The median of a graph $G$ with weighted vertices is the set of all vertices $x$ minimizing the sum of weighted distances from $x$ to the vertices of $G$. For any integer $p\ge 2$, we characterize the graphs in which, with respect to any non-negative weights, median sets always induce connected subgraphs in the $p$th power $G^p$ of $G$. This extends some characterizations of graphs with connected medians (case $p=1$) provided by Bandelt and Chepoi (2002). The characteristic conditions can be tested in polynomial time for any $p$. We also show that several important classes of graphs in metric graph theory, including bridged graphs (and thus chordal graphs), graphs with convex balls, bucolic graphs, and bipartite absolute retracts, have $G^2$-connected medians. Extending the result of Bandelt and Chepoi that basis graphs of matroids are graphs with connected medians, we characterize the isometric subgraphs of Johnson graphs and of halved-cubes with connected medians.

翻译：摘要：带权顶点图$G$的中位数是指所有最小化$x$到$G$各顶点加权距离之和的顶点$x$的集合。对任意整数$p\ge 2$，我们刻画了那些对于任意非负权重，中位数集在$G$的$p$次幂$G^p$中始终导出连通子图的图。这推广了Bandelt和Chepoi（2002）关于连通中位数图（$p=1$情形）的若干刻画特征。对于任意$p$，这些特征条件均可在多项式时间内检验。我们还证明了度量图论中的若干重要图类（包括桥接图（从而弦图）、凸球图、田园图以及二部绝对收缩图）均具有$G^2$连通中位数。通过推广Bandelt和Chepoi关于拟阵基图具有连通中位数的结论，我们刻画了Johnson图与半立方图的等距子图中具有连通中位数的图类。