A cut in a graph $G$ is called a {\em bond} if both parts of the cut induce connected subgraphs in $G$, and the {\em bond polytope} is the convex hull of all bonds. Computing the maximum weight bond is an NP-hard problem even for planar graphs. However, the problem is solvable in linear time on $(K_5 \setminus e)$-minor-free graphs, and in more general, on graphs of bounded treewidth, essentially due to clique-sum decomposition into simpler graphs. We show how to obtain the bond polytope of graphs that are $1$- or $2$-sum of graphs $G_1$ and $ G_2$ from the bond polytopes of $G_1,G_2$. Using this we show that the extension complexity of the bond polytope of $(K_5 \setminus e)$-minor-free graphs is linear. Prior to this work, a linear size description of the bond polytope was known only for $3$-connected planar $(K_5 \setminus e)$-minor-free graphs, essentially only for wheel graphs. We also describe an elementary linear time algorithm for the \MaxBond problem on $(K_5\setminus e)$-minor-free graphs. Prior to this work, a linear time algorithm in this setting was known. However, the hidden constant in the big-Oh notation was large because the algorithm relies on the heavy machinery of linear time algorithms for graphs of bounded treewidth, used as a black box.

翻译：图$G$中的割称为{\em 键}，若该割的两部分均在$G$中诱导出连通子图，而{\em 键多面体}是所有键的凸包。计算最大权键即使对于平面图也是NP难问题。然而，该问题在$(K_5 \setminus e)$-无 minors 图及更一般的树宽有界图上可在线性时间内求解，原因在于可通过团和分解将其简化为更简单的图。本文展示了如何从图$G_1$和$G_2$的键多面体出发，得到它们进行$1$-和或$2$-和运算后所得图的键多面体。利用这一结果，我们证明$(K_5 \setminus e)$-无 minors 图的键多面体的扩展复杂度是线性的。在此之前，仅对$3$-连通可平面$(K_5 \setminus e)$-无 minors 图（主要是轮图）已知键多面体的线性规模描述。我们还针对$(K_5 \setminus e)$-无 minors 图上的\MaxBond 问题描述了一个基础线性时间算法。在此之前，已知存在该情景下的线性时间算法，但由于该算法将树宽有界图的线性时间算法的重型机制作为黑箱使用，大$O$记号中的隐式常数较大。