Heavy paths, light stars, and big melons

A graph <f>H</f> is defined to be light in a family <f>H</f> of graphs if there exists a finite number <f>w(H,H)</f> such that each <f>G∈H</f> which contains <f>H</f> as a subgraph, contains also a subgraph <f>KH</f> such that the sum of degrees (in <f>G</f>) of the vertices of <f>K</f> (that is, the weight of <f>K</f> in <f>G</f>) is at most <f>w(H,H)</f>. In this paper we study the conditions related to the weight of fixed subgraphs of the plane graphs which can enforce the existence of light graphs in some families of plane graphs. For the families of plane graphs and triangulations whose edges are of weight <f>⩾w</f> we study the necessary and sufficient conditions for the lightness of certain graphs according to values of <f>w</f>. [Copyright &y& Elsevier]