AI 中文总结
本文在前期笛卡尔积、字典积隔离数研究基础上，研究四类图乘积的隔离数与$\boldsymbol{\textit{F}}$-隔离数，推导相关上下界并确定多族图乘积的精确值，扩展了笛卡尔积的对应研究结果。
AI 中文摘要
作为笛卡尔积和字典积中隔离数前期研究的延续，本文研究图的直积、强积、字典积和笛卡尔积中的隔离数，更一般地，研究其中的$\boldsymbol{\textit{F}}$-隔离数。对于直积，我们基于因子图的隔离数和全控制参数推导$\boldsymbol{\textit{F}}=\boldsymbol{\textit{K}}_{n_1,\boldsymbol{\textit{K}}_{n_d}}$时的$\boldsymbol{\textit{F}}$-隔离数上界，基于开 packing 建立下界；还确定了多族直积的精确值，包括$\boldsymbol{\textit{\u03b9}}(\boldsymbol{\textit{P}}_{4\boldsymbol{\u03b9}} \times \boldsymbol{\textit{C}}_{2\boldsymbol{\textit{k}}+1})=\boldsymbol{\textit{\u03b9}}(\boldsymbol{\textit{k}}+1)$。对于强积，证明涉及2-packing 数的$\boldsymbol{\textit{\u03b9}}(\boldsymbol{\textit{G}} \boxtimes \boldsymbol{\textit{H}},\boldsymbol{\textit{F}})$的通用下界，给出$\boldsymbol{\textit{\u03b9}}(\boldsymbol{\textit{G}} \boxtimes \boldsymbol{\textit{H}})$的上界。对于字典积，确定多类通用场景下的$\boldsymbol{\textit{F}}$-隔离数，得到基于第一个因子的控制数和全控制数的精确公式。最后，对于笛卡尔积，将前期工作的结果扩展到任意图族$\boldsymbol{\textit{F}}$，引入$\boldsymbol{\textit{F}}$-隔离图并利用$\boldsymbol{\textit{F}}$-横截集推导通用上界及对应下界。
英文摘要
As a continuation of a previous study of isolation numbers in Cartesian and lexicographic products, we investigate isolation numbers and, more generally, $\cal F$-isolation numbers in direct, strong, lexicographic, and Cartesian products of graphs. For direct products, we derive upper bounds for the $\{K_{n_1,\ldots,n_d}\}$-isolation number in terms of isolation and total domination parameters of the factors, and establish lower bounds based on open packings. We also determine exact values for several infinite families of direct products, including $ι(P_{4\ell}\times C_{2k+1})=\ell(k+1)$. For strong products, we prove a general lower bound on $ι(G\,\boxtimes\, H,{\cal F})$ involving the $2$-packing number and provide an upper bound on $ι(G\,\boxtimes\, H)$. For lexicographic products, we determine the $\cal F$-isolation number in several general settings, obtaining exact formulas in terms of domination and total domination numbers of the first factor. Finally, for Cartesian products, we extend results from our previous work to arbitrary graph families $\cal F$. We introduce $\cal F$-isolation graphs and use $\cal F$-transversals to derive general upper bounds, together with corresponding lower bounds.
Comments11 pages
