Abstract
A set D ⊆ V is called a dominating set of G = (V,E) if |N G [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 within ln n. However, we show that for any e > 0, the Minimum Domination problem does not admit a (1 − e)ln n-approximation algorithm for star-convex bipartite graphs with n vertices unless NP ⊆ DTIME(n O(loglogn)). 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.
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