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 within ln n. However, we show that for any epsilon > 0, the Minimum Domination problem does not admit a (1 - epsilon) ln n-approximation algorithm 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 polynomially reducing the Minimum Domination problem for these graph classes to the Minimum Domination problem for convex bipartite graphs.