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A structure theorem for graphs with no cycle with a unique chord and its consequences

Author

Listed:
  • Nicolas Trotignon

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

  • Kristina Vuskovic

    (School of Computing [Leeds] - University of Leeds)

Abstract

We give a structural description of the class C of graphs that do not contain a cycle with a unique chord as an induced subgraph. Our main theorem states that any connected graph in C is a either in some simple basic class or has a decomposition. Basic classes are cliques, bipartite graphs with one side containing only nodes of degree two and induced subgraph of the famous Heawood or Petersen graph. Decompositions are node cutsets consisting of one or two nodes and edge cutsets called 1-joins. Our decomposition theorem actually gives a complete structure theorem for C, i.e. every graph in C can be built from basic graphs that can be explicitly constructed, and gluing them together by prescribed composition operations ; and all graphs built this way are in C. This has several consequences : an O(nm)-time algorithm to decide whether a graph is in C, an O(n+m)-time algorithm that finds a maximum clique of any graph in C and an O(nm)-time coloring algorithm for graphs in C. We prove that every graph in C is either 3-colorable or has a coloring with ω colors where ω is the size of a largest clique. The problem of finding a maximum stable set for a graph in C is known to be NP-hard.

Suggested Citation

  • Nicolas Trotignon & Kristina Vuskovic, 2008. "A structure theorem for graphs with no cycle with a unique chord and its consequences," Post-Print halshs-00265957, HAL.
    • Handle: RePEc:hal:journl:halshs-00265957
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00265957
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    References listed on IDEAS

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    1. Benjamin Lévêque & David Y. Lin & Frédéric Maffray & Nicolas Trotignon, 2007. "Detecting induced subgraphs," Documents de travail du Centre d'Economie de la Sorbonne b07049, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    2. Benjamin Lévêque & David Y. Lin & Frédéric Maffray & Nicolas Trotignon, 2007. "Detecting induced subgraphs," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00180953, HAL.
    3. Benjamin Lévêque & David Y. Lin & Frédéric Maffray & Nicolas Trotignon, 2007. "Detecting induced subgraphs," Post-Print halshs-00180953, HAL.
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