Scattered classes of graphs

Abstract

For a class C of graphs G equipped with functions f(G) defined on subsets of E(G) or V (G), we say that C is k-scattered with respect to f(G) if there exists a constant .e such that for every graph G is an element of C, the domain of f(G) can be partitioned into subsets of size at most k so that the union of every collection of the subsets has f(G) value at most We present structural characterizations of graph classes that are k-scattered with respect to several graph connectivity functions. In particular, our theorem for cut-rank functions provides a rough structural characterization of graphs having no mK(1,n) vertex-minor, which allows us to prove that such graphs have bounded linear rank-width.