Some Results on Multithreshold Graphs

Abstract

Jamison and Sprague defined a graph G to be a k-threshold graph with thresholds $\theta _1 , \ldots , \theta _k$ (strictly increasing) if one can assign real numbers $(r_v)_{v \in V(G)}$, called ranks, such that for every pair of vertices v, w, we have $vw \in E(G)$ if and only if the inequality $\theta _i \le r_v + r_w$ holds for an odd number of indices i. When $k=1$ or $k=2$, the precise choice of thresholds $\theta _1, \ldots , \theta _k$ does not matter, as a suitable transformation of the ranks transforms a representation with one choice of thresholds into a representation with any other choice of thresholds. Jamison asked whether this remained true for $k \ge 3$ or whether different thresholds define different classes of graphs for such k, offering $50 for a solution of the problem. Letting C[t] for $t > 1$ denote the class of 3-threshold graphs with thresholds $-1, 1, t$, we prove that there are infinitely many distinct classes C[t], answering Jamison’s question. We also consider some other problems on multithreshold graphs, some of which remain open.