A Linear Kernel for Planar Total Dominating Set

A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general graphs when parameterized by the solution size. By the meta-theorem of Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how such a kernel can be effectively constructed, and how to obtain explicit reduction rules with reasonably small constants. Following the approach of Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating Set on planar graphs with at most $410k$ vertices, where $k$ is the size of the solution. This result complements several known constructive linear kernels on planar graphs for other domination problems such as Dominating Set, Edge Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue Dominating Set.Comment: 33 pages, 13 figure

Hitting Forbidden Induced Subgraphs on Bounded Treewidth Graphs

For a fixed graph H, the H-IS-Deletion problem asks, given a graph G, for the minimum size of a set S ? V(G) such that G? S does not contain H as an induced subgraph. Motivated by previous work about hitting (topological) minors and subgraphs on bounded treewidth graphs, we are interested in determining, for a fixed graph H, the smallest function f_H(t) such that H-IS-Deletion can be solved in time f_H(t) ? n^{?(1)} assuming the Exponential Time Hypothesis (ETH), where t and n denote the treewidth and the number of vertices of the input graph, respectively. We show that f_H(t) = 2^{?(t^{h-2})} for every graph H on h ? 3 vertices, and that f_H(t) = 2^{?(t)} if H is a clique or an independent set. We present a number of lower bounds by generalizing a reduction of Cygan et al. [MFCS 2014] for the subgraph version. In particular, we show that when H deviates slightly from a clique, the function f_H(t) suffers a sharp jump: if H is obtained from a clique of size h by removing one edge, then f_H(t) = 2^{?(t^{h-2})}. We also show that f_H(t) = 2^{?(t^{h})} when H = K_{h,h}, and this reduction answers an open question of Mi. Pilipczuk [MFCS 2011] about the function f_{C?}(t) for the subgraph version. Motivated by Cygan et al. [MFCS 2014], we also consider the colorful variant of the problem, where each vertex of G is colored with some color from V(H) and we require to hit only induced copies of H with matching colors. In this case, we determine, under the ETH, the function f_H(t) for every connected graph H on h vertices: if h ? 2 the problem can be solved in polynomial time; if h ? 3, f_H(t) = 2^{?(t)} if H is a clique, and f_H(t) = 2^{?(t^{h-2})} otherwise

How Much Does a Treedepth Modulator Help to Obtain Polynomial Kernels Beyond Sparse Graphs?

In the last years, kernelization with structural parameters has been an active area of research within the field of parameterized complexity. As a relevant example, Gajarsky et al. [ESA 2013] proved that every graph problem satisfying a property called finite integer index admits a linear kernel on graphs of bounded expansion and an almost linear kernel on nowhere dense graphs, parameterized by the size of a c-treedepth modulator, which is a vertex set whose removal results in a graph of treedepth at most c for a fixed integer c>0. The authors left as further research to investigate this parameter on general graphs, and in particular to find problems that, while admitting polynomial kernels on sparse graphs, behave differently on general graphs. In this article we answer this question by finding two very natural such problems: we prove that VERTEX COVER admits a polynomial kernel on general graphs for any integer c>0, and that DOMINATING SET does not for any integer c>1 even on degenerate graphs, unless NP is a subset of coNP/poly. For the positive result, we build on the techniques of Jansen and Bodlaender [STACS 2011], and for the negative result we use a polynomial parameter transformation for c>2 and an OR-cross-composition for c=2. As existing results imply that DOMINATING SET admits a polynomial kernel on degenerate graphs for c=1, our result provides a dichotomy about the existence of polynomial problems for DOMINATING SET on degenerate graphs with this parameter