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

Abstract

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, Gajarský et al. (J Comput Syst Sci 84:219–242, 2017) 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, where $c \ge 1$ is a fixed integer. 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 $c \ge 1$, and that Dominating Set does not for any integer $c \ge 2$ even on degenerate graphs, unless $\text {NP} \subseteq \text {coNP}/\text {poly}$. For the positive result, we build on the techniques of Jansen and Bodlaender (Proceedings of the 28th symposium on theoretical aspects of computer science (STACS), volume 9 of LIPIcs, pp 177–188, 2011), and for the negative result we use a polynomial parameter transformation for $c\ge 3$ 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 kernels for Dominating Set on degenerate graphs with this parameter.

Appendices

List of Problems Considered in this Article

Stronger Negative Results for Dominating Set

In this section we rule out the existence of polynomial kernels for Dominating Set on graphs of bounded expansion for a parameter that is smaller than the size of a c-treedepth modulator. Let $b: \mathbb {N} \rightarrow R$ be a function. A b-treedepth modulator of a graph $G=(V,E)$ is a subset of vertices $X \subseteq V$ such that $\mathsf{td}(G[V{\setminus } X]) \le b(|X|)$, and we denote by b-$\mathsf{tdmod} (G)$ the size of a smallest b-treedepth modulator of G. Note that, in the particular case where the function b is constantly equal to a positive integer c, b-treedepth modulators correspond exactly to c-treedepth modulators. In the following proposition we show that $\textsc {DS}$ does not admit polynomial kernels on graphs of bounded expansion parameterized by the size of a b-treedepth modulator with $b(x) = \Omega (\log x)$.

Proposition 4

$\textsc {DS}/\log$-$\mathsf{tdmod}$ does not admit a polynomial kernel on graphs of bounded expansion unless $\text {NP} \subseteq \text {coNP}/\text {poly}$.

Proof

As in the proof of Proposition 1, it is sufficient to prove that $\textsc {RBDS}/U \le _{\textsc {ppt}} \textsc {DS} _{C_{BE}}/\log$-tdmod, as $\textsc {RBDS}/U$ (and even $\textsc {RBDS}/(k+U)$) does not admit a polynomial kernel unless $\text {NP} \subseteq \text {coNP}/\text {poly}$ [11]. Let $(G=(U,W,E),k)$ be an instance of $\textsc {RBDS}/U$ with $u=|U|$, $w=|W|$, and $m=|E|$. Let $G'=G^{3u\text {-sub}} \cup \tilde{G}$, where $\tilde{G}$ is a square grid on $(3u+1)^4$ vertices plus a vertex $\alpha$ connected to all vertices of this grid. G has a RB-dominating set of size k if and only if $G'$ has a dominating set of size $k+um+1$ (as we also have to take $\alpha$ in the solutions of $G'$). Let $X=U \cup \tilde{V}$ where $\tilde{V}$ is the vertex set of $\tilde{G}$. Note that $G'[V'{\setminus } X]$ is a disjoint collection of spiders $S_v$ (one rooted at each $v \in W$) of height $3u+1$. As $\mathsf{td}(S_v) \le 1+\mathsf{td}(P_{3u}) \le 2+ \log (3u+1) \le 2\log (3u+1) = \mathcal {O}(\log |X|)$ and $\mathsf{td}(G') \ge \mathsf{td}(\tilde{G}) \ge (3u+1)^2$, we get that X is a log-treedepth modulator of $G'$, and that $\log$-tdmod$(G') \le |X| \le \text {poly}(|U|)$. To summarize, we added a large grid to artificially increase the treedepth of $G'$. Moreover, observe that we could not reduce directly from DS/VC as before, as we need a lower bound depending on VC of the form $\frac{1}{\text {poly}(u)}$. Let us finally verify that $G'$ has bounded expansion. As $\tilde{G}$ is an apex graph, it has bounded expansion (as, for instance, planar graphs are well-known to have bounded expansion, and the addition of an apex vertex preserves this property), and thus it remains to verify that $G^{3u\text {-sub}}$ has bounded expansion. Let $K=K^{3u\text {-sub}}_{u,w}$. As $G^{3u\text {-sub}} \subseteq K$ as a subgraph, it is sufficient to prove that K verifies the condition of bounded expansion.

To that end, we will prove that $\tilde{\nabla }_r(K)\le r+2$, where $\tilde{\nabla }_r(G)$ denotes the density of a depth-r topological minor using the notation of [26]. Let H be a depth-r topological minor of K. If $r<3u$, then H is clearly 2-degenerate. If $r\ge 3u$, observe that every vertex of K that was originally in W has degree at most u, and every subdivision vertex (i.e., a vertex which is not already a vertex of $K_{u,w}$) has degree 2. As in a topological minor a vertex cannot have a higher degree than in the original graph, and K is bipartite, we conclude that H is u-degenerate. Hence, taking into account both cases, we have that $\tilde{\nabla }_r(K)\le r+2$. This proves that the class $\{K_{u,w}^{3u\text {-sub}} : u,w\in \mathbb {N}\}$ has bounded expansion. Thus, this is a PPT reduction and we get the desired result. $\square$