Sufficient conditions for connected betweenness centers

Find sufficient conditions on a graph that guarantee its betweenness center is connected.

Background

The paper establishes several localization results for betweenness centers, including conditions under which the center lies within a single block, but its examples show that the betweenness center need not be connected. In particular, the examples demonstrate disconnected or delocalized centers, prompting a separate structural problem.

The unresolved question is to identify graph properties or hypotheses that ensure connectivity of the subgraph induced by all vertices attaining maximum betweenness.