Dominating induced matching in some subclasses of bipartite graphs

A subset M subset of E of edges of a graph G = (V, E) is called a matching if no two edges of M share a common vertex. An edge e epsilon E is said to dominate itself and all other edges adjacent to it. A matching M in a graph G = (V, E) is called a dominating induced matching (d.i.m.) if every edge of G is dominated by edges of M exactly once. The dominating induced matching decide (DIM-DECIDE) problem asks whether a graph G contains a dominating induced matching. The dominating induced matching (DIM) problem asks to compute a dominating induced matching (d.i.m.) in a graph G that admits a dominating induced matching. The DIM-Decide problem is known to be NP-complete for general graphs as well as for bipartite graphs. In this paper, we strengthen the NP-completeness result of the DIM-DECIDE problem by showing that this problem remains NP-complete for perfect elimination bipartite graphs, a proper subclass of bipartite graphs. On the positive side, we characterize the class of star-convex bipartite graphs admitting a d.i.m. This characterization leads to a linear time algorithm to test whether a star-convex bipartite graph admits a d.i.m. and, if so, constructs a d.i.m. in such a star-convex bipartite graph in linear time. We also propose polynomial time algorithms to construct a d.i.m. in long-k-star-convex bipartite graphs as well as in circular-convex bipartite graphs if the input graph admits a d.i.m. (C) 2021 Elsevier B.V. All rights reserved.