Domination in some subclasses of bipartite graphs

Abstract

A set D ⊆ V is called a dominating set of G = (V, E) if |NG[v] ∩ D| ≥ 1 for all v ∈ 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 ln n+1. However, we show that for any ϵ > 0, the Minimum Domination problem does not admit a (1−ϵ) ln n-approximation algorithm even for star-convex bipartite graphs with n vertices unless NP ⊆ 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