$P_3$ – Forcing in Honeycomb Networks

Abstract

In this paper we compute the $P_3$-forcing number of honeycomb network. 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 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 $G$ whose components are all paths of length 3, then $S$ is called a $P_3$-forcing set of $G$. A Ps-forcing set of $G$ of minimum cardinality is called the $P_3$-forcing number of G denoted by $Z{P}_3(G)$.