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A Structure Theorem for Graphs with No Cycle with a Unique Chord and Its Consequences

Trotignon, N and Vuskovic, K (2010) A Structure Theorem for Graphs with No Cycle with a Unique Chord and Its Consequences. Journal of Graph Theory, 63 (1). 31 - 67 . ISSN 0364-9024

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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 either in some simple basic class or has a decomposition. Basic classes are chordless cycles, cliques, bipartite graphs with one side containing only nodes of degree 2 and induced subgraphs 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.

Item Type: Article
Copyright, Publisher and Additional Information: © 2010, Wiley Blackwell. This is an author produced version of a paper published in Journal of Graph Theory. Uploaded in accordance with the publisher's self-archiving policy.
Keywords: cycle with a unique chord, decomposition, structure, detection, recognition, Heawood graph, Petersen graph, coloring, unimodular matrices, berge graphs, algorithms, decomposition, recognition
Institution: The University of Leeds
Academic Units: The University of Leeds > Faculty of Engineering (Leeds) > School of Computing (Leeds)
Depositing User: Symplectic Publications
Date Deposited: 21 Jun 2012 14:13
Last Modified: 09 Jun 2014 20:45
Published Version: http://dx.doi.org/10.1002/jgt.20405
Status: Published
Publisher: Wiley Blackwell
Identification Number: 10.1002/jgt.20405
URI: http://eprints.whiterose.ac.uk/id/eprint/74349

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