Vulnerability bounds on the number of spanning tree leaves

Abstract

Hamiltonicity and vulnerability of graphs are in a strong connection. A basic necessary condition states that a graph containing a 2-leaf spanning tree (that is a Hamiltonian path) cannot be split into more than k + 1 components by deleting k of its vertices. In this paper we consider a more general approach and investigate the connection between the number of spanning tree leaves and two vulnerability parameters namely scattering number sc( G ) and cut-asymmetry ca( G ). We prove that any spanning tree of a graph G has at least sc( G ) + 1 leaves. We also show that if X $\subset$ V is a maximum cardinality independent set of G = ( V , E ) such that the elements of X are all leaves of a particular spanning tree then | X | = ca( G ) + 1 = | V | − cvc( G ), where cvc( G ) is the size of a minimum connected vertex cover of G . As a consequence we obtain a new proof for the following results: any spanning tree with independent leaves provides a 2-approximation for both MaximumInternalSpanningTree and MinimumConnectedVertexCover problems. We also consider the opposite point of view by fixing the number of leaves to q and looking for a q -leaf subtree of G that spans a maximum number of vertices. Bermond proved that a 2-connected graph on n vertices always contains a path (a 2-leaf subtree) of length min {n,δ 2 }, where δ 2 is the minimum degree-sum of a 2-element independent set. We generalize this result to obtain a sufficient condition for the existence of a large q -leaf subtree.