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

摘要

We define the subvariance Sp(F) of a family of graphs F with respect to property P to be the infimum of the ratio |H1/|H2|, where H1 and H2 are any two maximal spanning subgraphs of G with property P, and where G is a member of F. It is shown that, for the family of all connected graphs, the subvariance when P is planar, outerplanar, and bipartite planar, is 1/3, 1/2, and 1/2, respectively.