The sizes of maximal planar, outerplanar, and bipartite planar subgraphs

Abstract

We define the subvariance S℘(ℱ) of a family of graphs ℱ with respect to property ℱ to be the infimum of the ratio |H1|/|H2|, where H1 and H2 are any two maximal spanning subgraphs of G with property ℘, and where G is a member of ℱ. It is shown that, for the family of all connected graphs, the subvariance when ℘ is planar, outerplanar, and bipartite planar, is 1/2, 1/2, and 1/2, respectively.