Graphs of Edge-Intersecting Non-Splitting Paths in a Tree: Towards Hole Representations-Part I

Abstract: Given a tree and a set ${\cal P}$ of non-trivial simple paths on it, $VPT({\cal P})$ is the VPT graph (i.e. the vertex intersection graph) of the paths ${\cal P}$ of the tree $T$, and $EPT({\cal P})$ is the EPT graph (i.e. the edge intersection graph) of ${\cal P}$. These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their \emph{split vertices} is the set of vertices having degree at least $3$ in their union. A pair of (edge) intersecting paths is termed \emph{non-splitting} if they do not have split vertices (namely if their union is a path). In this work, motivated by an application in all-optical networks, we define the graph $ENPT({\cal P})$ of edge-intersecting non-splitting paths of a tree, termed the ENPT graph, as the (edge) graph having a vertex for each path in ${\cal P}$, 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 ${\cal P}$ of $T$ such that $G=ENPT({\cal P})$, and we say that $<T,{\cal P}>$ is a \emph{representation} of $G$. We first show that cycles, trees and complete graphs are ENPT graphs. Our work follows the lines of Golumbic and Jamison's research in which they defined the EPT graph class, and characterized the representations of chordless cycles (holes). It turns out that ENPT holes have a more complex structure than EPT holes. In our analysis, we assume that the EPT graph corresponding to a representation of an ENPT hole is given. We also introduce three assumptions $(P1)$, $(P2)$, $(P3)$ defined on EPT, ENPT pairs of graphs. In this Part I, using the results of Golumbic and Jamison as building blocks, we characterize (a) EPT, ENPT pairs that satisfy $(P1)$, $(P2)$, $(P3)$, and (b) the unique minimal representation of such pairs.