤ k and (G∗, k∗) ∈ Π if and onlyif (G, k) ∈ Π.

Proof: Let S be a set of representatives for (Π, r)and let c = maxY ∈S |Y |. Similarly let S ′ be a set ofrepresentatives for (Π, 2r) and let c′ = maxY ∈S′ |Y |. If|X| > 3c′ we find an extended 2r-protrusion X ′ ⊆ X suchthat c′ < |X ′| ≤ 3c′ and work on X ′ instead of X . This canbe done in time O(|X|) since G[X] has treewidth at mostr. From now on, we assume that |X| ≤ 3c′. This initialstep is the only step of the algorithm that does not workwith constant size structures, and hence the running time ofthe algorithm is upper bounded by O(|X|). The algorithmproceeds as follows.

Because Π has finite integer index there is a graph H =(VH , EH) ∈ S such that H ≡Π G[X]. We show how tocompute H from X . Since k is given in unary there exists

634

an integer p such that k ≤ |V |p. Let kmax = (6c′)p. Forevery G1 = (V1, E1) ∈ S, G2 = (V2, E2) ∈ S and k′ ≤kmax we compute whether (G1 ⊕ G2, k

′) ∈ Π. For eachsuch triple the computation can be done in time O((|V1| +|V2|)p) since Π has finite integer index [10], [18]. Now, forevery G1 ∈ S and k′ ≤ (|X| + |V1|)p we compute whether(G[X]⊕G1, k

′) ∈ Π. When all these computations are done,the results are stored in a table.

It is not hard to see that H ≡Π G[X] if and only if thereexists a constant c such that for all G2 ∈ S and k′ ≤ kmax,(H⊕G2, k

′) ∈ Π ⇔ (G[X]⊕G2, k′+c) ∈ Π. Also, c is the

constant such that for all r-boundaried graphs G2 and andintegers k′, (H ⊕G2, k

′) ∈ Π ⇔ (G[X]⊕G2, k′ + c) ∈ Π.

For each H ∈ C we can check whether H ≡Π G[X] usingthis condition and the pre-computed table, and if H ≡Π

G[X], find the constant c.After we have found a H ∈ S and the corresponding

constant c, such that H ≡Π G[X], we make G∗ from G byreplacing the extended r-protrusion X with H . Also, we setk∗ = k − c. Since |X| > c and H has at most c vertices,|V ∗| < |V |. By the choice of H and c, (G∗, k∗) ∈ Π if andonly if (G, k) ∈ Π. This concludes the proof.

4. DECOMPOSITION THEOREMS

Definition 8. We say that a graph G = (V,E) is (α, β, γ)-structured around S if |S| ≤ α and V can be parti-tioned into S,C1, C2, . . . , Cγ such that NG(Ci) ⊆ S,and max{|NG(Ci)|, tw(G[Ci ∪ NG(Ci)])} ≤ β, for everyi = 1, . . . , γ.

Lemma 8. [�] Let G = (V,E) be a graph, embedded in asurface Σ of Euler genus g and let S ⊆ V , |S| ≥ 3 suchthat Rr

G(S) = V for some r ≥ 0. Then there exists a set S′

such that S ⊆ S′ and G is (h(r, g)·|S|, h(r, g), h(r, g)·|S|)-structured around S′, where h(r, g) = O(rg).

5. KERNELS

In this section we prove Theorems 1 and 2. We say thatan instance (G′, k′) of a problem Π is reduced with respectto a set Q of reduction rules if none of the reduction rulesin Q can be applied to (G′, k′).

Proof of Theorem 1: We first give a proof for the casewhen Π is compact and Πα is an annotated p-MIN-CMSOproblem. A proof for the case when Π is compact and Πα isan annotated p-EQ/MAX-CMSO problem is identical. SinceΠ is compact there exists an integer r such that for all (G =(V,E), k) ∈ Πα there is an embedding of G into a surfaceof genus at most g, and a set S ⊆ V such that Rr

G(S) = Vand |S| ≤ r ·k. We show that for any reduced YES instance(G′ = (V ′, E′), k) with respect to the reduction rule givenby Lemma 4, we have |V ′| = O(k2).

