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.

翻译：成对兼容图（PCGs）是一类源于系统发育学的树度量图类，此后在图论领域持续引发研究兴趣。为突破经典PCG的表达局限，学界已提出多种自然推广形式，包括$k$-区间PCG、$k$-或PCG及$k$-与PCG。本文引入$(k,t)$-阈值PCG这一基于阈值的统一框架：当$k$个底层PCG谓词中至少有$t$个接受某顶点对时，该顶点对即构成邻接关系。我们从构造性与渐近性双重视角探究该模型的表达能力。正面结论表明，对于任意$1 \le t \le n$，所有$n$顶点图均可表示为$(n,t)$-阈值PCG；负面结论则证明，对任意固定对$(k,t)$，$(k,t)$-阈值PCG类在全体图中渐近稀疏。由此推导出与既有模型的严格分离关系，包括与$k$-区间PCG间存在严格表达差距。我们还通过关联图研究显式障碍族，并对合取情形导出额外结构结论，涵盖$k$-与PCG层次的严格性及补运算下非封闭性等性质。