Forbidden pairs with a common graph generating almost the same sets

Shuya Chiba, Michitaka Furuyal, Jun Fujisawa, Hironobu Ikarashi

Research output: Contribution to journalArticlepeer-review


Let H be a family of connected graphs. A graph G is said to be H-free if G does not contain any members of H as an induced subgraph. Let F(H) be the family of connected H-free graphs. In this context, the members of H are called forbidden subgraphs. In this paper, we focus on two pairs of forbidden subgraphs containing a com- mon graph, and compare the classes of graphs satisfying each of the two forbidden subgraph conditions. Our main result is the following: Let H1;H2;H3be connected graphs of order at least three, and suppose that H1 is twin-less. If the symmetric difference of F(fH1;H2g) and F(fH1;H3g) is finite and the tuple (H1;H2;H3) is non-trivial in a sense, then H2and H3are obtained from the same vertex-transitive graph by successively replacing a vertex with a clique and joining the neighbors of the original vertex and the clique. Furthermore, we refine a result in [Combin. Probab. Comput. 22 (2013) 733-748] concerning forbidden pairs.

Original languageEnglish
JournalElectronic Journal of Combinatorics
Issue number2
Publication statusPublished - 2017 Apr 13


  • Forbidden subgraph
  • Star-free graph
  • Vertex-transitive graph

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics


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