The Lexicographic Method for the Threshold Cover Problem

TL;DR

This paper introduces a new, simpler proof for the threshold dimension of graphs using the lexicographic method, improving understanding and efficiency especially for split graphs.

Contribution

The paper applies the lexicographic method to provide a novel, simpler proof for the threshold dimension of graphs, notably simplifying the case for split graphs.

Findings

The lexicographic method can be used to prove threshold dimension results.

The proof for bipartite auxiliary graphs is simplified.

The method yields a shorter proof for split graphs.

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. Chv\'atal and Hammer, in 1977, defined the threshold dimension $\mathrm{th}(G)$ of a graph $G$ to be the least integer $k$ such that $G$ has a threshold cover of size $k$ and observed that $\mathrm{th}(G)\geq\chi(G^*)$, where $G^*$ is a suitably constructed auxiliary graph. Raschle and Simon~[Proceedings of the Twenty-seventh Annual ACM Symposium on Theory of Computing, STOC '95, pages 650--661, 1995] proved that $\mathrm{th}(G)=\chi(G^*)$ 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.