S-distance in trees

Abstract

For a connected graph G, a subset S of V(G) and vertices u, v of G, the S-distance d _s (u,v) from u to v is the length of a shortest u — v walk in G that contains every vertex of S. The S-eccentricity e _s (v) of a vertex v is the maximum S-distance from v to each vertex of G and the S-diameter diam _s G is the maximum S-eccentricity among the vertices of G. For a tree T, a formula is given for d _s (u,v) and for S ≠ ϕ, it is shown that diam _s T is even. Finally, the complexity of finding d _s (u,v) is discussed.