1 Introduction
The vertex cover problem plays a central role in parameterized complexity. In particular, it has been very important for the development of new kernelization techniques and the study of structural parameters. As a result of this work, there is now a solid understanding of which parameterizations of vertex cover lead to fixedparameter tractability or existence of a polynomial kernelization. This is motivated by the fact that parameterization by solution size leads to large parameter values on many types of easy instances. Thus, while there is a wellknown kernelization for instances of vertex cover() to at most vertices, it may be more suitable to apply a kernelization with a size guarantee that is a larger function but depends on a smaller parameter.
Jansen and Bodlaender [13] were the first to study kernelization for vertex cover under different, smaller parameters. Their main result is a polynomial kernelization to instances with vertices when is a feedback vertex set of the input graph, also called a modulator to the class of forests. Clearly, the size of is a lower bound on the vertex cover size (as any vertex cover is a modulator to an independent set). Since then, their result has been generalized and complemented in several ways. The two main directions of followup work are to use modulators to other tractable cases instead of forests (see below) and parameterization above lower bounds (see related work).
For any graph class , we can define a parameterization of vertex cover by distance to , i.e., by the minimum size of a modulator such that belongs to . For fixedparameter tractability and kernelization of the arising parameterized problem it is necessary that vertex cover is tractable on inputs from . For hereditary classes , this condition is also sufficient for fixedparameter tractability but not necessarily for the existence of a polynomial kernelization. Interesting choices for are various wellstudied hereditary graph classes, like forests, bipartite, or chordal graphs, and graphs of bounded treewidth, bounded treedepth, or bounded degree.
Majumdar et al. [16] studied vertex cover parameterized by (the size of) a modulator to a graph of maximum degree at most . For this problem is hard but for and they obtained kernels with and vertices, respectively. Their result motivated Fomin and Strømme [9] to investigate a parameter that is smaller than both a modulator to degree at most two and the size of a feedback vertex set: They consider being a modulator to a pseudoforest, i.e., with each connected component of having at most one cycle. For this they obtain a kernelization to vertices, generalizing (except for the size) the results of Majumdar et al. [16] and Jansen and Bodlaender [13]. They also prove that the parameterization by a modulator to socalled mock forests, where no cycles share a vertex, admits no polynomial kernelization unless (and the polynomial hierarchy collapses).
For their kernelization, Fomin and Strømme [9] prove that minimal blocking sets in a pseudoforest have size at most three, which requires a lengthy proof. (A minimal blocking set is a set of vertices whose deletion decreases the independence number by exactly one.)^{2}^{2}2Like previous work [13, 9] we prefer to work with independent set rather than vertex cover, but this makes no important difference. This allows to reduce the number of components of the pseudoforest such that one can extend the modulator to a sufficiently small feedback vertex set by adding one (cycle) vertex per component to . At this point, the kernelization of Jansen and Bodlaender [13] can be applied to get the result.
The results of Fomin and Strømme [9]
suggest that the border for existence of polynomial kernels for feedback vertex setlike parameters may be much more interesting than expected previously. Arguably, there is still quite some room between allowing a single cycle per component and allowing an arbitrary number of cycles so long as they share no vertices. Do larger numbers of cycles per component still allow a polynomial kernelization? Similarly, cycles in the lower bound proof have odd length and it is known that absence of odd cycles is sufficient, i.e., a kernelization for modulators to bipartite graphs is known. Could this be extended to allowing bipartite graphs with one or more odd cycles per connected component?
Our work.
We show that the answers to the above questions are largely positive and provide, essentially, a single elegant proof to cover them. To this end, it is convenient to take the perspective of feedback sets rather than the maximum size of a cycle packing. Say that a quasiforest is a graph such that each connected component has a feedback vertex set of size at most , whereas in a quasibipartite graph each connected component must have an odd cycle transversal (a feedback set for odd cycles) of size at most .
We show that vertex cover admits a kernelization with vertices when is a modulator to a quasiforest (Section 3). The case for strengthens the result of Fomin and Strømme [9] (as one cycle per component is stricter than feedback vertex set size one). For every fixed larger value of we obtain a polynomial kernelization, though of increasing size. The result is obtained by proving that minimal blocking sets in a quasiforest have size at most (and then applying [13]). Intuitively, having a large minimal blocking set implies getting a fairly small maximum independent set because there are optimal independent sets that avoid all but any chosen vertex of a minimal blocking set. In contrast, a quasiforest always has a large independent set because each connected component is almost a tree.
