无向图中无冲突边割的参数化复杂性
On the Parameterized Complexity of Conflict-Free Edge Cut in Undirected Graphs
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中文总结 AI 辅助
本文系统研究无冲突边割及其最小化变体的参数化复杂性与核化，证明解离数≤2时NP-hard、≤1时多项式可解，并给出MIN-CF-CUT的FPT与W[1]-困难结果及无多项式核的结论。
中文摘要 AI 辅助
本文研究无冲突边割（CF-CUT）问题，这是最近引入的最小割（MIN-CUT）问题的无冲突版本，其目标是找到断开连通图所需的最少边数。CF-CUT的输入是一个连通无向图 G = (V, E)，以及一个冲突图 \u005cwidehat{G}，使得 E(G) = V(\u005cwidehat{G})，目标是判定是否存在 F \u2286 E(G) 使得 G - F 不连通且 F 是 \u005cwidehat{G} 中的独立集。Rauch 等人 [IPL-2025] 证明了 CF-CUT 是 NP-完全的，并提供了关于 CF-CUT 参数化复杂性的一些结果。一个相关的变体最小无冲突边割（MIN-CF-CUT）以连通图 G、冲突图 \u005cwidehat{G}（其中 V(\u005cwidehat{G}) = E(G)）和整数 k 为输入，询问是否存在至多 k 条边的集合 F 使得 G - F 不连通且 F 是 \u005cwidehat{G} 中的独立集。本文扩展了 Rauch 等人 [IPL-2025] 的工作，从参数化复杂性和多项式核化的角度对 CF-CUT 和 MIN-CF-CUT 进行了系统研究。我们证明当输入图的解离数至多为 2 时，CF-CUT 是 NP-困难的。我们还通过证明当输入图的解离数至多为 1 时 CF-CUT 可在多项式时间内求解来补充这一结果。此外，对于 MIN-CF-CUT，我们同时考虑解的大小和输入图的各种结构参数作为参数，并在冲突图限制为各种图类时提供固定参数可处理性和 W[1]-困难性结果。我们还证明，除非 NP \u2286 coNP/poly，否则 MIN-CF-CUT 在参数化为输入图的顶点覆盖数时没有多项式核；在参数化为解的大小与输入图的顶点完整度之和时也没有多项式核。
英文摘要
In this paper, we study CONFLICT-FREE EDGE CUT (CF-CUT), which is a recently introduced conflict-free version of the MIN-CUT problem that asks to find the minimum number of edges to disconnect a connected graph. The CF-CUT takes as input a connected undirected graph G = (V, E), a conflict graph $\widehat{G}$ such that $E(G) = V(\widehat{G})$, and the objective is to decide whether there exists $F \subseteq E(G)$ such that $G - F$ is disconnected and $F$ is an independent set in $\widehat{G}$. Rauch et. al. [IPL-2025] proved that CF-CUT is NP-Complete and also provided some results on the parameterized complexity of CF-CUT. A related variant MIN CONFLICT-FREE EDGE CUT (MIN-CF-CUT) takes a connected graph $G$, a conflict graph $\widehat{G}$ such that $V(\widehat{G}) = E(G)$, and an integer $k$ as input and asks if there is a set $F$ of at most $k$ edges such that $G - F$ is disconnected and $F$ is an independent set in $\widehat{G}$. In this paper, we extend the work of Rauch et al. [IPL-2025] and provide a systematic study on the CF-CUT and MIN-CF-CUT from the perspective of parameterized complexity and polynomial kernelization. We prove that CF-CUT is NP-hard when the dissociation number of the input graph is at most two. We also complement it by proving that CF-CUT is poly-time solvable when the dissociation number of an input graph is at most one. Additionally, for MIN-CF-CUT, we consider both solution size and various structural parameters of the input graph as parameters, and provide fixed-parameter tractability and W[1]-hardness results when the conflict graph is restricted to various graph classes. We also prove that unless NP $\subseteq$ coNP/poly, MIN-CF-CUT admits no polynomial kernel when parameterized by the vertex cover number of the input graph; and also when parameterized by the sum of the solution size and the vertex integrity of the input graph.
发表机构
- Indraprastha Institute of Information Technology Delhi（德里印度理工学院）
- National Institute of Technology Calicut（卡利卡特国立技术学院）
机构由 AI 辅助整理，请以论文原文为准。
