发表机构
University of Haifa-Oranim; University of Ljubljana; Rudolfovo – Science and Technology Centre Novo Mesto; University of Novo Mesto（海法奥拉尼姆大学; 卢布尔雅那大学; 新梅斯托鲁多尔福沃科学与技术中心; 新梅斯托大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该文改进了极大外平面图公平支配数的上界至(7n-3)/8，并构造无穷族使比值趋于7/9，从而确定最佳渐近常数在7/9与7/8之间。
AI 中文摘要
图$G$的一个支配集$D$称为公平支配集，如果$D$外的任意两个顶点在$D$中的邻居数相同，公平支配数$\mathrm{fd}(G)$是这种集合的最小基数。引入该参数的Caro、Hansberg和Henning证明了对于每个阶数$n\geq3$的极大外平面图$G$，有$\mathrm{fd}(G)<17n/19$，并询问该界是否渐近最优。我们证明并非如此，通过证明对于每个阶数$n\geq3$的极大外平面图$G$，有$\mathrm{fd}(G)\leq(7n-3)/8<7n/8$，并构造了一个无穷极大外平面图族，使得$\mathrm{fd}(G)/n\rightarrow7/9$，因此最佳渐近常数介于$7/9$和$7/8$之间。
英文摘要
A dominating set $D$ of a graph $G$ is a \emph{fair dominating set} if every two vertices outside $D$ have the same number of neighbors in $D$, and the \emph{fair domination number} $\mathrm{fd}(G)$ is the minimum cardinality of such a set. Caro, Hansberg and Henning, who introduced this parameter, proved that $\mathrm{fd}(G)<17n/19$ for every maximal outerplanar graph $G$ of order $n\geq3$, and asked whether this bound is asymptotically best possible. We show that it is not the case by proving $\mathrm{fd}(G)\leq(7n-3)/8<7n/8$ for every maximal outerplanar graph $G$ of order $n\geq3$, and we exhibit an infinite family of maximal outerplanar graphs with $\mathrm{fd}(G)/n\rightarrow7/9$, so that the best asymptotic constant lies between $7/9$ and $7/8$.
Comments16 pages, 1 figure
