Graphs with $G^p$-connected medians

TL;DR

This paper characterizes graphs where median sets always induce connected subgraphs in the pth power, extending previous results for p=1 and including various important graph classes with polynomial-time testability.

Contribution

It generalizes the characterization of graphs with connected medians to any p≥2 and identifies several key classes with this property, providing polynomial-time testing methods.

Findings

Graphs with connected medians in G^p are characterized for p≥2.

Several important graph classes, including bridged and bipartite graphs, have G^2-connected medians.

Polynomial-time algorithms exist for testing the median connectivity conditions.

Abstract

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.