设$S\subseteq V$是一个初始着色顶点子集, 它的所有顶点都着黑色, $S$中的每个顶点称为$S$-着色, $G$中所有其他未着色的顶点称为$S$-未着色。若一个着黑色的顶点$v$恰好只有一个未着色的邻点$u$, 则$v$强迫顶点$u$着黑色, 这样的过程称为强迫过程。如果从一个初始顶点集$S$出发, 逐步运用强迫过程直到$G$中所有的顶点都变成黑色, 则称这个初始集$S$为$G$的零强迫集(强迫集)。$G$的零强迫集的最小基数用$F (G)$表示, 称为$G$的零强迫数。如果$S$在$G$中的导出子图$G[S]$不包含孤立顶点, 则强迫集$S$称为$G$的全强迫集, 全强迫集的最小基数用$F_t (G)$表示, 称为$G$的全强迫数。注意到零强迫数和全强迫数有如下关系, $F (G)\leq F_t (G)\leq 2F (G)$。刻画满足$F (G)＝F_t (G)$或者$2F (G)＝F_t (G)$的图$G$是有意义的。基于此, 本文在单圈图上刻画了满足$F (G)＝F_t (G)$的所有图$G$。此外, 本文研究了给定匹配数为$k$的单圈图$G$的零强迫数的上界, 即$F (G)\leq n-k$, 等号成立当且仅当$G\cong C_4, A_0$。此外, 当$G\not\cong C_4, A_0$时, 本文刻画了使得$F (G)=n-k-1$的所有图$G$。

Let $S\subseteq V$ be an initial vertex subset of coloring, and all vertices of $S\subseteq V$ are black. Each vertex in $S$ is called $S$-colored, and all other vertices in $G$ are called $S$-uncolored. If a vertex $v$ with black color has just one uncolored neighbor $u$, then $v$ forces the vertex $u$ to be black. Such a process is called forcing process. The initial set $S$ is called a zero forcing set (forcing set) of $G$ if, by iteratively applying the forcing process, every vertex in $G$ becomes black. The minimum cardinality of a zero forcing set in $G$ is zero forcing number, denoted $F(G)$. If the induced subgraph $G[S]$ does not contain isolated vertices, then $S$ is called a total forcing set of $G$, and the minimum cardinality of a total forcing set in $G$ is total forcing number, denoted $F_t(G)$. Note that zero forcing number and total forcing number of graph $G$ has the relation, $F(G)\leq F_t (G)\leq 2F(G)$. Hence, it is interesting and meaningful to characterize all graphs satisfying $F(G)=F_t (G)$ or $2F(G)=F_t (G)$. In this paper, we study all graphs $G$ satisfying $F(G)=F_t (G)$ in unicyclic graphs. In addition, we discuss the upper bound of zero forcing number in all unicyclic graphs with given matching number $k$, and show that $F(G)\leq n-k$ and equality holds if and only if $G\cong C_4, A_0$. Moreover, we characterize all graphs such that their forcing number equal to $n-k-1$ if $G\not\cong C_4, A_0$.