Dominating induced matchings in graphs without a skew star

We study the problem of determining whether a graph G has an induced matching that dominates every edge of the graph, which is also known as efficient edge domination. This problem is known to be NP-complete in general graphs, but it can be solved in polynomial time for graphs in some special classes, such as weakly chordal, P7P7-free or claw-free graphs. In the present paper we extend the polynomial-time solvability of the problem from claw-free graphs to graphs without a skew star, where a skew star is a tree with exactly three vertices of degree 1 being of distance 1,2,31,2,3 from the only vertex of degree 3.