### 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 |
---|---|

Journal | Electronic Journal of Combinatorics |

Volume | 24 |

Issue number | 2 |

Publication status | Published - 2017 Apr 13 |

### Fingerprint

### 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).

**Forbidden pairs with a common graph generating almost the same sets.** / Chiba, Shuya; Furuyal, Michitaka; Fujisawa, Jun; Ikarashi, Hironobu.

Research output: Contribution to journal › Article

*Electronic Journal of Combinatorics*, vol. 24, no. 2.

}

TY - JOUR

T1 - Forbidden pairs with a common graph generating almost the same sets

AU - Chiba, Shuya

AU - Furuyal, Michitaka

AU - Fujisawa, Jun

AU - Ikarashi, Hironobu

PY - 2017/4/13

Y1 - 2017/4/13

N2 - 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.

AB - 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.

KW - Forbidden subgraph

KW - Star-free graph

KW - Vertex-transitive graph

UR - http://www.scopus.com/inward/record.url?scp=85018526888&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85018526888&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85018526888

VL - 24

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 2

ER -