111 research outputs found

    Combinatorial Problems on HH-graphs

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    Bir\'{o}, Hujter, and Tuza introduced the concept of HH-graphs (1992), intersection graphs of connected subgraphs of a subdivision of a graph HH. They naturally generalize many important classes of graphs, e.g., interval graphs and circular-arc graphs. We continue the study of these graph classes by considering coloring, clique, and isomorphism problems on HH-graphs. We show that for any fixed HH containing a certain 3-node, 6-edge multigraph as a minor that the clique problem is APX-hard on HH-graphs and the isomorphism problem is isomorphism-complete. We also provide positive results on HH-graphs. Namely, when HH is a cactus the clique problem can be solved in polynomial time. Also, when a graph GG has a Helly HH-representation, the clique problem can be solved in polynomial time. Finally, we observe that one can use treewidth techniques to show that both the kk-clique and list kk-coloring problems are FPT on HH-graphs. These FPT results apply more generally to treewidth-bounded graph classes where treewidth is bounded by a function of the clique number

    Unit Interval Editing is Fixed-Parameter Tractable

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    Given a graph~GG and integers k1k_1, k2k_2, and~k3k_3, the unit interval editing problem asks whether GG can be transformed into a unit interval graph by at most k1k_1 vertex deletions, k2k_2 edge deletions, and k3k_3 edge additions. We give an algorithm solving this problem in time 2O(klogk)(n+m)2^{O(k\log k)}\cdot (n+m), where k:=k1+k2+k3k := k_1 + k_2 + k_3, and n,mn, m denote respectively the numbers of vertices and edges of GG. Therefore, it is fixed-parameter tractable parameterized by the total number of allowed operations. Our algorithm implies the fixed-parameter tractability of the unit interval edge deletion problem, for which we also present a more efficient algorithm running in time O(4k(n+m))O(4^k \cdot (n + m)). Another result is an O(6k(n+m))O(6^k \cdot (n + m))-time algorithm for the unit interval vertex deletion problem, significantly improving the algorithm of van 't Hof and Villanger, which runs in time O(6kn6)O(6^k \cdot n^6).Comment: An extended abstract of this paper has appeared in the proceedings of ICALP 2015. Update: The proof of Lemma 4.2 has been completely rewritten; an appendix is provided for a brief overview of related graph classe

    Subclasses of Normal Helly Circular-Arc Graphs

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    A Helly circular-arc model M = (C,A) is a circle C together with a Helly family \A of arcs of C. If no arc is contained in any other, then M is a proper Helly circular-arc model, if every arc has the same length, then M is a unit Helly circular-arc model, and if there are no two arcs covering the circle, then M is a normal Helly circular-arc model. A Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc graph is the intersection graph of the arcs of a Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc model. In this article we study these subclasses of Helly circular-arc graphs. We show natural generalizations of several properties of (proper) interval graphs that hold for some of these Helly circular-arc subclasses. Next, we describe characterizations for the subclasses of Helly circular-arc graphs, including forbidden induced subgraphs characterizations. These characterizations lead to efficient algorithms for recognizing graphs within these classes. Finally, we show how do these classes of graphs relate with straight and round digraphs.Comment: 39 pages, 13 figures. A previous version of the paper (entitled Proper Helly Circular-Arc Graphs) appeared at WG'0

    Isomorphism of graph classes related to the circular-ones property

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    We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc graphs, \Gamma-circular-arc graphs, proper circular-arc graphs and convex-round graphs.Comment: 25 pages, 9 figure

    Balancedness of subclasses of circular-arc graphs

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    A graph is balanced if its clique-vertex incidence matrix contains no square submatrix of odd order with exactly two ones per row and per column. There is a characterization of balanced graphs by forbidden induced subgraphs, but no characterization by mininal forbidden induced subgraphs is known, not even for the case of circular-arc graphs. A circular-arc graph is the intersection graph of a family of arcs on a circle. In this work, we characterize when a given graph G is balanced in terms of minimal forbidden induced subgraphs, by restricting the analysis to the case where G belongs to certain classes of circular-arc graphs, including Helly circular-arc graphs, claw-free circular-arc graphs, and gem-free circular-arc graphs. In the case of gem-free circular-arc graphs, analogous characterizations are derived for two superclasses of balanced graphs: clique-perfect graphs and coordinated graphs.Fil: Bonomo, Flavia. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Duran, Guillermo Alfredo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Universidad de Chile; ChileFil: Safe, Martin Dario. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Wagler, Annegret Katrin. Centre National de la Recherche Scientifique; Franci

    Proper circular arc graphs as intersection graphs of paths on a grid

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    In this paper we present a characterisation, by an infinite family of minimal forbidden induced subgraphs, of proper circular arc graphs which are intersection graphs of paths on a grid, where each path has at most one bend (turn)
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