A graph $G$ is a PCG if there exists an edge-weighted tree 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 a given interval. PCGs have different applications in phylogenetics and have been lately generalized to multi-interval-PCGs. In this paper we define two new generalizations of the PCG class, namely k-OR-PCGs and k-AND-PCGs, that are the classes of graphs that can be expressed as union and intersection, respectively, of $k$ PCGs. The problems we consider can be also described in terms of the \emph{covering number} and the \emph{intersection dimension} of a graph with respect to the PCG class. In this paper we investigate how the classes of PCG, multi-interval-PCG, OR-PCG and AND-PCG are related to each other and to 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-PCG, k-OR-PCG and k-AND-PCG classes. Furthermore, 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-PCG class, proving that there is no finite $k$ for which the k-interval-PCG class contains all the graphs. Finally, we use a Ramsey theory argument to show that for any $k$, there exist graphs that are not in k-AND-PCG, and graphs that are not in k-OR-PCG.

翻译：图$G$称为PCG，若存在一棵边赋权树，使得树的每个叶子对应图的一个顶点，且图$G$中存在边$\{ x, y \}$当且仅当树中连接$x$和$y$的路径权重落在给定区间内。PCG在系统发育学中有多种应用，并最近被推广为多区间PCG。本文定义了PCG类的两种新推广，即k-OR-PCG和k-AND-PCG，分别表示可由$k$个PCG的并集和交集表示的图类。所考虑的问题也可从图相对于PCG类的覆盖数和交维数角度描述。本文研究了PCG、多区间PCG、OR-PCG和AND-PCG类之间以及与文献中已知其他图类的关系。特别地，我们给出了任意图$G$属于k-区间PCG、k-OR-PCG和k-AND-PCG类所需最小$k$的上界。此外，针对特定图类，我们改进了这些一般上界。同时，我们证明对每个整数$k$，存在一个二分图不属于k-区间PCG类，从而表明不存在有限$k$使得k-区间PCG类包含所有图。最后，利用拉姆齐理论论证，证明对任意$k$，存在不属于k-AND-PCG的图，也存在不属于k-OR-PCG的图。