Checking global graph properties by means of local computations: the majority problem

Abstract This paper is a contribution to the study of the general problem of characterizing those properties which can be computed on a graph or a network by means of local transformations. By using an abstract model based on graph relabelling systems we consider the majority problem: let G be a graph whose vertices have label A or B ; we say that label A has the majority if the number of A -labelled vertices is strictly greater than the number of B -labelled vertices (∣ G ∣ A > ∣ G ∣ B ). We prove that there exists graph relabelling systems deciding for every connected graph G whether ∣ G ∣ A > ∣ G ∣ B (resp. ∣ G ∣ A = ∣ G ∣ B ) or not. On the other hand, we prove that no such system can decide if ∣ G ∣ A > ∣ G ∣ B – m (resp. ∣ G ∣ A = ∣ G ∣ B – m ), for any positive integer m.