Department of Applied Mathematics & Physics, Kyoto University

Technical Report 2014-002 (August 19, 2014)

Exact Algorithms for Dominating Induced Matching Based on Graph Partition
by Mingyu Xiao and Hiroshi Nagamochi

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A dominating induced matching, also called an efficient edge domination, of a graph $G=(V,E)$ with $n=|V|$ vertices and $m=|E|$ edges is a subset $F \subseteq E$ of edges in the graph such that no two edges in $F$ share a common endpoint and each edge in $E\setminus F$ is incident with exactly one edge in $F$. It is NP-hard to decide whether a graph admits a dominating induced matching or not. In this paper, we design a $1.1467 ^nn^{O(1)}$-time exact algorithm for this problem, improving all previous results. This problem can be redefined as a partition problem that is to partition the vertex set of a graph into two parts $I$ and $F$, where $I$ induces an independent set (a 0-regular graph) and $F$ induces a perfect matching (a 1-regular graph). After giving several structural properties of the problem, we show that the problem always contains some ``good vertices,'' branching on which by including them to either $I$ or $F$ we can effectively reduce the graph. This leads to a fast exact algorithm to this problem.