Zero forcing versus domination in cubic graphs

Abstract

In this paper, we study a dynamic coloring of the vertices of a graph G that starts with an initial subset S of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set S is a zero forcing set of G if, by iteratively5 applying the forcing process, every vertex in G becomes colored. The zero forcing number of G is the minimum cardinality of a zero forcing set of G. In this paper, we prove that if $G \ne K_4$ is a connected cubic graph, then the zero forcing number of G is bounded above by twice its domination number, where the domination number of G is the minimum cardinality of a set of vertices of G such that every vertex not in S is adjacent to some vertex in S.