The value is tight already for cliques of size , which are permissible connected components in a quasiforest. Such cliques also imply that our parameterization inherits a lower bound of from the lower bound of (assuming ) for being a modulator to a cluster graph with component size at most [16].
It turns out that our proof directly extends also to quasibipartite graphs, proving that their minimal blocking sets similarly have size at most (Section 4). Thus, when given a modulator such that is quasibipartite, we can extend it to an odd cycle transversal of size at most , which directly yields a randomized polynomial kernel by using a randomized polynomial kernelization for vertex cover parameterized by an odd cycle transversal [15].
Motivated by this, we explore also modulators to graphs in which each connected component has vertex cover size at most plus the size of a minimum fractional vertex cover, which we call quasiintegral (Section 4). This is stronger than the previous parameter because it allows connected components that have an odd cycle transversal of size at most . We show that minimal blocking sets in any quasiintegral graph have size at most . This bound is tight, as witnessed by the cliques with vertices, and the problem inherits a lower bound of from the lower bound for modulators to cluster graphs with clique size at most [16]. Using the upper bound of one can remove redundant connected components until the obtained instance has vertex cover size at most more than the best fractional vertex cover. In other words, one can reduce to an instance of vertex cover parameterized above LP with parameter value and apply the randomized polynomial kernelization of Kratsch and Wahlström [15] to get a randomized polynomial kernel.
Related work.
Recent work of Bougeret and Sau [5] shows that vertex cover admits a kernel of size when is a modulator to a graph of treedepth at most . Their result is incomparable to ours: Already the kernelization by feedback vertex set size [13], which we generalize, allows arbitrarily long paths in ; such paths are forbidden in a graph of bounded treedepth. Conversely, taking a star with leaves and appending a cycle at each leaf yields a graph with feedback vertex set and odd cycle transversal size equal to but constant treedepth; can be chosen arbitrarily large.
The fact that deciding whether a graph has a vertex cover of size at most is trivial when is lower than the size of a largest matching in has motivated the study of above lower bound parameters like . The strongest lower bound employed so far is , where denotes the minimum cost of a fractional vertex cover, and Garg and Philip [10] gave an time algorithm. Randomized polynomial kernels are known for parameters and [15] and for parameter [14]. Our present kernelizations are not covered even by the strongest parameter because already quasiforests for any can have a vertex cover size that is arbitrarily larger than : Consider, for example, a disjoint union of cliques with four vertices each, where but vertex cover size is three per component.
Regarding lower bounds for kernelization (all assuming ), it is of course well known that there are no polynomial kernels for vertex cover when parameterized by width parameters like treewidth, pathwidth, or treedepth (cf. [2]). Lower bounds similar to the one for modulators to mock forests by Fomin and Strømme [9] were already obtained by Cygan et al. [7] (modulators to treewidth at most two) and Jansen [12] (modulators to outerplanar graphs). Bodlaender et al. [3] showed that there is no polynomial kernelization in terms of the vertex deletion distance to a single clique, which is stronger than distance to cluster or perfect graphs for example. Majumdar et al. [16] ruled out kernels of size when is a modulator to a cluster graph with cliques of size bounded by .
2 Preliminaries and notation
Graphs.
We use standard notation mostly following Diestel [8]. Let be a graph. For a set , let denote the neighborhood of in , i.e., and let denote the neighborhood of in including , i.e., . We omit the subscript whenever the underlying graph is clear from the context. Furthermore, we use as shorthand for . For a graph we denote by the vertex cover number of and by the independence number of . Let , we call a blocking set of , if deleting the vertex set from the graph decreases the size of a maximum independent set, hence if . A blocking set is minimal, if no proper subset of is a blocking set of . We denote by the clique of size .
Linear Programming.
We denote the linear program relaxation for
vertex cover resp. independent set for a graph by resp. . Recall that and . A feasible solution to one of the above linear program relaxations is an assignment to the variables for all vertices which satisfies the conditions of the linear program. An optimum solution to resp. is a feasible solution which minimizes resp. maximizes the objective function value . It follows directly from the definition that is a feasible solution to if and only if is a feasible solution to ; thus . It is well known that there exists an optimum feasible solution to with ; we call such a solution half integral. The same is, of course, true for . Given a half integral solution (to or ), we define for each . Note that if is an optimum half integral solution to , then it holds that , whereas, it holds that , when is an optimum half integral solution to . We omit the subscript , when the solution is clear from the context.3 Vertex Cover parameterized by a modulator to a quasiforest
In this section we present a polynomial kernel for vertex cover parameterized by a modulator to a quasiforest. More precisely, we develop a polynomial kernel for independent set parameterized by a modulator to a quasiforest which, by the relation between these two problems, directly yields a polynomial kernel for vertex cover parameterized by a modulator to a quasiforest.
