Francis, MC and Jacob, D (2023) The lexicographic method for the threshold cover problem. In: Discrete Mathematics, 346 (6).
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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.
| Item Type: | Journal Article |
|---|---|
| Publication: | Discrete Mathematics |
| Publisher: | Elsevier B.V. |
| Additional Information: | The copyright for this article belongs to Elsevier B.V. |
| Keywords: | Chain subgraph cover; Lexicographic method; Threshold cover |
| Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |
| Date Deposited: | 21 Feb 2023 03:16 |
| Last Modified: | 21 Feb 2023 03:16 |
| URI: | https://eprints.iisc.ac.in/id/eprint/80538 |
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