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The majority strategy on graphs. (English) Zbl 0888.05025
An interval $I\left(u,v\right)$ in a graph $G$, where $u$, $v$ are vertices, is the set of all vertices $w$ of $G$ for which the equality $d\left(u,w\right)+d\left(w,v\right)=d\left(u,v\right)$ holds, where $d$ denotes the distance. If for any three vertices $u$, $v$, $w$ of $G$ the intersection $I\left(u,v\right)\cap I\left(v,w\right)\cap I\left(w,u\right)$ 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 ${v}_{1},{v}_{2},\cdots ,{v}_{p}$ of vertices of $G$; its median is a vertex $x$ of $G$ for which ${\sum }_{i=1}^{p}d\left(x,{v}_{i}\right)$ is minimal. The set of all medians of a profile $\pi$ is the median set $M\left(\pi \right)$. In the paper a strategy for finding $M\left(\pi \right)$ for a given profile $\pi$ 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 $M\left(\pi \right)$ for each profile $\pi$ of $G$, independently of the initial position, if and only if $G$ is a median graph.

##### MSC:
 05C12 Distance in graphs 05C75 Structural characterization of families of graphs 05C99 Graph theory 90B80 Discrete location and assignment
##### Keywords:
interval; median graph; profile; median set; majority strategy