Consider an instance of the problem, which asks whether graph , with is a quasiforest, has an independent set of size . Like Fomin and Strømme [9], we reduce the input instance until the quasiforest has at most polynomially many connected components in terms of ; see Reduction Rule 3. By adding for each component of the quasiforest a feedback vertex set of size to the modulator , we polynomially increase the size of the modulator . The resulting modulator is a feedback vertex set, hence we can apply the polynomial kernelization for independent set parameterized by a modulator to a feedback vertex set from Jansen and Bodlaender [13].
Let be an instance of independent set parameterized by a modulator to a quasiforest. Since is a constant we can compute in polynomial time a maximum independent set in . Choosing some vertices from the set to be in an independent set will prevent some vertices in to be part of the same independent set; thus it may be that we can add less than vertices from to an independent set that contains some vertices of . To measure this difference, we use the term of conflicts introduced by Jansen and Bodlaender [13]. Our definition is more general in order to use it also for modulators to quasibipartite resp. quasiintegral graphs.
Definition 1 (Conflicts).
Let be a graph and be a subset of , such that we can compute a maximum independent set in in polynomial time. Let be a subgraph of and let . We define the number of conflicts on which are induced by as .
Now we can state our reduction rule, which deletes some connected components of the quasiforest . More precisely, we delete connected components of which we know that there exists a maximum independent set in that contains a maximum independent set of the connected component .
 Reduction Rule 1:

If there exists a connected component of such that for all independent sets of size at most with it holds that , then delete from and reduce by .
The proof of safeness will be given in the sequel. In particular, we delete connected components that have no conflicts. The goal of Reduction Rule 3 is to delete connected components of the quasiforest such that we can bound the number of connected components by a polynomial in the size of . Thus, if we cannot apply this reduction rule any more we should be able to find a good bound for the number of connected components in the quasiforest . The following lemma yields such a bound.
Lemma 1.
Let be an instance of independent set parameterized by a modulator to a quasiforest where Reduction Rule 3 is not applicable. Then the number of connected components in is at most .
Proof.
Let be a connected component of the quasiforest . Since Reduction Rule 3 is not applicable, there exists an independent set of size at most such that and ; otherwise Reduction Rule 3 would delete (or another connected component with the same properties).
Observe, that there are at most connected components of the quasiforest that have a conflict with an independent set , when is the reason that we cannot apply Reduction Rule 3 to one of these connected components: Assume for contradiction that there are connected components of the quasiforest that have a conflict with the same independent set of size at most ; therefore it holds that for all . But now, for all
where the first inequality corresponds to summing over some connected components of . Thus, could not be the reason why the connected components are not reduced during Reduction Rule 3.
This leads to the claimed bound of at most connected components in , because for every independent set of size at most there are at most connected components for which is the reason that we cannot apply Reduction Rule 3. ∎
It remains to show that Reduction Rule 3 is safe; i.e. that there exists a solution for if and only if there exists a solution for , where , and is the connected component of we delete during Reduction Rule 3. The main ingredient for this is to prove that any minimal blocking set has size at most (Lemma 4). To bound the size of minimal blocking sets we need the existence of a half integral solution to for which every maximum independent set in fulfills . This is similar to the result of Nemhauser and Trotter [17] and other results about the connection between maximum independent sets (resp. minimum vertex covers) and their fractional solutions [1, 4, 6, 11].
Lemma 2.
Let be an undirected graph. There exists an optimum half integral solution to such that for all maximum independent sets in it holds that .
Proof.
Let be an optimum half integral solution to , such that is maximal; this means, that there exists no optimum half integral solution to such that . We will show that every independent set in with or is not a maximum independent set in .
First, we observe that for all subsets it must hold that the size of the neighborhood of in is larger than the size of , i.e. ; if this is not the case, then we can construct an optimum half integral solution to with (which contradicts the fact that is maximal), by assigning a value of to all vertices in . Obviously, it holds that and that
In order to show that is indeed a feasible solution to , it suffices to consider edges of that have at least one endpoint in , say , because these are the only vertices for which we increase the value of the half integral solution to obtain . Since , the constraint can only be violated if . But then must hold since the only changed values are in . This of course means that and ; a contradiction.
Now, we assume that there exists a maximum independent set that contains a vertex of the set . Let . We will show that deleting the set from the independent set and adding the set to the independent set leads to a larger independent of , i.e. . First we show that has larger cardinality than . Since is an independent set, we know that and hence that the cardinality of is . From the above observation, we know that and it follows that has larger cardinality than . To prove that is an independent set in , it is enough to show that any vertex has no neighbor in ; this holds because is an independent set, and . Thus, is an independent set which has larger cardinality than ; this contradicts the assumption that is a maximum independent set.
