### Abstract

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 H_{1};H_{2};H_{3}be connected graphs of order at least three, and suppose that H1 is twin-less. If the symmetric difference of F(fH_{1};H_{2}g) and F(fH_{1};H_{3}g) is finite and the tuple (H_{1};H_{2};H_{3}) is non-trivial in a sense, then H_{2}and H_{3}are 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 language | English |
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Journal | Electronic Journal of Combinatorics |

Volume | 24 |

Issue number | 2 |

Publication status | Published - 2017 Apr 13 |

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### Keywords

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

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Theory and Mathematics

### Cite this

*Electronic Journal of Combinatorics*,

*24*(2).