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The majority strategy on graphs. (English) Zbl 0888.05025
An interval I(u,v) in a graph G, where u, v are vertices, is the set of all vertices w of G for which the equality d(u,w)+d(w,v)=d(u,v) holds, where d denotes the distance. If for any three vertices u, v, w of G the intersection I(u,v)I(v,w)I(w,u) consists of one vertex only, then G is called a median graph. A profile of length p on a graph G is a finite sequence v 1 ,v 2 ,,v p of vertices of G; its median is a vertex x of G for which i=1 p d(x,v i ) is minimal. The set of all medians of a profile π is the median set M(π). In the paper a strategy for finding M(π) for a given profile π is described; it is called the majority strategy. It begins in a vertex of G and consists of certain successive moves from one vertex to another along an edge. It is proved that the majority strategy produces the median set M(π) for each profile π of G, independently of the initial position, if and only if G is a median graph.

MSC:
05C12Distance in graphs
05C75Structural characterization of families of graphs
05C99Graph theory
90B80Discrete location and assignment