Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. To be more formal, a graph $G$ is a median graph if, for all $\mu, u,v\in V(G)$, it holds that $|I(\mu,u)\cap I(\mu,v)\cap I(u,v)|=1$ where $I(x,y)$ denotes the set of all vertices that lie on shortest paths connecting $x$ and $y$. In this paper we are interested in a natural generalization of median graphs, called $k$-median graphs. A graph $G$ is a $k$-median graph, if there are $k$ vertices $\mu_1,\dots,\mu_k\in V(G)$ such that, for all $u,v\in V(G)$, it holds that $|I(\mu_i,u)\cap I(\mu_i,v)\cap I(u,v)|=1$, $1\leq i\leq k$. By definition, every median graph with $n$ vertices is an $n$-median graph. We provide several characterizations of $k$-median graphs that, in turn, are used to provide many novel characterizations of median graphs.

翻译：中位数图是连通图，其中对于任意三个顶点，存在唯一的一个顶点属于这三个顶点中每对顶点之间的最短路径。更形式化地说，图$G$是一个中位数图，如果对于所有$\mu, u,v\in V(G)$，有$|I(\mu,u)\cap I(\mu,v)\cap I(u,v)|=1$，其中$I(x,y)$表示连接$x$和$y$的所有最短路径上的顶点集合。在本文中，我们关注中位数图的一个自然推广，称为$k$-中位数图。图$G$是一个$k$-中位数图，如果存在$k$个顶点$\mu_1,\dots,\mu_k\in V(G)$，使得对于所有$u,v\in V(G)$，有$|I(\mu_i,u)\cap I(\mu_i,v)\cap I(u,v)|=1$，$1\leq i\leq k$。根据定义，每个具有$n$个顶点的中位数图都是一个$n$-中位数图。我们给出了$k$-中位数图的若干刻画，而这些刻画反过来又用于提供中位数图的许多新刻画。