Claw-free cubic graphs and zero forcing

A claw-free cubic graph is a cubic graph with no induced subgraph isomorphic to $K_{1,3}$. The zero forcing process begins with an initial set $S$ of colored vertices. At each step, a colored vertex with exactly one uncolored neighbor forces that neighbor to become colored. If repeated applications of this rule color every vertex of $G$, then $S$ is called a zero forcing set. The minimum cardinality of a zero forcing set is the zero forcing number, denoted by $Z(G)$. In this paper, we answer three open questions posed by Davila and Henning concerning upper bounds on the zero forcing number of claw-free cubic graphs.
We characterize the connected claw-free cubic graphs satisfying $Z(G)=\alpha(G)+1$, where $\alpha(G)$ is the independence number. In addition, we establish the improved upper bound $Z(G)\leq \frac{T}{2}+D+2$ for claw-free cubic graphs with Hamiltonian contraction multigraphs, where $D$ is the number of diamonds and $T$ is the number of triangles in $G$.