On Generalizations of Pairwise Compatibility Graphs
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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 covering number and the 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.
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