For S E V(G) the S-center and S-centroid of G are defined as the collection of vertices u E V(G) that minimize eJu) = max {d (u, v ) : V E S} and & ( u ) =IveS d (u, v), respectively. This generalizes the standard definition of center and centroid from the.special case of S = V(G). For
is defined to be the subset of vertices u in G for which fk(u) is a minimum. This also generalizes the standard definitions of center and centroid since C(G; 1) is the center and C ( G ; (V(G)() is the centroid. In this paper the structure of these sets for trees is examined. Generalizations of theorems of Jordan and Zelinka are included.
1. Introduction
Consider a finite, connected, undirected graph G with vertex set V(G) and edge set E(G). An edge e E E ( G ) is identified with the unordered pair (u, u) of vertices, where u and u are the vertices that e connects, and in this paper each edge will be assumed to have a length of one. The distance in G between vertices u and u, denoted d(u, u), is therefore the minimum number of edges in a u to u path.
Graphs are often used to model such things as street networks and communication networks, and many mathematical problems have arisen