Graphs with $G^p$-connected medians
Laurine B\'en\'eteau, J\'er\'emie Chalopin, Victor Chepoi, Yann, Vax\`es

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 with weighted vertices is the set of all vertices minimizing the sum of weighted distances from to the vertices of . For any integer , we characterize the graphs in which, with respect to any non-negative weights, median sets always induce connected subgraphs in the th power of . This extends some characterizations of graphs with connected medians (case ) provided by Bandelt and Chepoi (2002). The characteristic conditions can be tested in polynomial time for any . 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 -connected medians. Extending the result of Bandelt and Chepoi that basis graphs of matroids are graphs with connected medians, we characterize the…
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