Kernels for Structural Parameterizations of Vertex Cover - Case of Small Degree Modulators

Abstract

Vertex Cover is one of the most well studied problems in the realm of parameterized algorithms and admits a kernel with O(l^2) edges and 2*l vertices. Here, l denotes the size of a vertex cover we are seeking for. A natural question is whether Vertex Cover admits a polynomial kernel (or a parameterized algorithm) with respect to a parameter k, that is, provably smaller than the size of the vertex cover. Jansen and Bodlaender [STACS 2011, TOCS 2013] raised this question and gave a kernel for Vertex Cover of size O(f^3), where f is the size of a feedback vertex set of the input graph. We continue this line of work and study Vertex Cover with respect to a parameter that is always smaller than the solution size and incomparable to the size of the feedback vertex set of the input graph. Our parameter is the number of vertices whose removal results in a graph of maximum degree two.  While vertex cover with this parameterization can easily be shown to be fixed-parameter tractable (FPT), we show that it has a polynomial sized kernel. 

The input to our problem consists of an undirected graph G, S \subseteq V(G) such that |S| = k and G[V(G)\S] has maximum degree at most 2 and a positive integer l. Given (G,S,l), in polynomial time we output an instance (G',S',l') such that |V(G')|<= O(k^5), |E(G')|<= O(k^6) and G has a vertex cover of size at most l if and only if G' has a vertex cover of size at most l'. 
When G[V(G)\S] has maximum degree at most 1, we improve the known kernel bound from O(k^3) vertices to O(k^2) vertices (and O(k^3) edges).  In general, if G[V(G)\S] is simply a collection of cliques of size at most d, then we transform the graph in polynomial time to an equivalent hypergraph with O(k^d) vertices and show that, for d >= 3, a kernel with O(k^{d-epsilon}) vertices is unlikely to exist for any epsilon >0 unless NP is a subset of coNO/poly.