Finding Dominating Induced Matchings in (S2,2,3, S1,4,4)-Free Graphs in Polynomial Time
نویسندگان
چکیده
Let G = (V,E) be a finite undirected graph without loops and multiple edges. An edge set 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. In particular, this means that E is an induced matching, and every edge not in E shares exactly one vertex with an edge in E. Clearly, not every graph has a d.i.m. 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 in general, and even for very restricted graph classes such as planar bipartite graphs with maximum degree 3. However, DIM is solvable in polynomial time for claw-free (i.e., S1,1,1-free) graphs, for S1,2,3-free graphs, for S1,2,4-free graphs, as well as for S2,2,2-free graphs, in linear time for P7-free graphs, and in polynomial time for P8-free graphs. In a paper by Hertz, Lozin, Ries, Zamaraev and de Werra, it was conjectured that DIM is solvable in polynomial time for Si,j,k-free graphs for every fixed i, j, k. In this paper we solve it in polynomial time for (S2,2,3, S1,4,4)-free graphs (generalizing S1,2,3-free graphs and based on the solution for S2,2,2-free graphs).
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