Kernelization of Maximum Minimal Vertex Cover
KKernelization of Maximum Minimal Vertex Cover
Júlio Araújo
Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, [email protected]
Marin Bougeret
LIRMM, Université de Montpellier, CNRS, Montpellier, [email protected]
Victor A. Campos
Departamento de Computação, Universidade Federal do Ceará, Fortaleza, [email protected]
Ignasi Sau
LIRMM, Université de Montpellier, CNRS, Montpellier, [email protected]
Abstract
In the
Maximum Minimal Vertex Cover ( MMVC ) problem, we are given a graph G and a positiveinteger k , and the objective is to decide whether G contains a minimal vertex cover of size at least k .This problem has been considered in several articles in the last years. We focus on its kernelization,which had been almost unexplored so far. We prove that MMVC does not admit polynomialkernels parameterized by the size of a minimum vertex cover, even on bipartite graphs, unless NP ⊆ coNP / poly . Motivated by a question of Boria et al. [Discret. Appl. Math. 2015] about theexistence of subquadratic kernels for MMVC parameterized by k , we rule out their existence unless P = NP , if we restrict the kernelization algorithms to apply only a type of natural reduction rulesthat we call large optimal preserving rules . In particular, these rules contain the typical reductionrules to obtain linear kernels for Vertex Cover . On the positive side, we provide subquadratickernels on H -free graphs for several graphs H , such as the bull, the paw, or the complete graphs, bymaking use of the Erdős-Hajnal property in order to find an appropriate decomposition. Mathematics of computing → Graph algorithms.
Keywords and phrases maximum minimal vertex cover, parameterized complexity, kernelization,Erdős-Hajnal property, induced subgraphs, lower bound.
Funding
Júlio Araújo : CNPq-Pq 304478/2018-0, CAPES-PrInt 88887.466468/2019-00 and CAPES-STIC-AmSud 88881.569474/2020-01.
Victor A. Campos : FUNCAP - PNE-011200061.01.00/16.
Ignasi Sau : DEMOGRAPH (ANR-16-CE40-0028), ESIGMA (ANR-17-CE23-0010) and ELIT (ANR-20-CE48-0008-01). A vertex cover in a graph G is a subset of vertices containing at least one endpoint ofevery edge. In the associated optimization problem, called Minimum Vertex Cover , theobjective is to find, given an input graph G , a vertex cover in G of minimum size. Thisproblem has been one of the leitmotifs of the area of parameterized complexity [17,20], servingas a test bed for many of the most fundamental techniques. An instance of a parameterizedproblem is of the form ( x, k ), where x is the total input (typically, a graph) and k is a positiveinteger called the parameter . The crucial notion is that of fixed-parameter tractable algorithm, FPT for short, which is an algorithm deciding whether ( x, k ) is a positive instance in time f ( k ) · | x | O (1) , where f is a computable function depending only on k . In the parameterized Vertex Cover problem, we are given a graph G and an integer parameter k , and theobjective is to decide whether G contains a vertex cover of size at most k . One of the main a r X i v : . [ c s . D S ] F e b Kernelization of Maximum Minimal Vertex Cover fields within parameterized complexity is kernelization [24], where the objective is to decidewhether an instance ( x, k ) of a parameterized problem can be transformed in polynomialtime into an equivalent instance ( x , k ) whose total size is bounded by a function of k ; thereduced instance is called a kernel , and finding kernels of small size, typically polynomial oreven linear in k in the best case, is one of the most active areas of parameterized complexity.There are several techniques for obtaining linear kernels for the Vertex Cover problem [24].Considering the “max-min” version of minimization problems, that is, maximizing thesize of a minimal solution of the corresponding problem, is a natural approach that has beenapplied to several problems such as
Dominating Set [5, 21] (whose “max-min” version iscalled
Upper Domination ) or
Hitting Set [3, 18]. In this article we are interested inthe “max-min” version of
Minimum Vertex Cover , called
Maximum Minimal VertexCover , or just
MMVC for short. Surprisingly, there are relatively few articles in theliterature dealing with the
MMVC problem.
Previous work . In his habilitation, Fernau [23] presented
FPT algorithms for
MMVC aswell as some results about its kernelization parameterized by the solution size k . It is easy tonote, as observed in [23], that the problem admits a kernel with at most k vertices: if somevertex has degree at least k , we can safely answer “ yes ” (cf. Lemma 2 for a proof); otherwise,the maximum degree is at most k −
1, and it follows that every positive instance withoutisolated vertices (which may be safely removed) must have at most k vertices. Fernau [23]presented a kernel with at most 4 k vertices for MMVC restricted to planar instances usingthe algorithmic version of the Four Color Theorem [35], and claimed in [23, Corollary 4.25] akernel with at most 2 k vertices on general graphs using spanning trees. Unfortunately, thislatter kernelization algorithm is incorrect, as we discuss at the end of Section 4.Boria et al. [10] initiated a study of the complexity of MMVC and presented a number ofresults, in particular a polynomial-time approximation algorithm with ratio n / on n -vertexgraphs, and showed that, unless P = NP , no polynomial-time approximation algorithm withratio n / − ε exists for any ε >
0. They also presented
FPT algorithms for
MMVC for severalchoices of the parameters such as the treewidth, the size of a maximum matching, or the sizeof a minimum vertex cover of the input graph. The authors asked explicitly whether kernelsof size o ( k ) exist for MMVC parameterized by k .Zehavi [37] presented tight FPT algorithms, under the Strong Exponential Time Hypo-thesis, for
MMVC and its weighted version parameterized by the size of a minimum vertexcover. Recently, Bonnet and Paschos [8] and Bonnet et al. [7] considered the inapproximabilityof
MMVC in subexponential time.Note that the
MMVC problem is the dual of the well-studied
Minimum IndependentDominating Set problem (to see this, note that the complement of any minimal vertexcover is an independent dominating set), which has applications in wireless and ad-hocnetworks [32]. We refer to the survey of Goddard and Henning [27].
