紧（$k,0$）-稳定图的刻画
A characterization of tight ($ k, 0 $)-stable graphs
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中文总结 AI 辅助
本文对$k\geq4$的紧（$k,0$）-稳定图给出完整刻画，证明$k\geq5$时这类图为$K_{k+1}$和$K_{k+2}$，推广了相关结果并证明了Dong与Luo的猜想。
中文摘要 AI 辅助
设$k$和$l$为两个非负整数且满足$k>l$。图$G$是$(k,l)$-稳定的，当对$V(G)$的任意大小为$k$的子集$S$，都有$\alpha(G-S)\geq\alpha(G)-l$，其中$\alpha(G)$表示$G$的独立数。Dong和Wu证明了对$(k,l)$-稳定图$G$，有$\alpha(G)\leq\lfloor\frac{n-k+1}{2}\rfloor+l$，其中$n$是$G$的阶数。一个$(k,l)$-稳定图$G$是紧的，当$\alpha(G)=\lfloor\frac{n-k+1}{2}\rfloor+l$。本文对$k\geq4$的紧（$k,0$）-稳定图给出完整刻画，特别证明了$k\geq5$时紧（$k,0$）-稳定图为$K_{k+1}$和$K_{k+2}$，这不仅将Liu、Song和Wang的结果（原适用于$k\geq24$）推广到$k\geq5$，还再次证明了Dong和Luo的猜想。
英文摘要
Let k and l be two non-negative integers with k > l. A graph G is (k,l)-stable if alpha(G - S) >= alpha(G) - l for every subset S of V(G) with |S| = k, where alpha(G) denotes the independence number of G. Dong and Wu established that alpha(G) <= floor((n - k + 1)/2) + l for a (k, l)-stable graph G, where n is the order of G. A (k, l)-stable graph G is tight if alpha(G) = floor((n - k + 1)/2) + l. In this paper, we provide a complete characterization of tight (k, 0)-stable graphs for k >= 4. In particular, we prove that tight (k, 0)-stable graphs are K_{k+1} and K_{k+2} for k >= 5, which not only extends the result of Liu, Song and Wang [J. Graph Theory 110(2) (2025), 193-199] from k >= 24 to k >= 5, but also proves the conjecture of Dong and Luo [Electron. J. Comb. 32(4) (2025), 4-45] once more.
