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 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

On the Maximum Cardinality Cut Problem in Proper Interval Graphs and Related Graph Classes

Although it has been claimed in two different papers that the maximum cardinality cut problem is polynomial-time solvable for proper interval graphs, both of them turned out to be erroneous. In this paper, we give FPT algorithms for the maximum cardinality cut problem in classes of graphs containing proper interval graphs and mixed unit interval graphs when parameterized by some new parameters that we introduce. These new parameters are related to a generalization of the so-called bubble representations of proper interval graphs and mixed unit interval graphs and to clique-width decompositions

Graphs of Edge-Intersecting and Non-Splitting One Bend Paths in a Grid

The families EPT (resp. EPG) Edge Intersection Graphs of Paths in a tree(resp. in a grid) are well studied graph classes. Recently we introduced thegraph classes Edge-Intersecting and Non-Splitting Paths in a Tree ENPT, and ina Grid (ENPG). It was shown that ENPG contains an infinite hierarchy ofsubclasses that are obtained by restricting the number of bends in the paths.Motivated by this result, in this work we focus on one bend {ENPG} graphs. Weshow that one bend ENPG graphs are properly included in two bend ENPG graphs.We also show that trees and cycles are one bend ENPG graphs, and characterizethe split graphs and co-bipartite graphs that are one bend ENPG. We prove thatthe recognition problem of one bend ENPG split graphs is NP-complete even in avery restricted subfamily of split graphs. Last we provide a linear timerecognition algorithm for one bend ENPG co-bipartite graphs