Graphs of Edge-Intersecting Non-Splitting Paths in a Tree: Representations of Holes-Part II

Given a tree and a set P of non-trivial simple paths on it, VPT(P) is the VPT graph (i.e. the vertex intersection graph) of the paths P , and EPT(P) is the EPT graph (i.e. the edge intersection graph) of P . 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). We define the graph ENPT(P) of edge intersecting non-splitting paths of a tree, termed the ENPT graph, as the graph having a vertex for each path in P , and an edge between every pair of vertices representing two 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 P of T such that G = ENPT(P), and we say that 〈T,P〉 is a representation of G. Our work follows Golumbic and Jamison’s research, in which they defined the EPT graph class, and characterized the representations of chordless cycles (holes). Our main goal is the characterization of the representations of chordless ENPT cycles. To achieve this goal, we assume that the EPT graph induced by the vertices of an ENPT hole is given. We use the results of that research as building blocks in order to discover this characterisation, which turn out to have a more complex structure than in the case of EPT holes. In the first part of this work we have shown that cycles, trees and complete graphs are ENPT graphs. We also introduced three assumptions (P1), (P2), (P3) defined on EPT, ENPT pairs of graphs, and characterized the representations of ENPT holes that satisfy (P1), (P2), (P3). In this work we relax two of these three assumptions and characterize the representations of ENPT holes satisfying (P3). These two results are achieved by providing polynomial-time algorithms. Last we show that the problem of finding such a representation is NP-Hard in general, i.e. without assumption (P3). This result extends in some sense the NP-Hardness of EPT graph recognition shown in Golumbic and Jamison’s work.