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Linear-Time Algorithm for the Paired-Domination Problem in Convex Bipartite Graphs

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Abstract

A bipartite graph G=(U,W,E) with vertex set V=UW is convex if there exists an ordering of the vertices of W such that for each uU, the neighbors of u are consecutive in W. A compact representation of a convex bipartite graph for specifying such an ordering can be computed in O(|V|+|E|) time. The paired-domination problem on bipartite graphs has been shown to be NP-complete. The complexity of the paired-domination problem on convex bipartite graphs has remained unknown. In this paper, we present an O(|V|) time algorithm to solve the paired-domination problem on convex bipartite graphs given a compact representation. As a byproduct, we show that our algorithm can be directly applied to solve the total domination problem on convex bipartite graphs in the same time bound.

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Acknowledgements

The author gratefully acknowledges the helpful comments and suggestions of the reviewers, which have improved the presentation. We also deeply appreciate the anonymous reviewers for giving the comment that the proposed approach can be applied to the total domination problem in convex bipartite graphs.

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Correspondence to Ruo-Wei Hung.

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A preliminary version of this paper has appeared in: Proceedings of the International MultiConference of Engineers and Computer Scientists 2010 (IMECS’2010), Hong Kong, vol. I, pp. 365–369 (2010) [24].

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Hung, RW. Linear-Time Algorithm for the Paired-Domination Problem in Convex Bipartite Graphs. Theory Comput Syst 50, 721–738 (2012). https://doi.org/10.1007/s00224-011-9378-8

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