On graph contractions and induced minors

Abstract

The Induced Minor Containment problem takes as input two graphs $G$ and $H$, and asks whether $G$ has $H$ as an induced minor. We show that this problem is fixed parameter tractable in $\left|V_H\right|$ if $G$ belongs to any nontrivial minor-closed graph class and $H$ is a planar graph. For a fixed graph $H$, the $H$-Contractibility problem is to decide whether a graph can be contracted to $H$. The computational complexity classification of this problem is still open. So far, $H$ has a dominating vertex in all cases known to be solvable in polynomial time, whereas $H$ does not have such a vertex in all cases known to be NP-complete. Here, we present a class of graphs $H$ with a dominating vertex for which $H$-Contractibility is NP-complete. We also present a new class of graphs $H$ for which $H$-Contractibility can be solved in polynomial time. Finally, we study the $(H,v)$-Contractibility problem, where $v$ is a vertex of $H$. The input of this problem is a graph $G$ and an integer $k$, and the question is whether $G$ is $H$-contractible such that the “bag” of $G$ corresponding to $v$ contains at least $k$ vertices. We show that this problem is NP-complete whenever $H$ is connected and $v$ is not a dominating vertex of $H$.