Online Dominating Set and Coloring

In this paper, we present online deterministic algorithms for minimum coloring, minimum dominating set and its variants 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 obtain an algorithm having 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 \(2\zeta \) . In addition, 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 present an algorithm 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 a width in between [1, m], where \(\zeta '\) is the independent kissing number of the geometric intersection graph of bounded scaled \(\alpha \) -fat objects having a width in between [1, 2]. Finally, we investigate the value of \(\zeta \) for geometric intersection graphs of various families of geometric objects.