Since (G′ = (V ′, E′), k) ∈ Πα and Π is compact,there exists an embedding of G′ into a surface of genusat most g and a set S ⊆ V ′ such that Rr

G′(S) = V ′.Hence by applying Lemma 8 we obtain a set S′ such

that G′ is (α, β, γ)-structured around S′, where α, γ =O(rg|S|) and β = O(rg). This implies that V ′ can bepartitioned into S′, C1, C2, . . . , Cγ such that NG′(Ci) ⊆ S′,|NG′(Ci)| ≤ β and tw(G′[Ci ∪ NG(Ci)]) ≤ β for everyi ≤ γ. Observe that each Ci is a β-protrusion in G′ and|Ci ∪ NG(Ci)| ≤ ck, where c is a constant of Lemma 4,otherwise we could have applied Lemma 4. This implies that|V ′| ≤ |S′| + ∑γ

i=1 |Ci| = O (rg|S| + ∑γi=1 ck) = O(k2),

for some fixed g and r. Here constants hidden in big-Ohdepend only on r and g.

So given an input (G, k), if the size of the reduced graphis more than c∗k2 for some constant c∗ then we return NOelse we have G′ as the desired annotated kernel for G.

Now we give a proof for the case when Π is compactand Πα is an annotated p-MAX-CMSO problem. A prooffor the case when Π is compact and Πα is an annotatedp-MIN/EQ-CMSO problem is similar. Towards this end, weclaim that for all (G, k) ∈ Π

α, the equivalent instance (G′ =

(V ′, E′), k), reduced with respect to the reduction rule givenby Lemma 6, has |V ′| = O(k2). The proof for the claim isidentical to the one we gave above to bound all the YESinstance for an annotated compact p-MIN-CMSO problem.So given an input (G, k), if the equivalent instance (G′ =(V ′, E′), k), reduced with respect to the reduction rule givenby Lemma 6, is more than c∗k2 for some constant c∗ thenwe return YES else we have G′ as the desired annotatedkernel for G. The reason we return YES is that if (G, k)would have a NO instance then the size of |V ′| ≤ c∗k2 as(G, k) ∈ Π

α.

The proof of Corollary 1 follows by applying Theorem 1together with polynomial time many to one reduction fromΠα to Π. The idea of the proof of Theorem 2 is that if aproblem has finite integer index, then an instance reducedwith respect to the reduction rule given by Lemma 7 does nothave a large protrusion, which in turn implies compactness.

Lemma 9. [�] Let G = (V,E) be a graph embedded intoa surface of genus at most g and S ⊆ V such that tw(G \Rr

G(S)) ≤ r and all r-protrusions of G have size at mostm. Then there exists a S′ ⊆ V such that |S′| ≤ |S| + g ·(m + 2r + 1) and V = Rm+3r+1

G (S′).

The proof of Theorem 2 follows along the lines ofTheorem 1. We first reduce the input with respect to thereduction rule given by Lemma 7 and then apply Lemma 9together with Lemma 8 to argue that the reduced instancehas size linear in k.

6. IMPLICATIONS OF OUR RESULTS

In this section we mention a few parameterized problemsfor which we can obtain either polynomial or linear kernelusing either Theorem 1 or Theorem 2. Various other prob-lems for which we can obtain either polynomial or linearkernels using our results are mentioned in Section 8.

635

6.1. A Sufficient Condition for Finite Integer Index

We first give a sufficient condition which implies thata large class of p-MIN/MAX-CMSO problems has finiteinteger index. We prove it here for vertex versions of p-MIN/MAX-CMSO problems that is if Π is a p-MIN/MAX-CMSO problem then PΠ is a property of vertex sets. Theedge version can be dealt in a similar manner.

Let Π be a p-MIN-CMSO problem and Ft be the set ofpairs (G = (V,E), S) where G is a t-boundaried graph andS ⊆ V . For a t-boundaried graph G = (V,E) we definethe function ζG : Ft → N ∪ {∞} as follows. For a pair(G′ = (V ′, E′), S′) ∈ Ft, if there is no set S ⊆ V (S ⊆ E)such that PΠ(G⊕G′, S∪S′) holds, then ζG((G′, S′)) = ∞.Otherwise ζG((G′, S′)) is the size of the smallest S ⊆ V(S ⊆ E) such that PΠ(G ⊕ G′, S ∪ S′) holds.

Definition 9. A p-MIN-CMSO problem Π is said to bestrongly monotone if there exists a function f : N → N

such that the following condition is satisfied. For every t-boundaried graph G = (V,E), there is a subset S ⊆ Vsuch that for every (G′ = (V ′, E′), S′) ∈ Ft such thatζG((G′, S′)) is finite, PΠ(G ⊕ G′, S ∪ S′) holds and |S| ≤ζG((G′, S′)) + f(t).

Lemma 10. [�] Every strongly monotone p-MIN/MAX-CMSO problem has finite integer index.

For the definition of strong monotonicity of p-MAX-CMSO problems and for the proof of Lemma 10 we referto the extended version of the paper.

