https://hal.inria.fr/hal-00972327Călinescu, GruiaGruiaCălinescuUIC CS - Departement of Computer Science [UIC] - UIC - University of Illinois [Chicago] - University of Illinois SystemFernandes, Cristina G.Cristina G.FernandesIME-USP - Department of Computer Science - USP - University of São PauloOn the k-restricted structure ratio in planar and outerplanar graphsHAL CCSD2008[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]Inria Sophia Antipolis-Méditerranée / I3s, Service Ist2014-04-03 16:13:222019-08-01 14:12:402014-04-03 16:39:43enJournal articleshttps://hal.inria.fr/hal-00972327/document10.46298/dmtcs.423application/pdf1A 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.