Fast algorithms for some dominating induced matching problems

• Determine bounds on the number of edges of some graphs having DIMs.
• In O(n)O(n) time, find values of DIMs for several classes of graphs.
• Improve to O(n)O(n) time a known quadratic algorithm for claw-free graphs.

We describe O(n)O(n) time algorithms for finding the minimum weighted dominating induced matching of chordal, dually chordal, biconvex, and claw-free graphs. For the first three classes, we prove tight O(n)O(n) bounds on the maximum number of edges that a graph having a dominating induced matching may contain. By applying these bounds, and employing existing O(n+m)O(n+m) time algorithms we show that they can be reduced to O(n)O(n) time. For claw-free graphs, we describe a variation of the existing algorithm for solving the unweighted version of the problem, which decreases its complexity from O(n2)O(n2) to O(n)O(n), while additionally solving the weighted version. The same algorithm can be easily modified to count the number of DIM's of the given graph.