6.2. Minor Covering and Packing

Let H be a finite set of connected planar graphs. InVERTEX-H-COVERING, given a graph G = (V,E), anda positive integer k, the objective is to find a subset S ⊆ Vsuch that |S| ≤ k and G[V \S] does not contain any of thegraphs in H as a minor. In VERTEX-H-PACKING, given agraph G = (V,E), and a positive integer k, the objective isto find k vertex disjoint subgraphs G1, . . . , Gk of G suchthat each of them contain some graph in H as a minor.

Lemma 11. [�] Let H be a finite set of connected pla-nar graphs and let Π1 denote VERTEX-H-PACKING. ThenVERTEX-H-COVERING has finite integer index and is quasi-compact and VERTEX-H-PACKING has finite integer indexand Π1 is quasi-compact.

VERTEX-H-COVERING contains various problems as aspecial case, for example: (a) FEEDBACK VERTEX SET bytaking H = {�} where � is a cycle of length at most3; deleting at most k vertices to obtain a graph of fixedtreewidth t; deleting at most k vertices to obtain a graph intoa graph class M, where M is characterized by a finite set ofminors. Similarly VERTEX-H-PACKING contains problemlike CYCLE PACKING as a special case.

6.3. Subgraph Covering and Packing

Let S be a finite set of connected graphs. In VERTEX

(EDGE)-S-COVERING, given a graph G = (V,E), and apositive integer k, the objective is to find a subset S ⊆ V(S ⊆ E) such that |S| ≤ k and G[V \ S] does not containany of the graphs in S as a subgraph. In VERTEX (EDGE)-S-PACKING, given a graph G, and a positive integer k,the objective is to find k vertex (edge) disjoint subgraphsG1, . . . , Gk of G such that each of them is isomorphic to agraph in S.

We can not show that VERTEX/EDGE-S-COVERING orVERTEX/EDGE-S-PACKING or there no instances are com-pact unless we remove all the edges and the vertices thatare not part of any subgraph isomorphic to any graph inS. So given an input we first remove all the redundantvertices and edges in polynomial time, using an algorithmof Eppstein [20].

Lemma 12. [�] Let S be a finite set of connected graphsand Π1 and Π2 correspond to VERTEX-S-PACKING andEDGE-S-PACKING respectively. Then the following hold:(a) VERTEX-S-COVERING has finite integer index and iscompact; (b) VERTEX-S-PACKING has finite integer indexand Π1 is compact; (c) EDGE-S-COVERING is p-MIN-CMSO problem and is compact; and (d) EDGE-S-PACKING

is p-MAX-CMSO problem and Π2 is compact.

Our results can be applied to variety of domination andcovering problems and even on certain problems on directedgraphs. We refer to the extended version of the paper for anextensive details.

Finally by applying Theorem 2 together with Lemmata 10,11 and 12, we get the following corollary.

Corollary 2. For g ≥ 0, FEEDBACK VERTEX SET,EDGE DOMINATING SET, VERTEX COVER, DOMINAT-ING SET, r-DOMINATING SET, q-THRESHOLD DOMINAT-ING SET, CONNECTED DOMINATING SET, DIRECTED

DOMINATION, r-SCATTERED SET, CONNECTED VER-TEX COVER, MINIMUM-VERTEX FEEDBACK EDGE SET,MINIMUM MAXIMAL MATCHING, EFFICIENT DOMINAT-ING SET, INDEPENDENT SET, INDUCED d-DEGREE SUB-GRAPH, MIN LEAF SPANNING TREE, MAXIMUM FULL-DEGREE SPANNING TREE, CYCLE PACKING, INDUCED

MATCHING, TRIANGLE PACKING, CYCLE DOMINATION,VERTEX-H-PACKING, VERTEX-H-COVERING, VERTEX-S-COVERING and VERTEX-S-PACKING admit a linearkernel on graph of genus at most g.

Corollary 2 unifies and generalizes results presentedin [2], [3], [8], [9], [13], [21], [24], [25], [26], [29], [30].By applying Theorem 1, Corollary 1, and Lemmata 11, and12, we get the following corollary for problems which arenot finite integer index.

Corollary 3. For g ≥ 0, INDEPENDENT DOMINATING

636

SET, INDEPENDENT DIRECTED DOMINATION, MINIMUM

LEAF OUT-BRANCHING, ODD SET, EDGE-S-COVERING

and EDGE-S-PACKING admit a polynomial kernels ongraphs of genus at most g.

