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arXiv · 2201.12248

Graphs with $G^p$-connected medians

Laurine Bénéteau·Jérémie Chalopin·Victor Chepoi·Yann Vaxès
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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.

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Laurine Bénéteau, Jérémie Chalopin, Victor Chepoi, Yann Vaxès. 2022-01-28. Graphs with $G^p$-connected medians. https://doi.org/10.1007/s10107-023-01939-3

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