平面图和单位圆盘图的支配-打包比率
Domination-packing ratio for planar and unit disk graphs
浏览论文内容
中文总结 AI 辅助
研究平面图和单位圆盘图中支配数与打包数的比率，证明其上界并改进了相关结果。
中文摘要 AI 辅助
图G的支配数γ(G)是能够与G中每个半径为1的球相交的最小顶点集的大小，而打包数ρ(G)是最大数量的两两顶点不相交的半径为1的球。我们证明对于每一个平面图，γ(G)/ρ(G)≤5，而对于每一个单位圆盘图，γ(G)/ρ(G)≤18√3/π≈9.924，从而在两种图类中给出了半径为1的球的超图的埃罗-波萨型界。这一结果改进了Gutiérrez和Paul，以及Dúcz和Gujgiczer之前的结果，而他们又改进了Bonamy、Csikós、Gujgiczer和Yuditsky，以及Böhme和Mohar的下界。对于这两种图类，目前最佳已知的最优常数下界仍为3。
英文摘要
The domination number $γ(G)$ of a graph $G$ is the smallest possible size of a vertex set that intersects every radius-$1$ ball of $G$, and the packing number $ρ(G)$ is the maximum number of pairwise vertex-disjoint radius-$1$ balls. We prove that $\frac{γ(G)}{ρ(G)}\le 5$ for every planar graph and $\frac{γ(G)}{ρ(G)} \le \frac{18\sqrt3}π\approx 9.924$ for every unit disk graph, thus yielding Erdős-Pósa-type bounds for the hypergraph of radius-$1$ balls in the two graph classes. This improves upon results of Gutiérrez and Paul, and Dúcz and Gujgiczer, who in turn lowered bounds of Bonamy, Csikós, Gujgiczer and Yuditsky, and Böhme and Mohar. For both graph classes, the best known lower bound on the optimal constant remains $3$.
