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(Even hole, triangle)-free graphs revisited

Beatriz Martins, Nicolas Trotignon

Abstract

We revisit a classical paper about (even hole, triangle)-free graphs [Conforti, Cornuéjols, Kapoor and Vu\v sković, Triangle-free graphs that are signable without even holes, Journal of Graph Theory, 34(3), 204--220, 2000]. In fact, the previous study describes a more general class, the so called triangle-free odd signable graphs, and we further generalise the class to the (theta, triangle, wac)-free graphs (not worth defining in an abstract). We exhibit a stronger structure theorem, by precisely describing basic classes and separators. We prove that the separators preserve the treewidth and several properties. Some consequences are a recognition algorithm with running time $O(|V(G)|^4|E(G)|)$, a proof that the treewidth of graphs in the class is at most~4 (improving a previous bound of~5), and a simple criterion to decide if a graph in the class is planar.

(Even hole, triangle)-free graphs revisited

Abstract

We revisit a classical paper about (even hole, triangle)-free graphs [Conforti, Cornuéjols, Kapoor and Vu\v sković, Triangle-free graphs that are signable without even holes, Journal of Graph Theory, 34(3), 204--220, 2000]. In fact, the previous study describes a more general class, the so called triangle-free odd signable graphs, and we further generalise the class to the (theta, triangle, wac)-free graphs (not worth defining in an abstract). We exhibit a stronger structure theorem, by precisely describing basic classes and separators. We prove that the separators preserve the treewidth and several properties. Some consequences are a recognition algorithm with running time , a proof that the treewidth of graphs in the class is at most~4 (improving a previous bound of~5), and a simple criterion to decide if a graph in the class is planar.

Paper Structure

This paper contains 15 sections, 47 theorems, 6 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Basic graphs
  3. Separators
  4. Known results
  5. Lemmas
  6. Proof of Theorem \ref{['th:Decomp']}
  7. Consequences
  8. Even-hole-free, even-wheel-free and bipartite graphs
  9. Ear decomposition
  10. Planarity
  11. Treewidth and cliquewidth
  12. Testing for a wac
  13. Algorithms
  14. Structure theorem
  15. Open questions

Key Result

Theorem 1.1

If $G$ is a (theta, triangle, wac)-free graph, then $G$ is basic or $G$ has a clique separator, a proper 2-separator or a proper $P_3$-separator.

Figures (11)

  • Figure 1: A theta and a wheel
  • Figure 2: Two examples of (theta, triangle)-free wacs
  • Figure 3: The Vušković graph (the smallest non-planar daisy)
  • Figure 4: The cube
  • Figure 5: Graphs with separators
  • ...and 6 more figures

Theorems & Definitions (92)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 82 more