It remains to show that there exists no maximum independent set in with . Let . Since is a maximum independent set, there exists a vertex (otherwise would be a larger independent set in ). But and hence , which contradicts the assumption that . ∎
Using the above lemma, we can show that every minimal blocking set in a quasiforest has size at most . This generalizes the result of Fomin and Stromme [9], who showed that a minimal blocking set in a pseudoforest has size at most three. Furthermore, we can show that this bound is tight.
Theorem 1.
Minimal blocking sets have a tight upper bound of in quasiforests.
The crucial part of Theorem 1 is to prove the upper bound.
Lemma 3.
Let be a quasiforest and let be a feedback vertex set in of size at most . Then it holds that a minimal blocking set in the quasiforest has size at most .
Proof.
We consider an optimum half integral solution to which fulfills the properties of Lemma 2; let for . We know that every maximum independent set of contains the set and no vertex of the set (because fulfills the properties of Lemma 2).
Observe that for all vertices it holds that ; otherwise, the set would not be a minimal blocking set. Furthermore, from the above observation it follows that , because
The key observation of our proof is that ; this follows from the fact that is minimal: As observed above, we know that . Thus, for all vertices there exists a maximum independent set in that contains the vertex and no other vertex from the set . Consider the sets for all vertices . Obviously, the sets are independent sets in for all vertices , because is the only vertex of the set that is contained in . Furthermore, we know that the sets are maximum independent sets in because
The fact that is a maximum independent set for all vertices implies that (by the choice of the solution to ). Thus, for all vertices it holds that and therefore that which implies that (because ). Since this holds for all vertices it follows that , hence .
To bound the size of we try to find an upper bound for the size of a maximum independent set in and a lower bound for the size of a maximum independent set in . An obvious upper bound for the size of a maximum independent set in is the optimum value of which is equal to . This leads to an upper bound for :
(1) 
because .
Next, we try to find a lower bound for the size of a maximum independent set in . We will construct an independent set in and the size of this independent set is a lower bound for the size of a maximum independent set in . First of all, we add all vertices from the independent set to ; this will prevent every vertex from to be part of the independent set . Now, we can extend the independent by an independent set in . First, observe that , because and . From this follows that , because . Instead of adding an independent set of to , we add a maximum independent set of the forest to ; such an independent set has size at least . This leads to the following lower bound for :
(2) 
Using the equation together with the upper bound for and the lower bound for leads to the requested upper bound for the size of :
We showed that every minimal blocking set in a quasiforest has size at most . To proof Theorem 1 it remains to show that the bound is tight:
Proof of Theorem 1.
We show the remaining part of Theorem 1, namely that the bound is tight.
Consider the connected graph . It holds that is a quasiforest, because any vertices from are a feedback vertex set. It holds that the size of a maximum independent set in a clique is 1, hence for all subsets . Therefore, is the only, and hence a minimal, blocking set in . ∎
Recall that Reduction Rule 3 considers the conflicts that a connected component of the quasiforest has with subsets of . So far, we only talked about the size of minimal blocking sets instead of the size of minimal subset of that leads to a conflict. Since every independent set that has a conflict with , has some neighbors in this component, we know that these vertices are a blocking set of . Using Lemma 3 we can argue that only a subset of at most vertices (of the neighborhood of in ) is important. Like Jansen and Bodlaender [13] resp. Fomin and Strømme [9] we show how a smaller subset of leads to a smaller subset of that has a conflict with the connected component .
Lemma 4.
Let be an instance of independent set parameterized by a modulator to a quasiforest. Let be a connected component of and let be an independent set in . If , then there exists a set of size at most such that .
Proof.
Let be the neighborhood of in the connected component ; it holds that , because . Let be a minimal blocking set. It follows from Lemma 3 that .
We pick for every vertex an arbitrary neighbor in . Let . Clearly, it holds that and that , which implies that ; thus has the desired properties. ∎
We showed that if a connected component of has a conflict with a subset of the modulator, then there always exists a set of size at most that has a conflict with the connected component . Knowing this, we can show that Reduction Rule 3 is safe using Lemma 4 as well as some observations that where already used in earlier work [9, 13].
Lemma 5.
Proof.
Let be the connected component of that we delete by applying Reduction Rule 3. For the forward direction of the proof, we assume that has a solution, thus there exists an independent set of size at least in . Consider the set . Clearly, is an independent set of of size at least , because is an independent set in . Therefore, has a solution, namely .
