Călinescu, Gruia and Fernandes, Cristina G. - On the k-restricted structure ratio in planar and outerplanar graphs

dmtcs:423 - Discrete Mathematics & Theoretical Computer Science, January 1, 2008, Vol. 10 no. 3
On the k-restricted structure ratio in planar and outerplanar graphs

Authors: Călinescu, Gruia and Fernandes, Cristina G.

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.


Source : oai:HAL:hal-00972327v1
Volume: Vol. 10 no. 3
Section: Graph and Algorithms
Published on: January 1, 2008
Submitted on: March 26, 2015
Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]


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