## Abstract

Given a system G = (G_{1}, G_{2}, ..., G_{m}) of m graphs on the same vertex set V, define the "joint independence number" α_{∩} (G) as the maximal size of a set which is independent in all graphs G_{i}. Let also γ_{∪} (G) be the "collective domination number" of the system, which is the minimal number of neighborhoods, each taken from any of the graphs G_{i}, whose union is V. König's classical duality theorem can be stated as saying that if m = 2 and both graphs G_{1}, G_{2} are unions of disjoint cliques then α_{∩} (G_{1}, G_{2}) = γ_{∪} (G_{1}, G_{2}). We prove that a fractional relaxation of α_{∩}, denoted by α_{∩}^{*}, satisfies the condition α_{∩}^{*} (G_{1}, G_{2}) ≥ γ_{∪} (G_{1}, G_{2}) for any two graphs G_{1}, G_{2}, and α_{∩}^{*} (G_{1}, G_{2}, ..., G_{m}) > frac(2, m) γ_{∪} (G_{1}, G_{2}, ..., G_{m}) for any m > 2 and all graphs G_{1}, G_{2}, ..., G_{m}. We prove that the convex hull of the (characteristic vectors of the) independent sets of a graph contains the anti-blocker of the convex hull of the non-punctured neighborhoods of the graph and vice versa. This, in turn, yields α_{∩}^{*} (G_{1}, G_{2}, ..., G_{m}) ≥ γ_{∪}^{*} (G_{1}, G_{2}, ..., G_{m}) as well as a dual result. All these results have extensions to general simplicial complexes, the graphical results being obtained from the special case of the complexes of independent sets of graphs.

Original language | English |
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Pages (from-to) | 1259-1270 |

Number of pages | 12 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 98 |

Issue number | 6 |

DOIs | |

State | Published - Nov 2008 |

### Bibliographical note

Funding Information:E-mail addresses: ra@tx.technion.ac.il (R. Aharoni), berger@math.haifa.ac.il (E. Berger), holzman@tx.technion.ac.il (R. Holzman), orikfir@msn.com (O. Kfir). 1 The author’s research was supported by the Fund for the Promotion of Research at the Technion and by the P. and E. Nathan Research Fund.

## Keywords

- Anti-blocker
- Domination
- Fractional ISR
- Graph systems
- Independence
- König's duality
- Matroid intersection

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics