Paths to Trees and Cacti

Abstract

For a family of graphs \(\mathcal F\), the \(\mathcal F\)-Contraction problem takes as an input a graph G and an integer k, and the goal is to decide whether there exists \(F \subseteq E(G)\) of size at most k such that G/F belongs to \(\mathcal F\). When \(\mathcal F\) is the family of paths, trees or cacti, then the corresponding problems are Path Contraction, Tree Contraction and Cactus Contraction, respectively. It is known that Tree Contraction and Cactus Contraction do not admit a polynomial kernel unless NP \(\subseteq \) coNP/poly, while Path Contraction admits a kernel with \(\mathcal {O}(k)\) vertices. The starting point of this article are the following natural questions: What is the structure of the family of paths that allows Path Contraction to admit a polynomial kernel? Apart from the size of the solution, what other additional parameters should we consider so that we can design polynomial kernels for these basic contraction problems? With the goal of designing polynomial kernels, we consider the family of trees with bounded number of leaves (note that the family of paths are trees with at most two leaves). In particular, we study Bounded Tree Contraction (Bounded TC). Here, an input is a graph G, integers k and \(\ell \), and the goal is to decide whether or not, there exists a subset \(F \subseteq E(G)\) of size at most k such that G/F is a tree with at most \(\ell \) leaves. We design a kernel for Bounded TC with \(\mathcal {O}(k\ell )\) vertices and \(\mathcal {O}(k^2+k\ell )\) edges. Finally, we study Bounded Cactus Contraction (Bounded CC) which takes as input a graph G and integers k and \(\ell \). The goal is to decide whether there exists a subset \(F \subseteq E(G)\) of size at most k such that G/F is a cactus graph with at most \(\ell \) leaf blocks in the corresponding block decomposition. For Bounded CC we design a kernel with \(\mathcal {O}(k^2 + k\ell )\) vertices and \(\mathcal {O}(k^2 + k\ell )\) edges. We complement our results by giving kernelization lower bounds for Bounded TC, Bounded OTC and Bounded CC by showing that unless NP \(\subseteq \) coNP/poly the size of the kernel we obtain is optimal.