Let $G=(V,E)$ be a finite undirected graph. An edge set $E' \subseteq E$ is a { \em dominating induced matching} ({ \em d.i.m.}) in $G$ if every edge in $E$ is intersected by exactly one edge of $E'$. The \emph{ Dominating Induced Matching} (\emph{ DIM}) problem asks for the existence of a d.i.m.\ in $G$; this problem is also known as the \emph{ Efficient Edge Domination} problem; it is the Efficient Domination problem for line graphs. The DIM problem is \NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree 3 but is solvable in linear time for $P_7$-free graphs, and in polynomial time for $S_{ 1,2,4}$-free graphs as well as for $S_{ 2,2,2}$-free graphs and for $S_{ 2,2,3}$-free graphs. In this paper, combining two distinct approaches, we solve it in polynomial time for $S_{ 1,1,5}$-free graphs.
Categories: cs.DM math.CO