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On the k-restricted structure ratio in planar and outerplanar graphs

Abstract : 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.
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Gruia Călinescu, Cristina G. Fernandes. On the k-restricted structure ratio in planar and outerplanar graphs. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2008, 10 (3), pp.135--147. ⟨hal-00972327⟩

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