AI 中文总结
研究图中全外独立联盟相关概念，引入TOI联盟、TOI联盟划分及TOI联盟数等定义。通过研究其存在性，建立\(C_t^{oi}(G)\)的紧上界，最终得出一些图类的\(C_t^{oi}(G)\)精确值。
AI 中文摘要
在图\(G\)中，若\(G\)的每个顶点都与\(D\)中的至少一个顶点相邻，且\(V(G)\setminus D\)是\(G\)的独立集，则顶点集\(D\)是\(G\)的全外独立支配集（TOIDS）。\(G\)中的一个TOI联盟由\(G\)的两个不相交顶点集\(A\)和\(B\)组成，它们都不是TOIDS，但它们的并集\(A\cup B\)是\(G\)的TOIDS。我们称\(A\)和\(B\)形成一个TOI联盟，且是TOI联盟伙伴。\(G\)中的一个TOI联盟划分是一个顶点划分\(\Psi =\{V_1, V_2,..., V_k\}\)，其中每个集合都与\(\Psi\)中的另一个集合形成TOI联盟。TOI联盟数\(C_t^{oi}(G)\)是\(G\)的所有TOI联盟划分中的最大基数。本文引入并研究了上述概念，研究了TOI联盟划分的存在性，建立了\(C_t^{oi}(G)\)的几个紧上界，最后得到了一些图类的\(C_t^{oi}(G)\)的精确值。
英文摘要
A set $D$ of vertices graph $G$ is a total outer-independent dominating set (TOIDS) of $G$ if every vertex of $G$ is adjacent to at least one vertex in $D$, and $V(G)\setminus D$ is an independent set of $G$. A TOI-coalition in $G$ comprises two disjoint sets of vertices $A$ and $B$ of $G$, neither of which is a TOIDS but whose union $A\cup B$ is a TOIDS of $G$. We say that the sets $A$ and $B$ form a TOI-coalition, and are TOI-coalition partners. A TOI-coalition partition in $G$ is a vertex partition $Ψ=\{V_1, V_2,..., V_k\}$ in which every set forms a TOI-coalition with another set in $Ψ$. The TOI-coalition number $C_t^{oi}(G)$ is the maximum cardinality among all TOI-coalition partitions of $G$. In this work, the above-mentioned concepts are introduced and studied. The existence of a TOI-coalition partition is investigated. Several sharp upper bounds on $C_t^{oi}(G)$ are established. Finally, the exact values of $C_t^{oi}(G)$ for some graph classes are obtained.