7. OPEN PROBLEMS AND FURTHER DIRECTIONS

This paper gives for the first time meta theorems, ascalled by Grohe and Kreutzer [22], [27], where logicaland combinatorial conditions on problems lead to kernelsof polynomial or linear sizes. Our results are very generalin the sense that they can be applied to a large numberof combinatorial problems on graphs on fixed surfaces andgeneralize a large collection of published results. Still, thereare several directions in which our results could possibly beextended. We conclude the paper in this section with somenew problems and further research directions opened by ourresults.

Further extensions: The first natural question for furtherresearch is if our logical and combinatorial conditions canbe extended to larger classes of problems. The condition thatproblems should satisfy some kind of compactness or quasi-compactness cannot be omitted. For instance, even thoughthe problem of finding a path of length k is expressiblein first order logic, it does not admit a polynomial kernelon planar graphs, unless polynomial hierarchy collapses tothe third level, a collapse which is deemed unlikely [7].Two interesting questions for further research are: Can wereplace the compactness condition with the weaker notionof quasi-compactness in Theorem 1? Do all quasi-compactCMSO problems admit a linear kernel on graphs of boundedgenus? It is very natural to ask whether our results can beextended to more general classes of graphs. The most naturalcandidates for such extensions are graphs of bounded local-treewidth and graphs excluding some fixed graph as a minor.

Practical considerations: Our meta theorems provide sim-ple criteria to decide whether a problem admits a poly-nomial or linear kernel. Our proofs are constructive andessentially provide meta-algorithms that construct kernelsfor problems in an automated way. Of course, it is expectedthat for concrete problems, tailor-made kernels will havemuch smaller constant factors, than what would follow froma direct application of our results. However, our approachmight be useful for computer aided design of kernelizationalgorithms: with a Myhill-Nerode approach, a computerprogram can output a set of rules that transform each regionto a minimum size representative and estimate the obtainedkernel size. This seems an interesting and far from trivialalgorithm-engineering problem. In general, finding linearkernels with small constant factors for concrete problems onplanar graphs or graphs with small genus remains a worthytopic of further research.

Acknowledgments.

We thank Jiong Guo for sending us the complete versionof [24] and referees of FOCS 2009 for several valuablecomments.

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8. PROBLEM COMPENDIUM

We refer to problem compendium given in [19] orthe compendium of parameterized problems provided athttp://bravo.ce.uniroma2.it/home/cesati/research/compendium/ for the definitions of problemsgiven below.

8.1. Problems Having Finite Integer Index and Quasi-Compactness Property – Linear Kernels

DOMINATING SET, r-DOMINATING SET, q-THRESHOLD

DOMINATING SET, EFFICIENT DOMINATING SET,VERTEX COVER, FEEDBACK VERTEX SET, CONNECTED

DOMINATING SET, CONNECTED r-DOMINATING SET,CONNECTED VERTEX COVER, CYCLE DOMINATION,CLIQUE-TRANSVERSAL, ALMOST-OUTERPLANAR,MINIMUM-VERTEX FEEDBACK EDGE SET, EDGE

DOMINATING SET, MINIMUM MAXIMAL MATCHING,VERTEX-H-COVERING, VERTEX-S-COVERING.

8.2. Problem Having Finite Integer Index and No Instanceshaving Quasi-Compactness Property – Linear Kernels

INDEPENDENT SET, INDUCED d-DEGREE SUBGRAPH,r-SCATTERED SET, MIN LEAF SPANNING TREE, INDUCED

MATCHING, TRIANGLE PACKING, CYCLE PACKING,MAXIMUM FULL-DEGREE SPANNING TREE, VERTEX-H-PACKING, VERTEX-S-PACKING.

8.3. Problems that are not covered by 8.1 and 8.2 andare p-MIN/EQ/MAX-CMSO problems having CompactnessProperty – Polynomial Kernels

INDEPENDENT DOMINATING SET, INDEPENDENT

DIRECTED DOMINATION, MINIMUM LEAF OUT-BRANCHING, EDGE-S-COVERING, EDGE-S-PACKING,ODD SET.

8.4. Problems Having Finite Integer Index but neither theproblem nor its no instances satisfy Quasi-CompactnessProperty

MINIMUM PARTITION INTO CLIQUES, HAMILTONIN

PATH COMPLETION.

8.5. Problems that do not have Finite Integer Index

LONGEST PATH, LONGEST CYCLE, MAXIMUM CUT,MINIMUM COVERING BY CLIQUES, INDEPENDENT DOM-INATING SET, MINIMUM LEAF OUT-BRANCHING, ODD

SET.

The following lemma is shown in [18].

Lemma 13. [18] LONGEST PATH, LONGEST CYCLE,MAXIMUM CUT and MINIMUM COVERING BY CLIQUES

are not of finite integer index.

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