Chip firing and vertex cover of squares of graphs

Dominique Rossin (1) , Fedor V. Fomin (2)

1 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications
2 UiB - Department of Informatics [Bergen]

In the chip firing game a number of chips is initially distributed on some vertices of a graph G. Then at every step of the game a vertex v containing at least deg(v) chips is toppled (fired), i.e. v loses as many chips as its degree and each of its neighbors receives one chip from v. Let kappa(G) be the smallest number of chips needed to cover all vertices of G. We prove that kappa(G) is equal to the minimum cardinality of a vertex cover of G^2. Moreover, we show that the initial distribution of chips is given by a solution of the vertex cover problem.

Mots clés: Chip-firing game; vertex cover; graph square

Domaines: Combinatoire [math.CO]

Dominique Rossin, Fedor V. Fomin. Chip firing and vertex cover of squares of graphs. RAIRO - Theoretical Informatics and Applications (RAIRO: ITA), 2006, To be published. ⟨hal-00112026⟩