Graphs with Gp-connected medians

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 ≥ 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 (SIAM J Discrete Math 15(2):268–282, 2002. https://doi.org/10.1137/S089548019936360X ). 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.