Our results and techniques . In this article we focus on the kernelization of the
MMVC problem, which has been almost unexplored so far in the literature. We first show (Theorem 3)that
MMVC , parameterized by the size of a minimum vertex cover or of a maximum matchingof the input graph, does not admit a polynomial kernel unless NP ⊆ coNP / poly , even restrictedto bipartite graphs. This result complements the FPT algorithms for
MMVC under theseparameterizations given by Boria et al. [10] and Zehavi [37], and shows that, in what concernsthe existence of polynomial kernels for
MMVC , the most natural structural parameterssmaller than the solution size are not large enough to yield polynomial kernels (note thatthe treewidth of any graph is at most one more than its vertex cover number, hence ourresult rules out the existence of polynomial kernels for
MMVC parameterized by treewidth úlio Araújo, Marin Bougeret, Victor A. Campos, and Ignasi Sau 3 as well). The proof consists of a polynomial parameter transformation from
MonotoneSat parameterized by the number of variables. In particular, our reduction yields also the NP -hardness of MMVC on bipartite graphs, which provides an alternative proof to the oneof Boliac and Lozin [6] via the NP -hardness of Minimum Independent Dominating Set on bipartite graphs.Motivated by the question of Boria et al. [10] about the existence of subquadratic kernelsfor
MMVC parameterized by the solution size k , we rule out their existence unless P = NP (Theorem 8), if we restrict the kernelization algorithms to apply only a type of naturalreduction rules that we call large optimal preserving rule , or lop -rule for short. Informally(cf. Section 4 for the details), lop -rules either answer “ yes ” or identify a subset of vertices ofthe input graph that can be safely taken into the solution. Note that most of the classicalreduction rules for Vertex Cover , such as the high-degree rule, the crown decompositionrule, or the Nemhauser-Trotter rule [24], are lop -rules. Thus, while our result does notcompletely rule out the existence of subquadratic kernels for
MMVC , it tells that, if such akernel exists, it should consist of “non-standard” reduction rules. In order to prove Theorem 8,we show that the existence of a subquadratic kernel for
MMVC using lop -rules would implyan O ( n − ε )-approximation for MMVC , which is not possible unless P = NP [10].Given the above negative result on general graphs, we identify graph classes where MMVC is still NP -hard and admits a subquadratic kernel. In particular, we deal withgraph classes defined by excluding an induced subgraph H that satisfies the Erdős-Hajnalproperty [22], that is, for which there exists a constant δ > H -free graphwith n vertices contains either a clique or an independent set of size n δ . In particular, wepresent a kernel for MMVC with O ( k / ) vertices on bull-free graphs (Theorem 10), with O ( k t − t − ) vertices on K t -free graphs graphs for every t ≥ O ( k / )vertices on paw-free graphs (Theorem 15).Our strategy to obtain these subquadratic kernels on H -free graphs is as follows. Bythe high-degree rule mentioned above, given an instance ( G, k ), we may assume that themaximum degree of G is at most k −
1. We find greedily a minimal vertex cover X of G . If | X | ≥ k we are done, so we may assume that | X | ≤ k −
1, hence the goal is to bound thesize of S := V ( G ) \ X . Using that G [ X ] is also H -free, the Erdős-Hajnal property implies(Lemma 9) that X can be partitioned in polynomial time into a sublinear (in k ) numberof independent sets and cliques. Since S is an independent set and we may assume that G has no isolated vertices, in order to bound | S | by a subquadratic function of k , it isenough to show that, for each of the sublinearly many cliques or independent sets Y thatpartition X , its neighborhood in S has size O ( k ). This is easy if Y is an independent set: if | N S ( Y ) | ≥ k we can conclude that ( G, k ) is a yes -instance (Lemma 2), so we may assumethat | N S ( Y ) | ≤ k −
1. The case where Y is a clique is more interesting, and we need ad-hocarguments depending on each particular excluded induced subgraph H .Finally, we present several positive results for MMVC restricted to other particular graphclasses, such as K ,t -free graphs (Lemma 16), graph classes with bounded chromatic number(Lemma 17), or graphs classes with bounded cliquewidth (Observation 18). Organization . In Section 2 we provide some basic preliminaries about graphs and paramet-erized complexity. In Section 3 we rule out the existence of polynomial kernels for
MMVC parameterized by the size of a minimum vertex cover or a maximum matching. In Section 4we show that if there exists a subquadratic kernel for
MMVC parameterized by k , it shoulduse “non-standard” reduction rules, and we discuss the flaw in the linear kernel claimed byFernau [23]. Section 5 is devoted to the subquadratic kernels on particular graph classes. Weconclude the article in Section 6 with a discussion and some directions for further research. Kernelization of Maximum Minimal Vertex Cover
Graphs and functions.
We use standard graph-theoretic notation, and we refer the readerto [19] for any undefined notation. For an integer p ≥
1, we let [ p ] be the set containing allintegers i with 1 ≤ i ≤ p . We use ] to denote the disjoint union. We will only consider finiteundirected graphs without loops nor multiple edges, and we denote an edge between twovertices u and v by { u, v } . A subgraph H of a graph G is induced if H can be obtained from G by deleting a set of vertices D = V ( G ) \ S , and we denote H = G [ S ]. A graph G is H -free if it does not contain any induced subgraph isomorphic to H . If H is a collection of graphs,a graph G is H -free is it is H -free for every H ∈ H . For a graph G and a set S ⊆ V ( G ), weuse the notation G \ S = G [ V ( G ) \ S ], and for a vertex v ∈ V ( G ), we abbreviate G \ { v } as G \ v . A vertex v is complete to a set S ⊆ V ( G ) if v is adjacent to every vertex in S .The open (resp. closed ) neighborhood of a vertex v is denoted by N ( v ) (resp. N [ v ]),whenever the graph G is clear from the context. For vertex sets X, Y ⊆ V ( G ), we define N [ X ] = S v ∈ X N [ v ], N ( X ) = N [ X ] \ X , N Y [ X ] = N [ X ] ∩ Y , and N Y ( X ) = N Y [ X ] \ X .The degree of a vertex v in a graph G is defined as | N ( v ) | , and we denote it by deg G ( v ), orjust deg ( v ) of the graph is clear from the context. For an integer t ≥
1, we denote by P t (resp. I t , K t ) the path (resp. edgeless graph, complete graph) on t vertices. For two integers a, b ≥
1, we denote by K a,b the bipartite graph with parts of sizes a and b .A clique (resp. independent set ) of a graph G is a set of vertices that are pairwise adjacent(resp. not adjacent). A graph property is hereditary if whenever it holds for a graph G , itholds for all its induced subgraphs as well. Note that the properties of being an edgeless or acomplete graph or an independent set are hereditary. We denote by ∆( G ) (resp. ω ( G ) themaximum vertex degree (resp. clique size) of a graph G .A vertex cover of a graph G is a set of vertices containing at least one endpoint of everyedge, and it is minimal if no proper subset of it is a vertex cover. The problem we study inthe paper is formally stated as follows. We state it as a decision problem, since most of ourresults consider its parameterization by the solution size k . Maximum Minimal Vertex Cover (MMVC)
Input:
A graph G and a positive integer k . Question:
Does G contain a minimal vertex cover of size at least k ?The following observation has been already used in previous work [10, 37]. (cid:73) Observation 1.
Let G be a graph. A set X ⊆ V ( G ) is a minimal vertex cover of G if andonly if X is a vertex cover of G and, for every vertex v ∈ X , N [ v ] (cid:42) X . The next lemma provides a useful way to conclude that we are dealing with a yes -instancein our kernelization algorithms. (cid:73)
Lemma 2.
Let G be a graph and let S ⊆ V ( G ) be an independent set. There exists aminimal vertex cover of G containing N ( S ) . Proof.
First note that, since S is an independent set, the set V ( G ) \ S is a vertex cover of G . Hence, there exists a minimal vertex cover X of G such that X ⊆ V ( G ) \ S . We claimthat N ( S ) ⊆ X . Suppose for the sake of contradiction that there exists a vertex v ∈ N ( S )such that v / ∈ X . Since v has a neighbor u in S and S ∩ X = ∅ , the edge { u, v } would notbe covered by X , and the lemma follows. (cid:74) Note that, in particular, Lemma 2 implies that if (
G, k ) is an instance of the
MaximumMinimal Vertex Cover problem and v ∈ V ( G ) is a vertex of degree at least k , then we úlio Araújo, Marin Bougeret, Victor A. Campos, and Ignasi Sau 5 can conclude that ( G, k ) is a yes -instance. This will allow us to assume, in our kernelizationalgorithms, that ∆( G ) ≤ k − Parameterized complexity.
We refer the reader to [17, 20] for basic background onparameterized complexity, and we recall here only some basic definitions used in this article.A parameterized problem is a language L ⊆ Σ ∗ × N . For an instance I = ( x, k ) ∈ Σ ∗ × N , k is called the parameter .A parameterized problem is fixed-parameter tractable ( FPT ) if there exists an algorithm A , a computable function f , and a constant c such that given an instance I = ( x, k ), A (called an FPT algorithm ) correctly decides whether I ∈ L in time bounded by f ( k ) · | I | c .For instance, the Vertex Cover problem parameterized by the size of the solution is
FPT .For an instance ( x, k ) of a parameterized problem Q , a kernelization algorithm is analgorithm A that, in polynomial time, generates from ( x, k ) an equivalent instance ( x , k ) of Q such that | x | + k ≤ f ( k ), for some computable function f : N → N , where | x | denotesthe size of x . If f ( k ) is bounded from above by a polynomial of the parameter, we say that Q admits a polynomial kernel . In particular, if f ( k ) is bounded by a linear (resp. quadratic)function, then we say that Q admits a linear (resp. quadratic ) kernel.A polynomial parameter transformation , abbreviated as PPT , is an algorithm that, givenan instance ( x, k ) of a parameterized problem A , runs in time f ( k ) · | x | O (1) and outputsan instance ( x , k ) of a parameterized problem B such that k is bounded from above by apolynomial on k and ( x, k ) is positive if and only if ( x , k ) is positive. If a parameterizedproblem A does not admit a polynomial kernel unless NP ⊆ coNP / poly and there exists a PPT from A to a parameterized problem B , then B does not admit a polynomial kernelunless NP ⊆ coNP / poly either [17]. In this section we rule out, assuming that NP (cid:42) coNP / poly , the existence of polynomialkernels for MMVC for parameters smaller than the solution size k . As mentioned in theintroduction, the reduction given in Theorem 3 also provides an alternative proof of the NP -completeness of MMVC on bipartite graphs, which also follows from [6]. We notethat the existing NP -hardness reductions for MMVC , such as [6], do not seem to be easilymodifiable so to yield the non-existence of polynomial kernels, which we prove in Theorem 3. (cid:73)
Theorem 3.
