# 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.

## Keywords

Hamiltonian Cycle Maximum Clique Intersection Graph Outerplanar Graph Planar Embedding## Preview

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