A Min–Max Property of Chordal Bipartite Graphs with Applications

Authors:
Atif Abueida;Arthur H. Busch;R. Sritharan
Affiliations:
The University of Dayton, Department of Mathematics, 45469-2316, Dayton, OH, USA;The University of Dayton, Department of Mathematics, 45469-2316, Dayton, OH, USA;The University of Dayton, Computer Science Department, 45469, Dayton, OH, USA
Venue:
Graphs and Combinatorics
Year:
2010

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Abstract

We show that if G is a bipartite graph with no induced cycles on exactly 6 vertices, then the minimum number of chain subgraphs of G needed to cover E(G) equals the chromatic number of the complement of the square of line graph of G. Using this, we establish that for a chordal bipartite graph G, the minimum number of chain subgraphs of G needed to cover E(G) equals the size of a largest induced matching in G, and also that a minimum chain subgraph cover can be computed in polynomial time. The problems of computing a minimum chain cover and a largest induced matching are NP-hard for general bipartite graphs. Finally, we show that our results can be used to efficiently compute a minimum chain subgraph cover when the input is an interval bigraph.