On a modular domination game

We present a generalization of the so-called σ-game, introduced by Sutner [9], a combinatorial game played on a graph, with relations to cellular automata, as well as odd domination in graphs. A configuration on a graph is an assignment of values in {0, . . . , p− 1} (where p is an arbitrary positive integer) to all the vertices of G. One may think of a vertex v of G as a button the player can press at his discretion. If vertex v is chosen, the value of all the vertices adjacent to v increases by 1 modulo p. This defines an equivalence relation between the configurations: two configurations are in relation if it is possible to reach one from the other by a sequence of such operations. We investigate the number of equivalence classes that a given graph has, and we give formulas for trees and special regular graphs.