Parameterized complexity of three edge contraction problems with degree constraints

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.