Arman Boyacı ; Tınaz Ekim ; Mordechai Shalom ; Shmuel Zaks - Graphs of Edge-Intersecting Non-Splitting Paths in a Tree: Representations of Holes-Part II

dmtcs:1562 - Discrete Mathematics & Theoretical Computer Science, January 11, 2018, Vol. 20 no. 1 - https://doi.org/10.23638/DMTCS-20-1-2
Graphs of Edge-Intersecting Non-Splitting Paths in a Tree: Representations of Holes-Part IIArticle

Authors: Arman Boyacı ; Tınaz Ekim ; Mordechai Shalom ORCID; Shmuel Zaks

    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 goal is to characterize the representation of chordless ENPT cycles (holes). To achieve this goal, we first assume that the EPT graph induced by the vertices of an ENPT hole is given. In [2] we introduce three assumptions (P1), (P2), (P3) defined on EPT, ENPT pairs of graphs. In the same study, we define two problems HamiltonianPairRec, P3-HamiltonianPairRec and characterize the representations of ENPT holes that satisfy (P1), (P2), (P3). In this work, we continue our work by relaxing these three assumptions one by one. We characterize the representations of ENPT holes satisfying (P3) by providing a polynomial-time algorithm to solve P3-HamiltonianPairRec. We also show that there does not exist a polynomial-time algorithm to solve HamiltonianPairRec, unless P=NP.


    Volume: Vol. 20 no. 1
    Section: Graph Theory
    Published on: January 11, 2018
    Accepted on: December 19, 2017
    Submitted on: August 10, 2016
    Keywords: Computer Science - Discrete Mathematics
    Funding:
      Source : OpenAIRE Graph
    • Seçmeli Çizge Boyama Problemi; Funder: Türkiye Bilimsel ve Teknolojik Araştırma Kurumu; Code: 111M303

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