### Abstract

Strongly perfect graphs have been studied by several authors (e.g. Berge and Duchet (1984) [1], Ravindra (1984) [12] and Wang (2006) [14]). In a series of two papers, the current paper being the first one, we investigate a fractional relaxation of strong perfection. Motivated by a wireless networking problem, we consider claw-free graphs that are fractionally strongly perfect in the complement. We obtain a forbidden induced subgraph characterization and display graph-theoretic properties of such graphs. It turns out that the forbidden induced subgraphs that characterize claw-free graphs that are fractionally strongly perfect in the complement are precisely the cycle of length 6, all cycles of length at least 8, four particular graphs, and a collection of graphs that are constructed by taking two graphs, each a copy of one of three particular graphs, and joining them in a certain way by a path of arbitrary length. Wang (2006) [14] gave a characterization of strongly perfect claw-free graphs. As a corollary of the results in this paper, we obtain a characterization of claw-free graphs whose complements are strongly perfect.

Original language | English (US) |
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Pages (from-to) | 1971-1995 |

Number of pages | 25 |

Journal | Discrete Applied Mathematics |

Volume | 159 |

Issue number | 17 |

DOIs | |

State | Published - Oct 28 2011 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Keywords

- Claw-free graphs
- Forbidden induced subgraphs
- Strongly perfect graphs
- Structural graph theory
- Wireless networking

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## Cite this

*Discrete Applied Mathematics*,

*159*(17), 1971-1995. https://doi.org/10.1016/j.dam.2011.06.024