Discrete Mathematics & Theoretical Computer Science,Vol 10, No 3 (2008)

摘要

A planar k-restricted structure is a simple graph whose blocks are planarand each has at most k vertices. Planar k-restricted structures are usedby 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.