On constructive characterizations of -sparse graphs

In this paper we study constructive characterizations of graphs satisfying tree-connectivity requirements. The main result is the following: if k and l are positive integers and l ≤ k 2 , then a necessary and sufficient condition is proved for a node being the last node of a construction in a graph having at most k | X | − ( k + l ) induced edges in every subset X of nodes. The arguments and proofs extend those of Frank and Szegő for the case l = 1 [A. Frank, L. Szegő, Constructive characterizations on packing and covering by trees, Discrete Appl. Math. 131 (2) (2003) 347–371].