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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.11546 |
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| _version_ | 1866915918437679104 |
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| author | Marcilon, Thiago Silva, Murillo Inácio da Costa |
| author_facet | Marcilon, Thiago Silva, Murillo Inácio da Costa |
| contents | 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 $O(k \cdot Δ(G))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_11546 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Parameterized complexity of the f-Critical Set problem Marcilon, Thiago Silva, Murillo Inácio da Costa Computational Complexity 03D15 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 $O(k \cdot Δ(G))$. |
| title | Parameterized complexity of the f-Critical Set problem |
| topic | Computational Complexity 03D15 |
| url | https://arxiv.org/abs/2511.11546 |