Graphs without odd holes, parachutes or proper wheels: a generalization of Meyniel graphs and of line graphs of bipartite graphs

  • Authors:

  • Michele Conforti;Gérard Cornuéjols

  • Affiliations:

  • Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Belzoni 7, 35131 Padova, Italy;Graduate School of Industrial Administration, Carnegie Mellon University, Schenley Park, Pittsburgh, PA

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2003

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Abstract

We prove that the strong perfect graph conjecture holds for graphs that do not contain parachutes or proper wheels. This is done by showing the following theorem: If a graph G contains no odd hole, no parachute and no proper wheel, then G is bipartite or the line graph of a bipartite graph or G contains a star cutset or an extended strong 2-join or G is disconnected.To prove this theorem, we prove two decomposition theorems which are interesting in their own rights. The first is a generalization of the Burlet-Fonlupt decomposition of Meyniel graphs by clique cutsets and amalgams. The second is a precursor of the recent decomposition theorem of Chudnovsky, Robertson, Seymour and Thomas for Berge graphs that contain a line graph of a bipartite subdivision of a 3-connected graph.