A graph $G$ is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree and an interval $I$, such that each leaf of the tree is a vertex of the graph, and there is an edge $\{ x, y \}$ in $G$ if and only if the weight of the path in the tree connecting $x$ and $y$ lies within the interval $I$. Originating in phylogenetics, PCGs are closely connected to important graph classes like leaf-powers and multi-threshold graphs, widely applied in bioinformatics, especially in understanding evolutionary processes. In this paper we introduce two natural generalizations of the PCG class, namely $k$-OR-PCG and $k$-AND-PCG, which are the classes of graphs that can be expressed as union and intersection, respectively, of $k$ PCGs. These classes can be also described using the concepts of the covering number and the intersection dimension of a graph in relation to the PCG class. We investigate how the classes of OR-PCG and AND-PCG are related to PCGs, $k$-interval-PCGs and other graph classes known in the literature. In particular, we provide upper bounds on the minimum $k$ for which an arbitrary graph $G$ belongs to $k$-interval-PCGs, $k$-OR-PCG or $k$-AND-PCG classes. For particular graph classes we improve these general bounds. Moreover, we show that, for every integer $k$, there exists a bipartite graph that is not in the $k$-interval-PCGs class, proving that there is no finite $k$ for which the $k$-interval-PCG class contains all the graphs. This answers an open question of Ahmed and Rahman from 2017. Finally, using a Ramsey theory argument, we show that for any $k$, there exists graphs that are not in $k$-AND-PCG, and graphs that are not in $k$-OR-PCG.

翻译：若存在一棵带权树和一个区间$I$，使得该树的每个叶子都是图的一个顶点，并且当且仅当树中连接$x$和$y$的路径权值位于区间$I$内时，图$G$中存在边$\{ x, y \}$，则称图$G$为成对兼容图（PCG）。PCG起源于系统发育学，与叶幂图和多阈值图等重要图类密切相关，广泛应用于生物信息学，特别是在理解进化过程中。本文引入了PCG类的两种自然推广，即$k$-OR-PCG和$k$-AND-PCG，它们分别是能够表示为$k$个PCG的并集和交集的图类。这些类也可以用图关于PCG类的覆盖数和交维数的概念来描述。我们研究了OR-PCG类和AND-PCG类如何与PCG、$k$-区间-PCG以及文献中已知的其他图类相关联。特别地，我们给出了任意图$G$属于$k$-区间-PCG、$k$-OR-PCG或$k$-AND-PCG类所需的最小$k$的上界。对于特定的图类，我们改进了这些一般性上界。此外，我们证明对于每个整数$k$，都存在一个不属于$k$-区间-PCG类的二部图，从而证明不存在有限的$k$使得$k$-区间-PCG类包含所有图。这回答了Ahmed和Rahman于2017年提出的一个开放性问题。最后，利用Ramsey理论论证，我们证明了对于任意$k$，都存在不属于$k$-AND-PCG的图，以及不属于$k$-OR-PCG的图。