On Threshold Compatibility Graphs

Pairwise Compatibility Graphs (PCGs) form a tree-metric graph class that originated in phylogeny and has since attracted sustained interest in graph theory. Several natural generalizations have been proposed in order to overcome the expressive limitations of classical PCGs, including $k$-interval-PCGs, $k$-OR-PCGs, and $k$-AND-PCGs. In this paper, we introduce $(k,t)$-threshold-PCGs, a threshold-based framework that unifies these generalized notions: adjacency is determined by whether at least $t$ among $k$ underlying PCG predicates accept the vertex pair. We investigate the expressive power of this model from both constructive and asymptotic viewpoints. On the positive side, we show that every graph on $n$ vertices is a $(n,t)$-threshold-PCG for every $1 \le t \le n$. On the negative side, we prove that for every fixed pair $(k,t)$, the class of $(k,t)$-threshold-PCGs is asymptotically rare among all graphs. As a consequence, we obtain sharp separations from previously studied models, including a strict expressive gap relative to $k$-interval-PCGs. We also study explicit obstruction families through incidence graphs and derive additional structural consequences for the conjunction case, including the strictness of the $k$-AND-PCG hierarchy and the failure of closure under complement.