Abstract

This paper examines the problem of determining the threshold dimension of a graph. There exists numerous characterizations of threshold graphs, those graphs of threshold dimension one, as well as fast polynomial time algorithms to test if a graph is a threshold graph. Yannakakis [1982] proved that, in general, determining the threshold dimension is a hard problem by proving that for fixed $k\geqq 3$, determining if the threshold dimension of a graph is less than or equal to k is an NP-complete problem. In this paper we compute the threshold dimension of several classes of graphs and obtain upper and lower bounds on the threshold dimension of a graph. Counterexamples to a conjecture on threshold dimension are provided.

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