### Abstract

Let H be a family of connected graphs. A graph G is said to be H-free if G is H-free for every graph H in H. In Aldred et al. (2010) [1], it was pointed that there is a family of connected graphs H not containing any induced subgraph of the claw having the property that the set of H-free connected graphs containing a claw is finite, provided also that those graphs have minimum degree at least 2 and maximum degree at least 3. In the same work, it was also asked whether there are other families with the same property. In this paper, we answer this question by solving a wider problem. We consider not only claw-free graphs but the more general class of star-free graphs. Concretely, given t<3, we characterize all the graph families H such that every large enough H-free connected graph is K1_{,t}-free. Additionally, for the case t=3, we show the families that one gets when adding the condition |H|≤k for each positive integer k.

Original language | English |
---|---|

Pages (from-to) | 2475-2484 |

Number of pages | 10 |

Journal | Discrete Mathematics |

Volume | 311 |

Issue number | 21 |

DOIs | |

Publication status | Published - 2011 Nov 6 |

### Fingerprint

### Keywords

- Claw-free
- Forbidden subgraph
- Star-free

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*311*(21), 2475-2484. https://doi.org/10.1016/j.disc.2011.07.022

**Forbidden induced subgraphs for star-free graphs.** / Fujisawa, Jun; Ota, Katsuhiro; Ozeki, Kenta; Sueiro, Gabriel.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 311, no. 21, pp. 2475-2484. https://doi.org/10.1016/j.disc.2011.07.022

}

TY - JOUR

T1 - Forbidden induced subgraphs for star-free graphs

AU - Fujisawa, Jun

AU - Ota, Katsuhiro

AU - Ozeki, Kenta

AU - Sueiro, Gabriel

PY - 2011/11/6

Y1 - 2011/11/6

N2 - Let H be a family of connected graphs. A graph G is said to be H-free if G is H-free for every graph H in H. In Aldred et al. (2010) [1], it was pointed that there is a family of connected graphs H not containing any induced subgraph of the claw having the property that the set of H-free connected graphs containing a claw is finite, provided also that those graphs have minimum degree at least 2 and maximum degree at least 3. In the same work, it was also asked whether there are other families with the same property. In this paper, we answer this question by solving a wider problem. We consider not only claw-free graphs but the more general class of star-free graphs. Concretely, given t<3, we characterize all the graph families H such that every large enough H-free connected graph is K1,t-free. Additionally, for the case t=3, we show the families that one gets when adding the condition |H|≤k for each positive integer k.

AB - Let H be a family of connected graphs. A graph G is said to be H-free if G is H-free for every graph H in H. In Aldred et al. (2010) [1], it was pointed that there is a family of connected graphs H not containing any induced subgraph of the claw having the property that the set of H-free connected graphs containing a claw is finite, provided also that those graphs have minimum degree at least 2 and maximum degree at least 3. In the same work, it was also asked whether there are other families with the same property. In this paper, we answer this question by solving a wider problem. We consider not only claw-free graphs but the more general class of star-free graphs. Concretely, given t<3, we characterize all the graph families H such that every large enough H-free connected graph is K1,t-free. Additionally, for the case t=3, we show the families that one gets when adding the condition |H|≤k for each positive integer k.

KW - Claw-free

KW - Forbidden subgraph

KW - Star-free

UR - http://www.scopus.com/inward/record.url?scp=81155161915&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=81155161915&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2011.07.022

DO - 10.1016/j.disc.2011.07.022

M3 - Article

AN - SCOPUS:81155161915

VL - 311

SP - 2475

EP - 2484

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 21

ER -