Finding Dominating Induced Matchings in $$P_8$$ P 8 -Free Graphs in Polynomial Time

Abstract

Let $$G=(V,E)$$G=(V,E) be a finite undirected graph. An edge set $$E' \subseteq E$$E?⊆E is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of $$E'$$E?. The Dominating Induced Matching (DIM) problem asks for the existence of a d.i.m. in G; this problem is also known as the Efficient Edge Domination problem. The DIM problem is related to parallel resource allocation problems, encoding theory and network routing. It is $${\mathbb {NP}}$$NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three and is solvable in linear time for $$P_7$$P7-free graphs. However, its complexity was open for $$P_k$$Pk-free graphs for any $$k \ge 8$$k?8; $$P_k$$Pk denotes the chordless path with k vertices and $$k-1$$k-1 edges. We show in this paper that the weighted DIM problem is solvable in polynomial time for $$P_8$$P8-free graphs.