The
Maximum Minimal Vertex Cover problem parameterized by the sizeof a minimum vertex cover or of a maximum matching of the input graph does not admit apolynomial kernel unless NP ⊆ coNP / poly , even restricted to bipartite graphs. Proof.
We present a
PPT from
Monotone Sat parameterized by the number of variables,which is also an NP -completeness reduction. The Monotone Sat problem is the restrictionof the
Sat problem to formulas in which the literals in each clause are either all positiveor all negative. This problem is well-known to be NP -complete [26], and it is easy to seethat, when parameterized by the number of variables, it does not admit a polynomialkernel unless NP ⊆ coNP / poly . Indeed, Fortnow and Santhanam [25] proved that the Sat problem parameterized by the number of variables does not admit a polynomial kernel unless NP ⊆ coNP / poly , and the classical reduction from Sat to Monotone Sat that replaces eachvariable with a “positive” and a “negative” variable and adds extra clauses appropriately [26]is in fact a
PPT when the parameter is the number of variables.Given an instance φ of Monotone Sat , where the formula φ contains n variables and m clauses, we construct in polynomial time an instance ( G, k ) of
Maximum Minimal Vertex
Kernelization of Maximum Minimal Vertex Cover
Cover as follows. For each variable x i of φ , i ∈ [ n ], we add to G four vertices ‘ i , x + i , x − i , r i and three edges { ‘ i , x + i } , { x + i , x − i } , { x − i , r i } , hence inducing a P . We call the vertex x + i (resp. x − i ) a positive (resp. a negative ) vertex of G . For each clause C j of φ , j ∈ [ m ], weadd to G a vertex c j , which we connect to the positive or negative vertices corresponding tothe literals contained in C j . This concludes the construction of G , which is illustrated inFigure 1(a). Note that, since φ is a monotone formula, G is a bipartite graph. Note also thatthe set of vertices { x + i , x − i | i ∈ [ n ] } is a minimum vertex cover of G of size 2 n , and that theset of edges {{ ‘ i , x + i } , { x − i , r i } | i ∈ [ n ] } is a maximum matching of G of size 2 n . We claimthat φ is satisfiable if and only if G contains a minimal vertex cover of size k := 2 n + m . x +1 x − (cid:96) r x + n x − n (cid:96) n r n c j c j (cid:48) positiveclauses negativeclauses (a) x +1 x − (cid:96) r x + n x − n (cid:96) n r n c j c j (cid:48) positiveclauses negativeclauses (b) Figure 1 (a) Illustration of the graph G built from the formula φ in the proof of Theorem 3.(b) A minimal vertex cover X of G is shown with larger red vertices. Suppose first that φ is satisfiable, and let σ be an assignment of the variables that satisfiesall the clauses in φ . We proceed to define a minimal vertex cover X of G of size k . First,add to X all the clause vertices { c j | j ∈ [ m ] } . For every i ∈ [ n ], if σ ( x i ) = true (resp. σ ( x i ) = false ), add to X vertices x − i and ‘ i (resp. x + i and r i ). See Figure 1(b) for anillustration, where the set X is shown with larger red vertices. Clearly, X is a vertex coverof G . To see that it is minimal, by Observation 1 it is enough to verify that, for every vertex v ∈ X , N [ v ] (cid:42) X . This condition holds easily for all vertices in X that are in the P ’s,since for each P its vertices in X are not adjacent. Let c j be a clause vertex. Since σ isa satisfying assignment of the variables, there exists a variable x i such that if σ ( x i ) = true (resp. σ ( x i ) = false ) then x i ∈ C j (resp. ¯ x i ∈ C j ). By definition of X , if σ ( x i ) = true (resp. σ ( x i ) = false ) then x + i / ∈ X (resp. x − i / ∈ X ), and by construction of G we have that x + i ∈ N ( c j ) (resp. x − i ∈ N ( c j )), so in both cases N [ c j ] (cid:42) X .Conversely, suppose that G contains a minimal vertex cover X of size k , and we proceedto define a variable assignment σ as follows. For i ∈ [ n ], as { x + i , x − i } ∈ E ( G ) we have that X contains one or two vertices in the set { x + i , x − i } . If x + i / ∈ X (resp. x − i / ∈ X ) we set σ ( x i ) = true (resp. σ ( x i ) = false ), and if both x + i and x − i belong to X we set σ ( x i ) to true orto false arbitrarily. We claim that σ satisfies all the clauses in φ . For i ∈ [ n ], let P i be the P of G induced by the vertices ‘ i , x + i , x − i , r i . Since X is a vertex cover, clearly | X ∩ V ( P i ) | ≥ | X ∩ V ( P i ) | = 2. Indeed, if | X ∩ V ( P i ) | ≥
3, then { ‘ i , x + i } ⊆ X or { x − i , r i } ⊆ X (or both). But then N [ ‘ i ] ⊆ X or N [ r i ] ⊆ X (or both), contradicting Observation 1. Thus, | X ∩ V ( P i ) | = 2, which implies that | X ∩ S i ∈ [ n ] V ( P i ) | = 2 n , hence necessarily X containsthe whole set { c j | j ∈ [ m ] } of clause vertices. Consider an arbitrary clause vertex c j . Since X is minimal and c j ∈ X , by Observation 1 there exists a neighbor of c j in G that is not in X , and by definition of σ it follows that the literal corresponding to that neighbor of c j úlio Araújo, Marin Bougeret, Victor A. Campos, and Ignasi Sau 7 satisfies clause C j . Thus, σ is a satisfying assignment and the proof is complete.Finally, note that the above reduction is also an NP -completeness reduction from Mono-tone Sat to Maximum Minimal Vertex Cover on bipartite graphs. (cid:74) lop -rules
In this section we show that, if a subquadratic kernel for
MMVC parameterized by k exists,then, assuming that P = NP , it must use reduction rules that are somehow non-standard, inthe sense that they cannot either answer “ yes ”, or identify a subset of vertices that can besafely assumed to be part of the solution, as most reduction rules do. We do so by presentingsimple self-contained ad-hoc arguments relating the kernel size to the approximability of MMVC , without using the existing techniques for providing lower bounds on the sizesof polynomial kernels [13] or the more involved framework of lossy kernelization [33]. Inparticular, the techniques of [13] are designed to provide lower bounds on the coefficient oflinear kernels, and we do not see how they could be adapted so to derive lower bounds onthe degree of polynomial kernels.Let us first provide an informal description of how we prove our result. We show that, ifwe restrict ourselves to kernelization rules for
MMVC that, given an instance (
G, k ), eitheranswer “ yes ”, or identify a non-empty subset of vertices T ⊆ V ( G ) that can be safely takeninto the solution, then we cannot obtain a kernel with O ( k − ε ) vertices. Indeed, supposethat there exists such a subquadratic kernel for some ε >
0. On an input (
