Polynomial Kernels for Vertex Cover Parameterized by Small Degree Modulators

Vertex Cover is one of the most well studied problems in the realm of parameterized algorithms. It admits a kernel with \(\phantom {\dot {i}\!}\mathcal {O}(\ell ^{2})\) edges and \(\phantom {\dot {i}\!}2\ell \) vertices where \(\phantom {\dot {i}\!}\ell \) denotes the size of the vertex cover we are seeking for. A natural question is whether Vertex Cover is fixed-parameter tractable or admits a polynomial kernel 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 \(\phantom {\dot {i}\!}\mathcal {O}(f^{3})\), 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 kernel. The input to our problem consists of an undirected graph G, \(\phantom {\dot {i}\!}S \subseteq V(G)\) such that \(\phantom {\dot {i}\!}|S| = k\) and \(\phantom {\dot {i}\!}G[V(G)\setminus S]\) has maximum degree at most two and a positive integer \(\phantom {\dot {i}\!}\ell \). Given \(\phantom {\dot {i}\!}(G,S,\ell )\), in polynomial time we output an instance \(\phantom {\dot {i}\!}(G^{\prime },S^{\prime },\ell ^{\prime })\) such that \(\phantom {\dot {i}\!}|V(G^{\prime })|\) is \(\phantom {\dot {i}\!}\mathcal {O}(k^{5})\), \(\phantom {\dot {i}\!}|E(G^{\prime })|\) is \(\phantom {\dot {i}\!}\mathcal {O}(k^{6})\) and G has a vertex cover of size at most \(\phantom {\dot {i}\!}\ell \) if and only if \(\phantom {\dot {i}\!}G^{\prime }\) has a vertex cover of size at most \(\phantom {\dot {i}\!}\ell ^{\prime }\). When \(\phantom {\dot {i}\!}G[V(G)\setminus S]\) has maximum degree at most one, we improve the known kernel bound from \(\phantom {\dot {i}\!}\mathcal {O}(k^{3})\) vertices to \(\phantom {\dot {i}\!}\mathcal {O}(k^{2})\) vertices (and \(\phantom {\dot {i}\!}\mathcal {O}(k^{3})\) edges). More generally, given \(\phantom {\dot {i}\!}(G, S, \ell )\) where every connected component of \(\phantom {\dot {i}\!}G \setminus S\) is a clique of at most d vertices (for constant d), in polynomial time, we output an equivalent instance \(\phantom {\dot {i}\!}(G^{\prime }, S^{\prime }, \ell ^{\prime })\) for the same problem where \(\phantom {\dot {i}\!}|V(G^{\prime })|\) is \(\phantom {\dot {i}\!}\mathcal {O}(k^{d})\). We also show that for this problem, when \(\phantom {\dot {i}\!}d \geq 3\), a kernel with \(\phantom {\dot {i}\!}\mathcal {O}(k^{d-\varepsilon })\) bits cannot exist for any \(\phantom {\dot {i}\!}\varepsilon >0\) unless \(\phantom {\dot {i}\!}\textsf {NP} \subseteq \textsf {coNP}/\textsf {poly}\).