%0 Journal Article
%T On the k-Structure Ratio in Planar and Outerplanar Graphs
%A Gruia Calinescu
%A Cristina G. Fernandes
%J Discrete Mathematics & Theoretical Computer Science
%D 2008
%I Discrete Mathematics & Theoretical Computer Science
%X A planar k-restricted structure is a simple graph whose blocks are planar and each has at most k vertices. Planar k-restricted structures are used by approximation algorithms for Maximum Weight Planar Subgraph, which motivates this work. The planar k-restricted ratio is the infimum, over simple planar graphs H, of the ratio of the number of edges in a maximum k-restricted structure subgraph of H to the number edges of H. We prove that, as k tends to infinity, the planar k-restricted ratio tends to 1/2. The same result holds for the weighted version. Our results are based on analyzing the analogous ratios for outerplanar and weighted outerplanar graphs. Here both ratios tend to 1 as k goes to infinity, and we provide good estimates of the rates of convergence, showing that they differ in the weighted from the unweighted case.
%U http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/961