Total Forcing and Zero Forcing in Claw-Free Cubic Graphs

Abstract

A dynamic coloring of the vertices of a graph G 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 called a forcing set (zero forcing set) of G if, by iteratively applying the forcing process, every vertex in G becomes colored. If the initial set S has the added property that it induces a subgraph of G without isolated vertices, then S is called a total forcing set in G. The total forcing number of G, denoted $F_t(G)$, is the minimum cardinality of a total forcing set in G. We prove that if G is a connected, claw-free, cubic graph of order $n \ge 6$, then $F_t(G) \le \frac{1}{2}n$, where a claw-free graph is a graph that does not contain $K_{1,3}$ as an induced subgraph. The graphs achieving equality in these bounds are characterized.