  In this paper, we investigate the graphs in which all balls are convex and
the groups acting on them geometrically (which we call CB-graphs and
CB-groups). These graphs have been introduced and characterized by Soltan and
Chepoi (1983) and Farber and Jamison (1987). CB-graphs and CB-groups generalize
systolic (alias bridged) and weakly systolic graphs and groups, which play an
important role in geometric group theory.
  We present metric and local-to-global characterizations of CB-graphs. Namely,
we characterize CB-graphs $G$ as graphs whose triangle-pentagonal complexes
$X(G)$ are simply connected and balls of radius at most $3$ are convex.
Similarly to systolic and weakly systolic graphs, we prove a dismantlability
result for CB-graphs $G$: we show that their squares $G^2$ are dismantlable.
This implies that the Rips complexes of CB-graphs are contractible. Finally, we
adapt and extend the approach of Januszkiewicz and Swiatkowski (2006) for
systolic groups and of Chalopin et al. (2020) for Helly groups, to show that
the CB-groups are biautomatic.
