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Medians in median graphs. (English) Zbl 0536.05057
If G is an undirected graph, V(G) is its vertex set and AV(G), then a median of A is any vertex of G which has the minimal sum of distance from all vertices of A. If each subset of V(G) having three vertices has exactly one median, then G is called a median graph. The properties of median graphs are studied. The interrelation between median graphs and median semilattices is shown; a median semilattice is a meet semilattice (X,) such that every principle ideal { x|xa} is a distributive lattice and any three elements have an upper bound whenever each pair of them does. At the end of the paper the concept of a local median is introduced and interrelations between medians and Condorcet vertices are described.
Reviewer: B.Zelinka
MSC:
05C99Graph theory
05C38Paths; cycles
06A12Semilattices