G, k ), it eitheranswers “ yes ”, or identifies a non-empty subset T ⊆ V ( G ), then another subset T and so on,until T x , and then it stops, producing an equivalent instance ( G , k ), where k = k − P | T i | .If we denote by mmvc ( G ) the size of a maximum minimal vertex cover of G , it follows that mmvc ( G ) ≤ mmvc ( G ) + P | T i | ≤ mmvc ( G ) + k ≤ | V ( G ) | + k = O ( k − ε ). Thus, we haveobtained a polynomial-time algorithm for MMVC that certifies that either mmvc ( G ) ≥ k or mmvc ( G ) = O ( k − ε ). We prove that this easily yields an O ( n − ε )-approximation for MMVC for some ε >
0, which is not possible unless P = NP by the results of Boria etal. [10]. In what follows me make this intuition precise.In order to define lop -rules, we consider an arbitrary vertex-maximization problem Π suchthat the input is a graph G , the output is a subset S ⊆ V ( G ) satisfying some conditions,and the goal is to maximize | S | . Given a graph G and an integer k , we say that ( G, k ) is a yes -instance of Π if opt Π ( G ) ≥ k , where opt Π ( G ) denotes the maximum size of a solution ofΠ in G . (cid:73) Definition 4. A large optimal preserving reduction rule, or lop -rule for short, for a vertex-maximization problem Π is an algorithm R that, given a graph G and an integer k , computes R ( G, k ) = ( G , k ) such that ∆ := k − k ≥ and k ≥ , if ( G , k ) is a yes -instance of Π , then ( G, k ) is a yes -instance of Π , and for any solution S of G such that | S | ≥ k , there exists a solution S of G such that | S | ≥ | S | − ∆ . Definition 4 can be easily adapted to minimization problems. Item 3 corresponds to a strongerversion of the implication “if (
G, k ) is a yes -instance of Π, then ( G , k ) is a yes -instance of Π”.Indeed, when we consider how this latter implication is generally proved in classical safenessproofs, the following scenarios often occur: (a) Sometimes we prove that for any solution S of G , there exists a solution S of G suchthat | S | ≥ | S | − ∆. Kernelization of Maximum Minimal Vertex Cover (b)
Sometimes we prove that for any solution S of G such that | S | ≥ k , there exists a solution S of G such that | S | ≥ | S | − ∆. (c) Sometimes we prove that if there exists a solution S of G such that | S | ≥ k , then thereexists a solution S of G such that | S | ≥ k − ∆ = k .In Case (a), the rule preserves all optimal solutions, and it implies that opt Π ( G ) ≥ opt Π ( G ) − ∆. In Case (b), the rule preserves only large optimal solutions, and it impliesthat if opt Π ( G ) ≥ k , then opt Π ( G ) ≥ opt Π ( G ) − ∆ (note that if opt Π ( G ) < k , then opt Π ( G )and opt Π ( G ) are not necessarily related). Case (c) corresponds to the weaker and classicalimplication “if ( G, k ) is a yes -instance of Π, then ( G , k ) is a yes -instance of Π”.A typical example of a lop -rule is when we can identify a “dominant” set of vertices thatwe can always take into a solution. Generally, consider a rule that finds a subset T ⊆ V ( G )and a graph G such thatthere exists an optimal solution S ∗ of G such that S ∗ = T ∪ S , where S is a solution of G , andfor any solution S of G , S ∪ T is a solution of G .Then, this rule is a lop -rule, as we even fall into Case (a) described above. Known examplesof lop -rules are the classical reduction rules for Vertex Cover such as the high-degree rule,the crown decomposition rule, or the Nemhauser-Trotter rule [24].The following observation is an immediate consequence of the definition of a lop -rule. (cid:73)
Observation 5. lop -rules can be composed. Formally, consider two lop -rules R and R .Then, the rule R that, given a instance ( G, k ) , returns R ( R ( G, k )) is a lop -rule. Our next objective is to define a lop -kernel, which is, informally speaking, a kernel whichonly uses lop -rules. However, if we only allow a lop -kernel to use lop -rules, this would excludefrom being a lop -kernel many natural kernels that either use lop -rules or conclude that theinput is a yes -instance (by finding a solution of size at least k ). For example, for MMVC , ifwe find a vertex v with deg ( v ) ≥ k , we know by Lemma 2 that N ( v ) can be completed intoa minimal vertex cover, and thus that we are dealing with a yes -instance. However, this doesnot satisfy item 3 in Definition 4, which justifies the choice of the next definition. (cid:73) Definition 6.
Given an instance ( G, k ) of a parameterized problem whose non-parameterizedversion corresponds to a vertex-maximization problem Π , a lop -kernel is a polynomial-timealgorithm that, either determines that ( G, k ) is a yes -instance, or outputs an equivalentinstance ( G , k ) that is obtained from ( G, k ) using only lop -rules. The size of such a kernelis | V ( G ) | . The following definition is inspired by a similar notion introduced by Hochbaum andShmoys [31]. (cid:73)
Definition 7.
Given a function f : N → N , an f -dual-approximation algorithm A is apolynomial-time algorithm that, given an instance ( G, k ) of a parameterized problem whosenon-parameterized version corresponds to a vertex-maximization problem Π ,either determines that ( G, k ) is a yes -instance,or concludes that opt Π ( G ) ≤ f ( k ) . It is well-known that we can obtain a classical approximation algorithm from an f -dual-approximation by performing a binary search. In particular, observe that if f ( k ) = O ( k − ε )for some ε >
0, then an f -dual-approximation algorithm A yields an O ( n − ε )-approximationfor some ε > ε . Indeed, given an n -vertex graph G , using linear search in úlio Araújo, Marin Bougeret, Victor A. Campos, and Ignasi Sau 9 k , we can find the largest k such that A does not return the second possible output , andreturn k . If k ≥ n (1+ ε ) then if we output k we clearly obtain the desired approximationratio, as opt Π ( G ) ≤ n . Otherwise, as opt Π ( G ) = O ( k + 1) − ε = O (( k ) − ε ), it follows that opt Π ( G ) k = O ( k (1 − ε )0 ) = O ( n (1+ ε )(1 − ε ) ) = O ( n (1 − ε ) ) , and we can choose ε = ε / O ( k − ε ) (typically linear) for MMVC , a natural approachwould be to try to adapt some of the rules providing linear kernels for
Vertex Cover , suchas the crown decomposition rule or the Nemhauser-Trotter rule. The adaptation of theserules typically leads to lop -rules, as they identify “dominant” sets of vertices that can bealways taken into a solution. However, as we will see in the next theorem, lop -rules cannotlead to subquadratic kernels for
MMVC . This implies that, in order to obtain a subquadratickernel, we would need to design kernels using at least one rule that only satisfies the propertyin Case (c). (cid:73)
Theorem 8.
Unless P = NP , there cannot exist a lop -kernel for Maximum MinimalVertex Cover with O ( k − ε ) vertices for any positive constant ε > . Proof.
