We present online deterministic algorithms for minimum coloring and minimum dominating set problems in the context of geometric intersection graphs. We consider a graph parameter: the independent kissing number $\zeta$, which is a number equal to `the size of the largest induced star in the graph $-1$'. For a graph with an independent kissing number at most $\zeta$, we show that the famous greedy algorithm achieves an optimal competitive ratio of $\zeta$ for the minimum dominating set and the minimum independent dominating set problems. However, for the minimum connected dominating set problem, we obtain a competitive ratio of at most $2\zeta$. To complement this, we prove that for the minimum connected dominating set problem, any deterministic online algorithm has a competitive ratio of at least $2(\zeta-1)$ for the geometric intersection graph of translates of a convex object in $\mathbb{R}^2$. Next, for the minimum coloring problem, we obtain algorithms having a competitive ratio of $O\left({\zeta'}{\log m}\right)$ for geometric intersection graphs of bounded scaled $\alpha$-fat objects in $\mathbb{R}^d$ having widths in the interval $[1,m]$, where $\zeta'$ is the independent kissing number of the geometric intersection graph of bounded scaled $\alpha$-fat objects having widths in the interval $[1,2]$. Finally, we investigate the value of $\zeta$ for geometric intersection graphs of various families of geometric objects.

翻译：我们提出了针对最小着色和最小支配集问题的在线确定性算法，研究对象为几何交图。考虑图参数：独立亲吻数 $\zeta$，该参数等于“图中最大诱导星图的规模减1”。对于独立亲吻数不超过 $\zeta$ 的图，我们证明经典贪心算法在最小支配集和最小独立支配集问题上实现了最优竞争比 $\zeta$。然而，对于最小连通支配集问题，我们得到的竞争比至多为 $2\zeta$。作为补充，我们证明在 $\mathbb{R}^2$ 中凸对象平移形成的几何交图上，任何确定性在线算法的最小连通支配集问题竞争比至少为 $2(\zeta-1)$。接下来，针对最小着色问题，我们获得了竞争比为 $O\left({\zeta'}{\log m}\right)$ 的算法，该算法适用于 $\mathbb{R}^d$ 中宽度在区间 $[1,m]$ 内的有界缩放 $\alpha$-胖对象的几何交图，其中 $\zeta'$ 为宽度在区间 $[1,2]$ 内的有界缩放 $\alpha$-胖对象几何交图的独立亲吻数。最后，我们研究了各类几何对象族对应的几何交图中 $\zeta$ 的取值。