Parameterized complexity of three edge contraction problems with degree constraints

Abstract

For any graph class $\mathcal{H}$, the $\mathcal{H}$-Contraction problem takes as input a graph $G$ and an integer $k$, and asks whether there exists a graph $H\in \mathcal{H}$ such that $G$ can be modified into $H$ using at most $k$ edge contractions. We study the parameterized complexity of $\mathcal{H}$-Contraction for three different classes $\mathcal{H}$: the class $\mathcal{H}_{\le d}$ of graphs with maximum degree at most $d$, the class $\mathcal{H}_{=d}$ of $d$-regular graphs, and the class of $d$-degenerate graphs. We completely classify the parameterized complexity of all three problems with respect to the parameters $k$, $d$, and $d+k$. Moreover, we show that $\mathcal{H}$-Contraction admits an $O(k)$ vertex kernel on connected graphs when $\mathcal{H}\in \{\mathcal{H}_{\le 2},\mathcal{H}_{=2}\}$, while the problem is $\mathsf{W}[2]$-hard when $\mathcal{H}$ is the class of $2$-degenerate graphs and hence is expected not to admit a kernel at all. In particular, our results imply that $\mathcal{H}$-Contraction admits a linear vertex kernel when $\mathcal{H}$ is the class of cycles.