Let us first prove that a lop -kernel A with O ( k − ε ) vertices yields a O ( k − ε )-dual-approximation. Let ( G, k ) be an input instance of A . If A ( G, k ) determines that (
G, k ) isa yes -instance, then we fall into the first case of f -dual-approximation for f ( k ) = O ( k − ε ).Otherwise, let us prove that mmvc ( G ) = O ( k − ε ). As A is a lop -kernel, it only uses lop -rulesto produce its output ( G , k ). As according to Observation 5 lop -rules compose, it followsthat A is equivalent to a single lop -rule R that transforms ( G, k ) into ( G , k ). Let ∆ = k − k .If mmvc ( G ) < k then we are done. Otherwise, let S be an optimal solution of G . As | S | ≥ k , Property 3 of lop -rules implies that mmvc ( G ) ≥ mmvc ( G ) − ∆. This implies that mmvc ( G ) ≤ mmvc ( G ) + ∆ ≤ | V ( G ) | + k = O ( k − ε ) + k = O ( k − ε ).Thus, if A is a lop -kernel with O ( k − ε ) vertices, we can obtain from it an O ( k − ε )-dual-approximation. Now, by the discussion after Definition 7, this O ( k − ε )-dual-approximationyields in turn an O ( n − ε )-approximation algorithm for MMVC , where ε = ε /
2, which isnot possible unless P = NP [10]. (cid:74) To conclude this section, we briefly explain the flaw in the linear kernel for
MMVC claimed by Fernau [23, Corollary 4.25], and that is based on joint unpublished work withDehne, Fellows, Prieto, and Rosamond. The kernelization algorithm is a small modificationof a linear kernel for the
Nonblocker Set problem presented by Ore [34]. A set of vertices S of a graph G is a nonblocker if its complement is a dominating set of G , that is, for every u ∈ S there exists v / ∈ S with { u, v } ∈ E ( G ). In the Nonblocker Set problem, we aregiven a graph G and an integer parameter k , and the goal is to decide whether G containsa nonblocker of size at least k . Suppose for simplicity that G is connected. The idea is toconsider an arbitrary spanning tree T of G , root it arbitrarily at a vertex r , and partition V ( G ) = V ] V such that the vertices in V (resp. V ) are within even (resp. odd) distancefrom r in T . By construction, each of V and V is a nonblocker in G , so if one of them hassize at least k , we can answer “ yes ”, and otherwise | V ( G ) | ≤ k and we are done.Back to MMVC , it is observed in [23, Reduction rule 24] that a simple reduction ruleallows to assume that no connected component of G is a clique (in particular, an isolated We consider here the problem of computing an approximation of the optimal value, and not constructingthe corresponding solution, but the definitions could be easily adapted. vertex). Assume again for simplicity that G is connected. It is then claimed in [23] that,using the same algorithm as for Nonblocker Set , the largest of V and V , say V , canbe always completed into a minimal vertex cover of G , which would immediately yield akernel of size at most 2 k for MMVC . Unfortunately, this claim is not true: when adding newvertices to V in order to make it a vertex cover of G , we may lose the minimality property,and some vertices may need to be removed. For instance, let G be the graph obtained froma triangle on vertices u, v, w by adding p ≥ u, v , and w . Let T be the spanning tree obtained from G by removing the edge { v, w } , and root T at vertex u . Then | V | = 1 + 2 p and | V | = 2 + p , so | V | > | V | , and note that the edge { v, w } is theonly edge of G not covered by V . But adding either of v or w to V , say v , results in anon-minimal vertex cover of G , and therefore the p pendant vertices adjacent to v have tobe removed from V , which yields a set of size 2 + p < | V ( G ) | = p , where we have usedthat p ≥
2. In fact, deciding whether a set S ⊆ V ( G ) can the extended to a minimal vertexcover of G is an NP -complete problem [12].It is natural to think whether the above algorithm could be modified so to obtain aminimal vertex cover of size at least n/ n -vertex graph. This is also not possible:take a clique of size k/ k/ k , but it has Ω( k ) vertices. Notealso that if this were possible, it would contradict the ( n / − ε )-inapproximability result for MMVC of Boria et al. [10].Finally, an indirect argument to conclude that the claimed kernelization algorithm [23] isflawed, using the results that we presented in this section, is to realize that it would constitutea linear lop -kernel for
MMVC , whose existence would imply, by Theorem 8, that P = NP . In this section we present subquadratic kernels for
Maximum Minimal Vertex Cover restricted to particular graph classes when the parameter is the solution size k . Namely, inSubsection 5.1 we provide kernels using the Erdős-Hajnal property, and in Subsection 5.2 weprovide further observations about other graph classes. For a constant δ >
0, a graph H is said to satisfy the Erdős-Hajnal property with constant δ if every H -free graph G with n vertices contains either a clique or an independent set of size n δ . The (still open) Erdős-Hajnal conjecture [22] states that every graph H satisfies theErdős-Hajnal property. As reported by Chudnovsky [14], the Erdős-Hajnal conjecture hasbeen verified for only a small number of graphs, namely all graphs on at most four vertices,the bull (i.e., the graph obtained by adding a pendant vertex to two different vertices of atriangle), the complete graphs, and every graph that can be constructed from them usingthe so-called substitution operation [2], which we define later.Since our goal is to use the Erdős-Hajnal property in order to obtain kernels for MaximumMinimal Vertex Cover , we need an algorithmic version of it. As defined by Bonnetet al. [9], for a constant δ >
0, a graph H is said to satisfy the constructive Erdős-Hajnalproperty with constant δ if there exists an algorithm that takes as input an H -free graph G on n vertices, and outputs in polynomial-time a clique or an independent set of G of size atleast n δ . Fortunately for our purposes, all the graphs H shown to satisfy the Erdős-Hajnalproperty so far, also satisfy its constructive version [9]. úlio Araújo, Marin Bougeret, Victor A. Campos, and Ignasi Sau 11 In the following lemma we show that, if H is a graph satisfying the constructive Erdős-Hajnal property, then the vertex set of an H -free graph can be partitioned in polynomialtime into “few” cliques or independent sets. This partition will then be used to obtainsubquadratic kernels on H -free graphs for several graphs H . (cid:73) Lemma 9.
Let H be a graph satisfying the constructive Erdős-Hajnal property withconstant δ . The vertex set of any H -free graph G on n vertices can be partitioned inpolynomial time into a collection of cliques C and a collection of independent sets I suchthat |C| + |I| ≤ (cid:16) (1 − δ ) − (cid:17) · n − δ . Proof.
Let G be an H -free graph on n vertices. We initialize X = V ( G ) , C = I = ∅ , andwe run the following procedure as far as | X | ≥ Y in G [ X ] with | Y | ≥| X | δ . Note that this is possible since G [ X ] is an H -free graph for any X ⊆ V ( G ).Add Y to C or to I depending on whether Y is a clique or an independent set,respectively (if | Y | = 1, choose C or I arbitrarily). Update X ← X \ Y .Clearly, the above algorithm terminates in polynomial time. It remains to bound |C| + |I| ,which is equal to the number of iterations of the algorithm. To this end, for a positive integer i , we say that an iteration belongs to step i of the algorithm if the current set X at thestart of the iteration satisfies n i < | X | ≤ n i − . We denote by t i the number of iterationsof the algorithm within step i . By definition, |C| + |I| = P ∞ i =1 t i . Let Y be a clique or anindependent set found by the algorithm within step i . Since the current set X satisfies | X | > n i , we have that | Y | > (cid:0) n i (cid:1) δ . And since the sum of the sizes of the sets found beforethe last iteration of step i is at most n i , it follows that t i ≤ (cid:0) n i (cid:1) − δ . Note that, in particular, t i = 0 for i > d log n e . Therefore, we conclude that |C| + |I| = ∞ X i =1 t i ≤ ∞ X i =1 (cid:16) n i (cid:17) − δ = n − δ · ∞ X i =1 (cid:18) − δ (cid:19) i = n − δ · (cid:18) (1 − δ ) − (cid:19) , and the lemma follows. (cid:74) We are now ready to present the subquadratic kernel on bull-free graphs. Note that, sincebipartite graphs are bull-free,
MMVC restricted to bull-free graphs is NP -hard by [6] (orby Theorem 3). In the kernels presented in this section, since we can easily obtain explicitconstants, we decided not to use the big-O notation. (cid:73) Theorem 10.
The
Maximum Minimal Vertex Cover problem parameterized by k restricted to bull-free graphs admits a kernel with at most c ( k − / + k − vertices, where c = − < . Proof.
