S-distance in trees
For a connected graph G, a subset S of V(G) and vertices u, v of G, the S-distance ds(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 es(v) of a vertex v is the maximum S-distance from v to each vertex of G and the S-diameter diamsG is the maximum S-eccentricity among the vertices of G. For a tree T, a formula is given for ds(u,v) and for S ≠ ϕ, it is shown that diamsT is even. Finally, the complexity of finding ds(u,v) is discussed.
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