Domination in some subclasses of bipartite graphs

A set D subset of V is called a dominating set of G = (V, E) if vertical bar N-G[v] boolean AND D vertical bar >= 1 for all v is an element of V. The MINIMUM DOMINATION problem is to find a dominating set of minimum cardinality of the input graph. In this paper, we study the MINIMUM DOMINATION problem for star-convex bipartite graphs, circular-convex bipartite graphs and triad-convex bipartite graphs. It is known that the MINIMUM DOMINATION PROBLEM for a graph with n vertices can be approximated with an approximation ratio of In n+1. However, we show that for any epsilon > 0, the MINIMUM DOMINATION problem does not admit a (1 - epsilon) In n-approximation algorithm even for star-convex bipartite graphs with n vertices unless NP subset of DTIME(n(O(log log) (n))). On the positive side, we propose polynomial time algorithms for computing a minimum dominating set of circular-convex bipartite graphs and triad-convex bipartite graphs, by making polynomial time Turing reductions from the MINIMUM DOMINATION problem for these graph classes to the MINIMUM DOMINATION problem for convex bipartite graphs. (C) 2018 Elsevier B.V. All rights reserved.