Let (
G, k ) be an instance of the
Maximum Minimal Vertex Cover problem,where G is a bull-free graph. Recall that by Lemma 2 we can assume that the maximumdegree of G is at most k −
1. We start by finding greedily, starting from V ( G ), a minimalvertex cover X of G . Note that X can be easily found in polynomial time by Observation 1.If | X | ≥ k , we conclude that ( G, k ) is a yes -instance, so we can assume that | X | ≤ k −
1. Let S = V ( G ) \ X and note that S is an independent set.Since the bull satisfies the constructive Erdős-Hajnal property with constant δ = [9, 15],we can apply Lemma 9 to the bull-free graph G [ X ] and obtain in polynomial time apartition of X into a collection of cliques C and a collection of independent sets I such that |C| + |I| ≤ d · | X | / ≤ d · ( k − / , where d = − < .
47. Since we can assume that G has no isolated vertices, as they can be safely removed without affecting the type of theinstance, it follows that S = [ C ∈C N S ( C ) ∪ [ I ∈I N S ( I ) . (1)Hence, our objective is to bound | N S ( Y ) | for every Y ∈ C ∪ I . Suppose first that I ∈ I isan independent set. From Observation 2, if | N S ( I ) | ≥ k we can conclude that ( G, k ) is a yes -instance, so we can assume henceforth thatfor every independent set I ∈ I , it holds | N S ( I ) | ≤ ( k − . (2)Suppose now that C ∈ C is a clique. We partition N S ( C ) = S C ] S C as follows. Let S C bean inclusion-wise maximal set of vertices in N S ( C ) such that for any two (not necessarilydistinct) vertices x, y ∈ S C , | N C ( x ) ∪ N C ( y ) | ≤ | C | −
1. That is, S C is a maximal set in N S ( C ) such that the neighborhoods of its vertices pairwise do not cover the whole clique C .We let S C = N S ( C ) \ S C . The following is the crucial property of the set S C . (cid:66) Claim 11.
The vertices in S C can be ordered x , . . . , x p so that N C ( x i ) ⊆ N C ( x j ) whenever i ≤ j . Proof.
In order to prove the claim, it is sufficient to prove that, for any two vertices x, y ∈ S C ,either N C ( x ) ⊆ N C ( y ) or N C ( y ) ⊆ N C ( x ). Suppose for the sake of contradiction that thereexist two vertices u ∈ N C ( x ) \ N C ( y ) and v ∈ N C ( y ) \ N C ( x ). By definition of the set S C ,there exists a vertex w ∈ C \ ( N C ( x ) ∪ N C ( y )). But then the vertices x, y, u, v, w induce abull as illustrated in Figure 2, contradicting the hypothesis that G is bull-free. (cid:74) x yu vw zS C S C C Figure 2
Configuration considered in the proof of Claim 11 and a vertex z ∈ T x ∈ S C N C ( x ). Claim 11 implies in particular that, unless S C = ∅ , there exists a vertex u ∈ T x ∈ S C N C ( x ).Since u has degree at most k − G , and each vertex x ∈ S C is adjacent to u , it followsthat | S C | ≤ k − S C . The definition of the set S C together with Claim 11 implythat there exists a vertex z ∈ C \ S y ∈ S C N C ( y ). Consider now an arbitrary vertex x ∈ S C .Since x could not be added to S C , there exists a vertex y ∈ S C such that N C ( x ) ∪ N C ( y ) = C .But since z ∈ C \ S y ∈ S C N C ( y ), necessarily z ∈ N C ( x ). It follows that z ∈ T x ∈ S C N C ( x )(see Figure 2). Using again the fact that z has degree at most k − G , we obtain that | S C | ≤ k −
1. Summarizing, we have thatfor every clique C ∈ C , it holds | N S ( C ) | = | S C | + | S C | ≤ k − . (3)Putting all pieces together, Equations (1), (2), and (3) and the fact that | X | ≤ k − |C| + |I| ≤ d · | X | / imply that, unless we have already concluded that ( G, k ) is a yes -instance, úlio Araújo, Marin Bougeret, Victor A. Campos, and Ignasi Sau 13 we have that | V ( G ) | = | X | + | S | = | X | + | [ C ∈C N S ( C ) | + | [ I ∈I N S ( I ) |≤ | X | + ( |C| + |I| ) · max Y ∈C∪I | N S ( Y ) | ≤ k − d · ( k − / · k − d · ( k − / + k − , and the theorem follows. (cid:74) It is easy to prove that, for every integer t ≥
2, every K t -free graph G on n verticeshas an independent set of size n t − , by induction on t : for t = 2 the statement is trivial,and if t ≥
3, then either ∆( G ) < n t − t − , and an independent set of size n t − can be foundgreedily by adding any vertex to it and deleting its neighborhood, or there exists a vertex v ∈ V ( G ) of degree at least n t − t − , in which case an independent set of size n t − can be foundapplying the inductive hypothesis to the K t − -free graph G [ N ( v )]. Clearly, this proof canbe translated to a polynomial-time algorithm to find an independent set of the appropriatesize. Therefore, for any integer t ≥ K t satisfies the constructive Erdős-Hajnal propertywith constant δ = t − . The proof of the following theorem is a simplified version of thatof Theorem 10. Note that, since bipartite graphs are K t -free for every t ≥ MMVC is NP -hard on K t -free graphs [6]. (cid:73) Theorem 12.
For every integer t ≥ , the Maximum Minimal Vertex Cover problemparameterized by k restricted to K t -free graphs admits a kernel with at most c t ( k − t − t − + k − vertices, where c t = t − t − t − − . Proof.
As in the proof of Theorem 10, given an instance (
G, k ) of
Maximum MinimalVertex Cover , where G is a K t -free graph, we partition V ( G ) = X ] S , where X is aminimal vertex cover of G with | X | ≤ k −
1, and we use Lemma 9 to partition X into twocollections C and I of cliques and independent sets, respectively, with |C| + |I| ≤ d t · | X | t − t − ,where d t = t − t − − . Equations (1) and (2) still hold, but now we have a much simpler versionof Equation (3): if C ∈ C is a clique then, since G is K t -free, necessarily | C | ≤ t −
1, whichtogether with the fact that ∆( G ) ≤ k − C ∈ C , it holds | N S ( C ) | = ( t − k − . (4)Combining Equations (1), (2), and (4) we get | V ( G ) | ≤ | X | + ( |C| + |I| ) · max Y ∈C∪I | N S ( Y ) | ≤ k − d t · ( k − t − t − · ( t − k − , and the theorem follows. (cid:74) We now extend the results of Theorem 10 and Theorem 12 to more general excludedinduced graphs H , by making use of the aforementioned substitution operation. As definedby Alon et al. [2], for two graphs H and H on disjoint vertex sets, we say that H is obtainedfrom H by substituting H for v ∈ V ( H ) (or just obtained from H by substituting H ifthe vertex v in question is not important) if V ( H ) = ( V ( H ) \ { v } ) ∪ V ( H ), H [ V ( H )] = H , H [ V ( H ) \ { v } ] = H \ v , and u ∈ V ( H ) is adjacent in H to w ∈ V ( H ) if and only if u is adjacent in H to v . Alon et al. [2] proved that if two graphs H and H satisfy Erdős-Hajnal property and H is obtained from H by substituting H , then H satisfies the Erdős-Hajnal property as well.More precisely, by following the details in the proof of [2, Theorem 2.1], we can derive thatif H and H satisfy Erdős-Hajnal property with constants δ and δ , respectively, then H satisfies the Erdős-Hajnal property with constant δ = δ δ + | V ( H ) |· δ . The same applies to theconstructive version of the Erdős-Hajnal property.For an integer t ≥
2, we define the t -bull as the graph obtained from K t by adding apendant vertex to two different vertices of the clique. Note that the 2-bull is equal to P andthat the 3-bull is equal to the bull. Note also that, for every t ≥
3, the t -bull is obtainedfrom the bull by substituting K t − for the vertex of degree two of the bull. Therefore, bythe discussion in the above paragraph, since the bull and K t − satisfy the constructiveErdős-Hajnal property with constants and t − , respectively, it follows that, for every t ≥ t -bull satisfies the constructive Erdős-Hajnal property with constant δ t = t − + t − = 4 t + 17 . The proof of the next theorem follows again (and generalizes) that of Theorem 10. Notethat Theorem 13 corresponds to the particular case t = 3 of Theorem 12. Note also thatbipartite graphs are t -bull-free for t ≥
3, hence
MMVC is NP -hard on t -bull-free graphsfor t ≥ P -free graphs, also calledcographs, which have cliquewidth at most two, hence by Observation 18 (proved later inSubsection 5.2) MMVC can be solved in polynomial time on this class. (cid:73)
Theorem 13.
