The lexicographic method for the threshold cover problem
Article Type
Research Article
Publication Title
Discrete Mathematics
Abstract
Threshold graphs are a class of graphs that have many equivalent definitions and have applications in integer programming and set packing problems. A graph is said to have a threshold cover of size k if its edges can be covered using k threshold graphs. Chvátal and Hammer, in 1977, defined the threshold dimension th(G) of a graph G to be the least integer k such that G has a threshold cover of size k and observed that th(G)≥χ(G⁎), where G⁎ is a suitably constructed auxiliary graph. Raschle and Simon (1995) [9] proved that th(G)=χ(G⁎) whenever G⁎ is bipartite. We show how the lexicographic method of Hell and Huang can be used to obtain a completely new and, we believe, simpler proof for this result. For the case when G is a split graph, our method yields a proof that is much shorter than the ones known in the literature. Our methods give rise to a simple and straightforward algorithm to generate a 2-threshold cover of an input graph, if one exists.
DOI
https://10.1016/j.disc.2023.113364
Publication Date
6-1-2023
Recommended Citation
Francis, Mathew C. and Jacob, Dalu, "The lexicographic method for the threshold cover problem" (2023). Journal Articles. 3714.
https://digitalcommons.isical.ac.in/journal-articles/3714
Comments
Open Access, Green