Given a graph $G=(V,E)$, and a function $f:V(G) \rightarrow \mathbb{N}$, an $f$-reversible process on $G$ is a dynamical system such that, given an initial vertex labeling $c_0 : V(G) \rightarrow \{0,1\}$, every vertex $v$ changes its label if and only if it has at least $f(v)$ neighbors with the opposite label. The updates occur synchronously in discrete time steps $t=0,1,2,\ldots$. An $f$-critical set of $G$ is a subset of vertices of $G$ whose initial label is $1$ such that, in an $f$-reversible process on $G$, all vertices reach label $1$ within one time step and then remain unchanged. The critical set number $r^c_f(G)$ is the minimum size of an $f$-critical set of $G$. Given a graph $G$, a threshold function $f$, and an integer $k$, the $f$-Critical Set problem asks whether $r^c_f(G) \leq k$. We prove that this problem is NP-complete for planar subcubic bipartite graphs with maximum threshold $m(f) = 2$ and W[1]-hard when parameterized by the treewidth $tw(G)$ of $G$. Additionally, we show that the problem is FPT when parameterized by $tw(G)+m(f)$, $tw(G)+Δ(G)$, and $k$, where $Δ(G)$ denotes the maximum degree of $G$. Finally, we present two kernels of sizes $O(k \cdot m(f))$ and $O(k \cdot Δ(G))$.

翻译：给定图$G=(V,E)$及函数$f:V(G) \rightarrow \mathbb{N}$，图$G$上的$f$-可逆过程是一个动力学系统：给定初始顶点标记$c_0 : V(G) \rightarrow \{0,1\}$，每个顶点$v$当且仅当至少有$f(v)$个邻居具有相反标记时改变其标记。更新过程在离散时间步$t=0,1,2,\ldots$内同步进行。图$G$的$f$-关键集是初始标记为$1$的顶点子集，使得在$G$的$f$-可逆过程中，所有顶点在一个时间步内全部达到标记$1$并保持稳定。关键集数$r^c_f(G)$是$G$中$f$-关键集的最小规模。给定图$G$、阈值函数$f$和整数$k$，$f$-关键集问题询问是否满足$r^c_f(G) \leq k$。我们证明该问题在最大阈值为$m(f)=2$的平面次立方二分图上为NP完全，且当以树宽$tw(G)$为参数时属于W[1]-难。此外，我们证明该问题在以$tw(G)+m(f)$、$tw(G)+Δ(G)$和$k$为参数时属于FPT，其中$Δ(G)$表示$G$的最大度数。最后，我们给出两个大小分别为$O(k \cdot m(f))$和$O(k \cdot Δ(G))$的核。