For every integer t ≥ , the Maximum Minimal Vertex Cover problemparameterized by k restricted to t -bull-free graphs admits a kernel with at most c t ( k − − δ t + k − vertices, where δ = and δ t = t +17 for t ≥ , and c t = t − (1 − δt ) − . Proof.
As in the proof of Theorem 10, given an instance (
G, k ) of
Maximum MinimalVertex Cover , where G is a t -bull-free graph, we partition V ( G ) = X ] S , where X is aminimal vertex cover of G with | X | ≤ k −
1, and we use Lemma 9 to partition X into twocollections C and I of cliques and independent sets, respectively, with |C| + |I| ≤ d t · | X | − δ t ,where δ = and δ t = t +17 for t ≥
4, and d t = (1 − δt ) − for every t ≥
3. Equations (1)and (2) still hold for every integer t ≥
3, but now we need slightly more involved argumentsto obtain an appropriate version of Equation (3) for every t ≥ C ∈ C is a clique. We partition N S ( C ) = S C ] S C as follows.Let S C be an inclusion-wise maximal set of vertices in N S ( C ) such that for any two (notnecessarily distinct) vertices x, y ∈ S C , | N C ( x ) ∪ N C ( y ) | ≤ | C | − ( t − S C is amaximal set in N S ( C ) such that the neighborhoods of its vertices pairwise leave at least t − C . We let S C = N S ( C ) \ S C . The set S C satisfies exactlythe same crucial property as for the case t = 3 (see Claim 11). (cid:66) Claim 14.
For every integer t ≥
3, the vertices in S C can be ordered x , . . . , x p so that N C ( x i ) ⊆ N C ( x j ) whenever i ≤ j . Proof.
In order to prove the claim, it is sufficient to prove that, for any two vertices x, y ∈ S C ,either N C ( x ) ⊆ N C ( y ) or N C ( y ) ⊆ N C ( x ). Suppose for the sake of contradiction that thereexist two vertices u ∈ N C ( x ) \ N C ( y ) and w ∈ N C ( y ) \ N C ( x ). By definition of the set S C , there exist t − w , . . . , w t − ∈ C \ ( N C ( x ) ∪ N C ( y )). But then the vertices x, y, u, v, w , . . . , w t − induce a t -bull, contradicting the hypothesis that G is t -bull-free. (cid:74) úlio Araújo, Marin Bougeret, Victor A. Campos, and Ignasi Sau 15 Claim 14 implies in particular that, unless S C = ∅ , there exists a vertex u ∈ T x ∈ S C N C ( x ).Since u has degree at most k − G , and each vertex x ∈ S C is adjacent to u , it followsthat | S C | ≤ k − S C . The definition of the set S C together with Claim 14imply that there exist at least t − z , . . . , z t − ∈ C \ S y ∈ S C N C ( y ). Consider nowan arbitrary vertex x ∈ S C . Since x could not be added to S C , there exists a vertex y ∈ S C such that | N C ( x ) ∪ N C ( y ) | ≥ | C | − ( t − z , . . . , z t − ∈ C \ S y ∈ S C N C ( y ), thereexists an index j ∈ [ t −
2] such that z j ∈ N C ( x ). That is, every vertex x ∈ S C is adjacentto at least one of the vertices z , . . . , z t − . Using again the fact that each of the vertices z , . . . , z t − has degree at most k − G , we obtain that | S C | ≤ ( t − k − C ∈ C , it holds | N S ( C ) | = | S C | + | S C | ≤ ( t − k − . (5)Putting all pieces together, Equations (1), (2), and (5) and the fact that | X | ≤ k − |C| + |I| ≤ d t · | X | − δ t imply that, unless we have already concluded that ( G, k ) is a yes -instance, we have that | V ( G ) | ≤ | X | + ( |C| + |I| ) · max Y ∈C∪I | N S ( Y ) | ≤ k − d t · ( k − − δ t · ( t − k − , and the theorem follows. (cid:74) Let the paw be the graph obtained from a triangle by adding a pendant edge. Gyárfás [29]showed that the paw satisfies the constructive Erdős-Hajnal property with constant δ = .Note that bipartite graphs are paw-free, hence MMVC is NP -hard on paw-free graphs [6]. (cid:73) Theorem 15.
The
Maximum Minimal Vertex Cover problem parameterized by k restricted to paw-free graphs admits a kernel with at most c ( k − / + k − vertices, where c = / − < . . Proof.
Given an instance (
G, k ) of
Maximum Minimal Vertex Cover , where G is apaw-free graph, we again partition V ( G ) = X ] S , where X is a minimal vertex cover of G with | X | ≤ k −
1, and we use Lemma 9 to partition X into two collections C and I ofcliques and independent sets, respectively, with |C| + |I| ≤ d · | X | / , where d = / − .Equations (1) and (2) still hold, and we can again obtain in a simpler way an appropriateversion of Equation (3). Indeed, let C ∈ C be a clique, and our goal is to bound | N S ( C ) | . If | C | = 1 then by the fact that ∆( G ) ≤ k − | N S ( C ) | ≤ k −
1, so assume that | C | ≥
2. Suppose for the sake of contradiction that there exists a vertex v ∈ N S ( C ) suchthat | N C ( v ) | ≤ | C | −
2. Let w ∈ N C ( v ) and let z , z be two vertices in C \ N C ( v ). Then thevertices v, w, z , z induce a paw, contradicting the hypothesis that G is paw-free. Therefore,for every vertex v ∈ N S ( C ) it holds that | N C ( v ) | ≥ | C | −
1. Hence, the number of edges in G between C and N S ( C ) is at least | N S ( C ) | · ( | C | −
1) and, since ∆( G ) ≤ k −
1, at most | C | · ( k − | C | ≥
2, it follows thatfor every clique C ∈ C , it holds | N S ( C ) | ≤ | C || C | − · ( k − ≤ k − . (6)Putting all pieces together, Equations (1), (2), and (6) and the fact that | X | ≤ k − |C| + |I| ≤ d · | X | / imply that, unless we have already concluded that ( G, k ) is a yes -instance,we have that | V ( G ) | ≤ | X | + ( |C| + |I| ) · max Y ∈C∪I | N S ( Y ) | ≤ k − d · ( k − / · k − , and the theorem follows. (cid:74) In this subsection we provide additional observations about the complexity of the
MaximumMinimal Vertex Cover problem restricted to special graph classes. (cid:73)
Lemma 16.
For every integer t ≥ , the Maximum Minimal Vertex Cover problemparameterized by k restricted to K ,t -free graphs admits a kernel with at most t ( k − vertices. Proof.
Given an instance (
G, k ) of
Maximum Minimal Vertex Cover , where G is a K ,t -free graph, we again partition V ( G ) = X ] S , where X is a minimal vertex cover of G with | X | ≤ k −
1. Since G is K ,t -free and S is an independent set, it holds that for every v ∈ X , | N S ( v ) | ≤ t −
1, and since we can assume that G contains no isolated vertex, weobtain that | V ( G ) | = | X | + | S v ∈ X | N S ( v ) | ≤ k − t − k −
1) = t ( k − (cid:74) Let C be a graph class such that there exists a polynomial-time algorithm that, givena graph G ∈ C , outputs a proper coloring of the vertices of G using at most c colors, forsome integer c ≥
1. We say that such a graph class C is poly - χ - c -bounded . Examples of poly - χ - c -bounded classes are planar graphs, minor-free graphs, or, more generally, graphs ofbounded expansion. We note that Fernau [23, Corollary 4.14] provides a similar observationfor the particular case of planar graphs. (cid:73) Lemma 17.
