On Extremal Graphs for Zero Forcing Number

Abstract

A subset S of vertex set V(G) of a graph G is a zero forcing set of G if iteratively adding vertices to S from $V(G){\setminus }S$ that are the unique neighbor in $V(G){\setminus } S$ of some vertex in S, results in the entire V(G) of G. Additionally, if the subgraph induced by S is connected, then S is a connected forcing set of G. The zero (resp., connected) forcing number, denoted by F(G) (resp., $F_c(G)$), of G is the minimum cardinality of a zero (resp., connected) forcing set of G. Davila and Kenter [Theory Appl. Graphs, 2(2) (2015) Article 1] proved that $F(G)\le n-g+2$ for graphs G of finite girth g and order $n (\ge g)$. In this paper, first, we restrict G to be connected and improve the upper bound according to the value of g: $F(G)\le n-g+2$ when $g=3, 4$ or n; $F(G)\le n-g+1$ when $g=5, 6$ or $n-1$ $(n \ge 6)$ and $F(G)\le n-g$ when $7\le g\le n-2$. Further, the extremal graphs are characterized, respectively. Davila et al. [Graphs Combin. 34 (2018) 1159-1174] proved a similar upper bound on $F_c(G)$ for 2-connected graphs G with n vertices and finite girth g: $F_c(G)\le n-g+2$. Secondly, we prove that $F_c(G)=n-g+2$ if and only if G is $C_{n}$, or $K_{n}$, or $K_{a,b}$ for integers $a, b \ge 2$, $a+b=n$. Finally, we also characterize all connected triangle-free or subcubic graphs G of order n with $F(G)=n-3$.