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The median procedure on median graphs. (English) Zbl 0906.05023
An ordered $k$-tuple $\pi =\left({x}_{1},\cdots ,{x}_{k}\right)$ of elements of a finite metric space $\left(X,d\right)$ is called a profile. An element $x\in X$ for which ${\sum }_{i=1}^{k}d\left(x,{x}_{i}\right)$ is minimum is called a median of $\pi$. The median procedure is the function $\text{Med}\left(\pi \right)=\left\{x\mid x$ is a median of $\pi \right\}$. These concepts are studied in the case when $X$ is the vertex set of a graph with the usual distance. In particular, they are studied for median graphs, i.e. connected graphs $G$ in which for any three vertices $x$, $y$, $z$ there is a unique vertex $w$ on the geodesic between each pair of $x$, $y$, $z$. Special attention is paid to cube-free median graphs.

MSC:
 05C12 Distance in graphs
Keywords:
profile; median; distance