a b s t r a c tGiven an undirected connected graph G we consider the problem of finding a spanning tree of G which has a maximum number of internal (non-leaf) vertices among all spanning trees of G. This problem, called Maximum Internal Spanning Tree problem, is clearly NP-hard since it is a generalization of the Hamiltonian Path problem. From the optimization point of view the Maximum Internal Spanning Tree problem is equivalent to the Minimum Leaf Spanning Tree problem. However, the two problems have different approximability properties. Lu and Ravi proved that the latter has no constant factor approximation -unless P = NP -, while Salamon and Wiener gave a linear-time 2-approximation algorithm for the Maximum Internal Spanning Tree problem.In this paper, we improve this approximation ratio by giving an O(|V (G)| 4 )-time 7/4-approximation algorithm for graphs without pendant vertices. Our approach is based on the successive execution of local improvement steps. We use a linear programming formulation and a primal-dual technique to prove the approximation ratio. We also investigate the vertex-weighted case, that is to find a spanning tree of a vertex-weighted graph G in which the weight sum of internal vertices is maximal among all spanning trees of G. For this problem we present a (2∆(G) − 3)-approximation algorithm, where ∆(G) is the maximum vertex-degree of G. A slight modification of this algorithm ensures a 2-approximation whenever the input graph is claw-free. Both algorithms run in O(|V (G)| 4 ) time for graphs with no pendant vertices.
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)[10] and cut-asymmetry ca(G) [16]. We prove that any spanning tree of a graph G has at least sc(G) + 1 leaves. We also show that if X ⊂ 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 the MAXIMUM INTERNAL SPANNING TREE [16] and the MINIMUM CONNECTED VERTEX COVER [17] 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 [2] 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.
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