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.

翻译：根据一个图形 $G= (V,E) $G 最大债券问题, 最大债券问题寻找最大削减 $delta(S)\ subseteq E$ $S = subseteq E$ $S = subseteque V$S 美元。 最大债券问题与已知的最大削减问题密切相关, 以诸如最大债券、 最大最小削减和最大连接( 侧面) 等不同名称为名。 债券聚点是所有债券事件矢量的锥体。 类似相应的优化问题, 债券聚点与切断聚点密切相关。 虽然对切断点的多点和 $ G[V\ setminus S] 美元有密切关联。 这个问题与已知的最大最大削减问题密切相关。 我们调查了切断点和债券多点( 侧面) 的图形修改效果。 此外, 我们研究了债券的边缘和周期( 美元 的边缘和周期 。 特别是, 在债券- 直径 的直径的债券的直径平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面图上, 。