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.