For the backward direction of the proof, we assume that has a solution, thus there exists an independent set of size at least in . First, we will show that there always exists an independent set which can only contain an entire set if this set induces strictly less than conflicts in . This was already shown by Jansen and Bodlaender [13].
Let be an arbitrary independent set in of size at least . Assume that there exists a set such that . We will construct an independent set of the same size that contains no vertex from and therefore fulfills the desired property. Since , we know that . The set is an independent set in , because is an independent set in and which implies that . Thus, we know that . Combining all this leads to
Thus, every maximum independent set of has at least the cardinality of and fulfills the desired properties.
Now, we can assume that is an independent set that can only contain an entire set when this set induces strictly less than conflicts in . We will show that we can extend to an independent set of size at least by adding a maximum independent set of to . More precisely, we will show that ; note that it suffices to show that . Observe that, if , then there exists a maximum independent set in that uses no vertex from the set , therefore is an independent set in of size at least .
Now, we assume for contradiction that which implies that . From Lemma 4 it follows that there exists a set of size at most such that . Since we assumed that can only contain subsets of that induce less than conflicts in (and ) it holds that . But this contradicts the requirements of Reduction Rule 3: is an independent set of size at most with and . Thus, the assumption is wrong and we have . ∎
Recall that if we have an instance of independent set parameterized by a modulator to a quasiforest where Reduction Rule 3 is not applicable then has at most connected components. To apply the kernelization for independent set parameterized by a modulator to a forest from Jansen and Bodlaender [13], we have to add vertices from each connected component of the quasiforest to the modulator , getting a set , such that the connected components of are trees.
We know that every connected component of the quasiforest has a feedback vertex set of size at most , which we can find in polynomial time, since is a constant. Let be the union of these feedback vertex sets; it holds that . Now, the instance with , and is an instance of independent set parameterized by a modulator to feedback vertex set. Obviously, it holds that has a solution if and only if has a solution. Applying the following result of Jansen and Bodlaender [13] will finish our kernelization.
Proposition 1 ([13, Theorem 2]).
independent set parameterized by a modulator to a feedback vertex set has a kernel with a cubic number of vertices: there is a polynomialtime algorithm that transforms an instance into an equivalent instance such that and .
Theorem 2.
independent set parameterized by a modulator to a quasiforest admits a kernel with vertices.
Proof.
Given an input instance of independent set parameterized by a modulator to a quasiforest, we first apply Reduction Rule 3 exhaustively to obtain an equivalent instance of independent set parameterized by a modulator to a quasiforest where the quasiforest has at most connected components (Lemma 1). The fact that the instances are equivalent follows from Lemma 5. Furthermore, we can compute the instance in polynomial time: Every application of Reduction Rule 3 deletes a connected component of and decreases the value of appropriately, hence we apply this rule at most times. To apply Reduction Rule 3, we have to find a connected component of that only has a conflict with an independent set , when the set has a large conflict in the quasiforest . Thus, for every connected component of (at most ) and the quasiforest we have to compute for at most sets the value resp. . Since is a constant we can compute and for all sets in polynomial time and we can easily check whether a connected component fulfills the properties of Reduction Rule 3. Summarized, instance is equivalent to instance and we can compute this instance in polynomial time.
Now, we add for each of the at most connected components of a feedback vertex set of size at most to ; let be the union of these feedback vertex sets. We add the vertex set to the modulator to obtain an instance of independent set parameterized by a modulator to a forest.
The instances and are obviously equivalent. To prove that we can construct in polynomial time, we only have to show that we can find the set in polynomial time; this holds, because we can find a feedback vertex of constant size in polynomial time.
Finally, we apply the kernelization algorithm of Proposition 1 to the instance of independent set parameterized by a modulator to a forest and obtain an equivalent instance in polynomial time.
So far, we know that we can compute instance of independent set parameterized by a modulator to a feedback vertex set, which is equivalent to the instance of independent set parameterized by a modulator to a quasiforest, in polynomial time. It remains to bound the size of , and .
We never increase the size of , we only decrease the size of in Reduction Rule 3 and the application of Proposition 1, hence . Next, we bound the size of . We increase the cardinality of the set twice, once by adding the feedback vertex set of the quasiforest to the modulator and once (by a factor of two) by applying Proposition 1. This leads to the following bound for the size of : .
Finally, we have to bound the number of vertices in . It follows from applying Proposition 1 to the instance of independent set parameterized by a modulator to a forest that the reduced instance has at most vertices. This leads to the desired bound for :