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：在图 $G$ 中，若割的两个部分均在 $G$ 中诱导出连通子图，则该割称为一个{\em 键}，而{\em 键多面体}是所有键的凸包。计算最大权重键是一个NP难问题，即使对于平面图也是如此。然而，该问题在$(K_5 \setminus e)$-无子式图上可在线性时间内求解，更一般地，在有界树宽图上也可求解，这本质上归因于通过团和分解为更简单的图。我们展示了如何从图$G_1$和$G_2$的键多面体，获得$G_1$与$G_2$进行$1$-和或$2$-和所得图的键多面体。利用这一点，我们证明了$(K_5 \setminus e)$-无子式图的键多面体的扩展复杂度是线性的。在此工作之前，键多面体的线性规模描述仅已知于$3$-连通平面$(K_5 \setminus e)$-无子式图，本质上仅已知于轮图。我们还描述了一个针对$(K_5\setminus e)$-无子式图上\MaxBond问题的初等线性时间算法。在此工作之前，该设定下的线性时间算法是已知的。然而，大O表示法中的隐藏常数很大，因为该算法依赖于有界树宽图的线性时间算法这一重型工具，并将其作为黑盒使用。