  Given a graph $G=(V,E)$, the maximum bond problem searches for a maximum cut
$\delta(S) \subseteq E$ with $S \subseteq V$ such that $G[S]$ and $G[V\setminus
S]$ are connected. This problem is closely related to the well-known maximum
cut problem and known under a variety of names such as largest bond, maximum
minimal cut and maximum connected (sides) cut. The bond polytope is the convex
hull of all incidence vectors of bonds. Similar to the connection of the
corresponding optimization problems, the bond polytope is closely related to
the cut polytope. While cut polytopes have been intensively studied, there are
no results on bond polytopes. We start a structural study of the latter.
  We investigate the relation between cut- and bond polytopes and study the
effect of graph modifications on bond polytopes and their facets. Moreover, we
study facet-defining inequalities arising from edges and cycles for bond
polytopes. In particular, these yield a complete linear description of bond
polytopes of cycles and $3$-connected planar $(K_5-e)$-minor free graphs.
Moreover we present a reduction of the maximum bond problem on arbitrary graphs
to the maximum bond problem on $3$-connected graphs. This yields a linear time
algorithm for maximum bond on $(K_5-e)$-minor free graphs.
