AI 中文总结
研究弱饱和问题的判定版本，通过新的图论和拓扑思想技术，证明对于固定整数\(r\geq3\)，当\(H\in\{K_r,K_{r,r}\}\)时，判定\(\mathrm{wsat}(F,H)\leq k\)是NP完全的。
AI 中文摘要
对于图\(F\)和\(H\)，若\(F\)的生成子图\(G\)满足\(E(F)\setminus E(G)\)中的边可逐个添加且每次添加创建一个新的\(H\)副本，则\(G\)在\(F\)中是弱\(H\)饱和的。近期Tancer和Tyomkyn证明判定\(\mathrm{wsat}(F,K_3)=n - 1\)是NP难的。本文研究该问题的判定版本，证明对于固定\(r\geq3\)，当\(H\in\{K_r,K_{r,r}\}\)时，判定\(\mathrm{wsat}(F,H)\leq k\)是NP完全的。我们的方法使用新的图论和拓扑思想及技术，基于Tancer和Tyomkyn的构造给出新构造，揭示了弱饱和与拓扑中旗无平方性质的联系。
英文摘要
For graphs $F$ and $H$, a spanning subgraph $G$ of $F$ is weakly $H$-saturated in $F$ if the edges in $E(F)\setminus E(G)$ can be added one at a time, each addition creating a new copy of $H$. Recently, Tancer and Tyomkyn proved that, given an $n$-vertex graph $F$, deciding whether $\mathrm{wsat}(F,K_3)=n-1$ is NP-hard. In this paper, we study the decision version of the weak saturation problem and show that, for every fixed integer $r\ge 3$, given a graph $F$ and an integer $k$, deciding whether $\mathrm{wsat}(F,H)\le k$ is NP-complete when $H\in\{K_r,K_{r,r}\}$. Our approach uses novel graph-theoretic and topological ideas and techniques, yielding new constructions that build on the construction of Tancer and Tyomkyn. In particular, our proofs bring the flag-no-square property, a fundamental property in topology that is of independent interest, into the study of weak saturation problem.
Comments16 pages, 1 figure
