  Betweenness centrality---measuring how many shortest paths pass through a
vertex---is one of the most important network analysis concepts for assessing
the relative importance of a vertex. The well-known algorithm of Brandes [J.
Math. Sociol.~'01] computes, on an $n$-vertex and $m$-edge graph, the
betweenness centrality of all vertices in $O(nm)$ worst-case time. In later
work, significant empirical speedups were achieved by preprocessing degree-one
vertices and by graph partitioning based on cut vertices. We contribute an
algorithmic treatment of degree-two vertices, which turns out to be much richer
in mathematical structure than the case of degree-one vertices. Based on these
three algorithmic ingredients, we provide a strengthened worst-case running
time analysis for betweenness centrality algorithms. More specifically, we
prove an adaptive running time bound $O(kn)$, where $k < m$ is the size of a
minimum feedback edge set of the input graph.
