Finding dominating induced matchings in P10-free graphs in polynomial time

Let G = (V, E) be a finite undirected graph. An edge set E ' c E is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of 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; 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 polynomial time for P9 -free graphs [and in linear time for P7 -free graphs] as well as for S1,2,4 -free, for S2,2,2 -free, and for S2,2,3 -free graphs. In this paper, combining two distinct approaches, we solve it in polynomial time for P10 -free graphs and introduce a partial result for the general case.