Lightness, heaviness and gravity

The gravity g(H,H)g(H,H) of a graph H   in the family of graphs HH is the greatest integer n with the property that for every integer m  , there exists a supergraph G∈HG∈H of H such that each subgraph of G, which is isomorphic to H, contains at least n   vertices of degree ⩾m⩾m in G. We study the basic properties of the gravity function for various families of plane graphs. We also introduce and study the almost-light graphs and the absolutely heavy graphs. The paper concludes with few open problems.