The structure of bull-free graphs I-Three-edge-paths with centers and anticenters

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Abstract

The bull is the graph consisting of a triangle and two disjoint pendant edges. A graph is called bull-free if no induced subgraph of it is a bull. This is the first paper in a series of three. The goal of the series is to explicitly describe the structure of all bull-free graphs. In this paper we study the structure of bull-free graphs that contain as induced subgraphs three-edge-paths P and Q, and vertices c∉ V(P) and a∉ V(Q), such that c is adjacent to every vertex of V(P) and a has no neighbor in V(Q). One of the theorems in this paper, namely 1.2, is used in Chudnovsky and Safra (2008) [9] in order to prove that every bull-free graph on n vertices contains either a clique or a stable set of size n14, thus settling the Erdös-Hajnal conjecture (Erdös and Hajnal, 1989) [17] for the bull.

Original languageEnglish (US)
Pages (from-to)233-251
Number of pages19
JournalJournal of Combinatorial Theory. Series B
Volume102
Issue number1
DOIs
StatePublished - Jan 1 2012
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Keywords

  • Bull-free graphs
  • Decomposition theorems
  • Graph structure
  • Induced subgraphs

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