Some properties of graph centroids

The concept of a branch weight centroid has been extended in [12] so that it can be defined for an arbitrary finite set X with a distinguished family 𝒞 of ‘convex’ subsets of X. In particular, the centroid of a graph G was defined for X to be the vertex set V(G) of G and UℝV(G) is convex if it is the vertex set of a chordless path in G. In this paper, which is an extended version of [13], the authors give necessary and sufficient conditions for a graph to be a centroid of another graph as well as of itself. Then, they apply these results to some particular classes of graphs: chordal, Halin, series-parallel and outerplanar.