For every integer c ≥ , the Maximum Minimal Vertex Cover problemparameterized by k restricted to the class of poly - χ - c -bounded graphs admits a kernel with atmost c ( k − vertices. Proof.
Given an instance (
G, k ) of
MMVC , where G belongs to a poly - χ - c -bounded class,we first compute in polynomial time a proper vertex coloring of G using at most c colors. Wemay clearly assume that G has no isolated vertices, as such vertices can be safely removed.Let V ( G ) = S ] · · · ] S c be the corresponding partition of V ( G ) into independent sets. ByLemma 2, for every i ∈ [ c ] there exists a minimal vertex cover of G that contains N ( S i ).Hence, if for some i ∈ [ c ] we have that | N ( S i ) | ≥ k , we can safely answer “ yes ”, so wemay assume that, for every i ∈ [ c ], | N ( S i ) | ≤ k −
1. Since G has no isolated vertices andevery set S i is an independent set, it follows that V ( G ) = S i ∈ [ c ] N ( S i ), so we have that | V ( G ) | ≤ P i ∈ [ c ] | N ( S i ) | ≤ c ( k − (cid:74) Another graph class K that allows for linear kernels is defined such that, for every graph G ∈ K , the minimum size of a dominating set of G is equal to the size of a minimum independent dominating set of G . We furthermore ask K to be hereditary. Such graphs havebeen studied, for instance, in [1, 36], and include in particular K , -free graphs (note that ageneralization to K ,t -graphs is given in Lemma 16). Let us see why the class K allows for alinear kernel. As discussed at the end of Section 3, the complement of a dominating set iscalled a nonblocker, and the Nonblocker Set problem admits a linear kernel [23]. On theother hand, the complement of an independent dominating set is a minimal vertex cover.Hence, if G ∈ K , an instance ( G, k ) of
Nonblocker Set is positive if and only if (
G, k ) isa positive instance of
MMVC . Note the linear kernel for the
Nonblocker Set problemdiscussed at the end of Section 3 outputs a subgraph G of G , and we have that G ∈ K since K is hereditary. Hence, the equivalence between Nonblocker Set and
MMVC also holdsfor G , and it follows that this kernel is also a linear kernel for MMVC restricted to graphsin K .Our last contribution in this section concerns graph classes of bounded cliquewidth .Cliquewidth, which we do not need to define here, is a graph parameter that is “smaller” thantreewidth in the sense that graph classes of bounded treewidth have also bounded cliquewidth úlio Araújo, Marin Bougeret, Victor A. Campos, and Ignasi Sau 17 (the opposite is not true, as cliques have cliquewidth one but unbounded treewidth); see [16]for the formal definition.The variation of monadic second order logic of graphs called MSO is defined by a syntaxthat includes the logical connectives ∨ , ∧ , ¬ , variables for vertices, edges, sets of vertices(but not sets of edges), the quantifiers ∀ , ∃ that can be applied to these variables, and thebinary relations expressing whether a vertex belongs to a set, whether an edge is incident tovertex, whether two vertices are adjacent, and whether two sets are equal. It is well-knownthat finding a minimum or maximum weight vertex set that satisfies a given graph propertyexpressed in MSO can be solved in linear time on graphs of cliquewidth bounded by aconstant [4, 16]. (cid:73) Observation 18.
The
Maximum Minimal Vertex Cover problem can be expressed in
MSO , and therefore it can be solved in linear time when restricted to any graph class ofcliquewidth bounded by a constant. Proof.
Given a graph G , we can express the property of a vertex set S being a minimalvertex cover of G in the syntax of MSO as follows: for every pair of vertices u, v such that u is adjacent to v , u ∈ S or v ∈ S (this guarantees that S is a vertex cover of G ), and forevery vertex v ∈ V ( G ), v / ∈ S or there exists a vertex u adjacent to v such that u / ∈ S (thisguarantees, by Observation 1, that S is minimal). (cid:74) Let the diamond be the graph obtained from K by removing an edge. Since Brand-städt [11] proved that { P , diamond } -free graphs have bounded cliquewidth, from Observa-tion 18 we immediately get the following corollary. (cid:73) Corollary 19.
The
Maximum Minimal Vertex Cover problem restricted to { P , diamond } -free graphs can be solved in linear time. We provided positive and negative results about the kernelization of the
Maximum MinimalVertex Cover problem. In particular, we proved (Theorem 8) that a subquadratic kernelthat uses only what we called lop -rules (or that answers “ yes ”) would imply that P = NP .It seems plausible that this restriction could be replaced by a more standard complexityassumption. On the other hand, we believe that the simple argument that we developed inorder to prove Theorem 8 might be applied to provide lower bounds for the kernel sizes ofother parameterized problems.We presented (Section 5) subquadratic kernels on H -free graphs for some graphs H satisfying the (constructive) Erdős-Hajnal property, such as the bull, the complete graphs, orthe paw. It would be interesting to obtain subquadratic kernels for other graphs H satisfyingthe Erdős-Hajnal property, such as C , the diamond, P , or C . Note that, from [29], C andthe diamond satisfy the constructive Erdős-Hajnal property with constant δ ≥ /
3. Note alsothat the graphs constructed in the reduction of Theorem 3 are { C , diamond } -free, as theyare bipartite, hence MMVC is NP -hard on this class, in contrast to the fact (Corollary 19)that MMVC can be solved in linear time on { P , diamond } -free graphs. To the best of ourknowledge, the complexity on P -graphs is open, as well as on K ,t graphs for t ≥ P -free graphs have unbounded cliquewidth, becauseco-bipartite graphs, which are P -free, have unbounded cliquewidth.As defined in Section 4, for a graph G we denoted by mmvc ( G ) the maximum size of aminimal vertex cover of G . Boria et al. [10] proved that if G is an n -vertex graph without isolated vertices, then mmvc ( G ) ≥ b n / c . Note that this immediately yields a quadratickernel for MMVC : if k ≤ b n / c we answer “ yes ”, otherwise n ≤ k . By the same argument,if C is a graph class such that every n -vertex graph G ∈ C without isolated vertices satisfies mmvc ( G ) ≥ n / ε , for some ε >
0, then
MMVC restricted to C admits a (subquadratic)kernel with at most k ε vertices. It might be possible that this is the case for someof the H -free graph classes for which we provided subquadratic kernels in Section 5: wewere not able to find any counterexample, that is, a family of n -vertex H -free graphs G forwhich mmvc ( G ) = Θ( n / ). In particular, the case of triangle-free graphs seems particularlyinteresting. Haviland [30] and Goddard and Lyle [28] established upper bounds on the size ofa minimum independent dominating set (that is, the complement of a minimal vertex cover) oftriangle-free graphs. It follows from their results [28,30] that there exist n -vertex triangle-freegraphs G with mmvc ( G ) = Θ( n / · log n ), hence if such a constant ε > (cid:15) ≤ − = . Therefore, the smallest kernelthat we may obtain in this way on triangle-free graphs would have k ε ≤ k / vertices,which matches the size of the kernel that we obtained in Theorem 12 for the particular case t = 3, disregarding lower order terms and multiplicative constants. Finding such a constant ε > H -graphs for small graphs H , in particular on triangle-free graphs, looks like achallenging problem, having interesting connections with the Ramsey numbers [28, 30]. References Robert B. Allan and Renu C. Laskar. On domination and independent domination numbersof a graph.
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