Graphs of Edge-Intersecting Non-splitting Paths in a Tree: Towards Hole Representations

Abstract

Given a tree and a set $\mathcal{P}$ of non-trivial simple paths on it, Vpt($\mathcal{P}$) is the VPT graph (i.e. the vertex intersection graph) of $\mathcal{P}$, and Ept($\mathcal{P}$) is the EPT graph (i.e. the edge intersection graph) of the paths $\mathcal{P}$ of the tree T. These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their split vertices is the set of vertices having degree at least 3 in their union. A pair of (edge) intersecting paths is termed non-splitting if they do not have split vertices (namely if their union is a path). In this work, we define the graph Enpt($\mathcal{P}$) of edge intersecting non-splitting paths of a tree, termed the ENPT graph, as the (edge) graph having a vertex for each path in $\mathcal{P}$, and an edge between every pair of paths that are both edge-intersecting and non-splitting. A graph G is an ENPT graph if there is a tree T and a set of paths $\mathcal{P}$ of T such that G = Ept $\mathcal{P}$, and we say that 〈T, ,$\mathcal{P}$〉 is a representation of G. We show that trees, cycles and complete graphs are ENPT graphs. We characterize the representations of chordless ENPT cycles that satisfy a certain assumption. Unlike chordless EPT cycles which have a unique representation, these representations turn out to be multiple and have a more complex structure. Therefore, in order to give this characterization, we assume the EPT graph induced by the vertices of a chordless ENPT cycle is given, and we provide an algorithm that returns the unique representation of this EPT, ENPT pair of graphs. These representations turn out to have a more complex structure than chordless EPT cycles.

This work was supported in part by the Israel Science Foundation grant No. 1249/08, by TUBITAK PIA BOSPHORUS grant No. 111M303, by BAP grant No. 6461 and by Catedra de Excelencia of Universidad Carlos III de Madrid.