Bond Polytope of Graphs

• Bond polytope is defined as the convex hull of incidence vectors of bonds, capturing minimal edge cuts that preserve graph connectivity.
• Its structure is characterized by facet-defining inequalities from cycles and specialized graph decompositions such as 1-sum and 2-sum operations.
• Efficient algorithms, including dynamic programming and recursive decomposition, enable linear-time solutions for the NP-hard MaxBond problem in specific graph classes.

A bond in a graph is an edge cut whose removal separates the vertex set into two connected induced subgraphs. The bond polytope of a graph $G=(V,E)$, denoted $P_{\rm bond}(G)$, is the convex hull of the incidence vectors of all bonds of $G$ in $\mathbb{R}^{E}$. The study of the bond polytope, motivated by its connections to the maximum bond problem and polyhedral combinatorics, provides deep insight into graph connectivity, decomposition theory, and the extension complexity of network design polytopes. The bond polytope is intimately related to, but more constrained than, the classical cut polytope. Recent work has produced a complete structural theory, classified its defining inequalities in key graph classes, characterized its behavior under standard graph compositions, and established optimal algorithms and extended formulations for its maximization problem in $(K_5\setminus e)$-minor-free graphs (Chimani et al., 2020, Kolman et al., 16 Jan 2026).

1. Formal Definitions and Fundamental Properties

Let $G=(V,E)$ be a finite undirected graph. For any $S \subseteq V$, the edge-cut $\delta_G(S)$ is

$$\delta_G(S) = \{ e = uv \in E : |\{u,v\}\cap S| = 1 \}.$$

A bond is a cut $\delta_G(S)$ such that both $G[S]$ and $G[V\setminus S]$ are connected. In connected graphs, bonds are precisely the inclusion-minimal edge cuts, forming the unique minimal separating edge sets between any two connected parts.

Associate to each bond a $0/1$ incidence vector $x^\delta \in \{0,1\}^E$ where $x^\delta_e=1$ if $e\in \delta$, $0$ otherwise. The bond polytope is then

$$P_{\rm bond}(G) = \mathrm{conv} \left\{ x^\delta : \delta \text{ is a bond of } G \right\} \subset \mathbb{R}^E.$$

Similarly, the cut polytope is $$\mathrm{Cut}(G) = \mathrm{conv}\left\{x^\delta : \delta \text{ is a cut of } G\right\}$$. The polyhedral optimization problem MaxBond seeks, for given weights $c_e \ge 0$, a bond maximizing $\sum_{e \in \delta} c_e$. The MaxBond problem is NP-hard, even for planar graphs, but enjoys tractability on $(K_5\setminus e)$-minor-free graphs due to specific decomposition properties (Chimani et al., 2020, Kolman et al., 16 Jan 2026).

2. Bond Polytope versus Cut Polytope

The bond polytope is always contained in the cut polytope, i.e., $$P_{\rm bond}(G)\subseteq \mathrm{Cut}(G) \subseteq [0,1]^E$$. Every vertex of $P_{\rm bond}(G)$ corresponds to a vertex of $\mathrm{Cut}(G)$ which arises from an actual bond (and the zero vector). Homogeneous facets of the cut polytope remain facet-defining for the bond polytope (Chimani et al., 2020).

Key properties of the relationship include:

• Edge contraction: Contraction of edge $e$ in $G$ corresponds to intersecting $P_{\rm bond}(G)$ with the hyperplane $x_e=0$.
• Edge deletion: Unlike in the cut polytope, deletion of edge $e$ does not correspond to projection of $P_{\rm bond}(G)$ along $x_e$ in general.
• Symmetry and switching operations: The switching symmetry of the cut polytope—corresponding to flipping sides of a cut—does not straightforwardly transfer to the bond polytope except under compatibility conditions involving homogeneous inequalities and preserving the bond structure.

The dimension of the bond polytope is $|E|$.

3. Graph Operations and Polytope Decomposition

The bond polytope admits descriptions under canonical graph sum constructions:

• 1-sum (vertex sum): If $G=G_1 +_1 G_2$ (sharing a vertex $u$), denoting $B_i = P_{\rm bond}(G_i)$, one obtains the subdirect sum:

$$P_{\rm bond}(G) = \mathrm{conv}\left(\{(x,0):x \in B_1\} \cup \{(0,y): y\in B_2\}\right).$$

• 2-sum (edge sum): For $G=G_1 +_{2\bar{}} G_2$ glued along edge $e=\{u,v\}$, if $B_i$ is partitioned by $b_e=0,1$, the bond polytope satisfies:

$$P_{\rm bond}(G) = \mathrm{conv}\left( B_1^0\cup B_2^0 \cup \{ b_1^1 + b_2^1 : b_i^1 \in B_i^1 \} \right).$$

Bonds with $x_e=0$ originate from one summand, while those with $x_e=1$ require "gluing" bonds in both $G_1$ and $G_2$ that separate $u,v$ (Kolman et al., 16 Jan 2026).

Standard graph modifications such as node splitting, node replacement by triangles, edge subdivision, and path compression are compatible with the facet structure of $P_{\rm bond}(G)$ under suitable lifting of inequalities, enhancing its composability and making recursive decomposition efficient (Chimani et al., 2020).

4. Facet Structure and Linear Descriptions

The characterization of the facets of the bond polytope is central. In general, the following families of valid and, under additional hypotheses, facet-defining inequalities are established (Chimani et al., 2020):

• Edge inequalities: $x_e \le 1$ for each $e\in E$, facet-defining if $e$ does not lie on any non-interleaved cycle.
• Non-interleaved cycle inequalities: For a cycle $C \subseteq G$ with no two node-disjoint internal $C$-paths crossing (i.e., non-interleaved), the inequality $\sum_{e\in E(C)} x_e \le 2$ is both valid and facet-defining if $G$ is 3-connected.
• Cycle polytopes: For the cycle $C_n$, the bond polytope is cut out exactly by

$$-x_e \le 0, \quad x_e - \sum_{f\neq e} x_f \le 0, \quad \sum_{e\in E(C_n)} x_e \le 2.$$

• Wagner graphs and special cycles: In $V_n$, even interleaved cycles have valid and facet-defining inequalities, e.g., $\sum_{e\in \text{outer cycle}} x_e \le 4$.

For 3-connected, planar, $(K_5\setminus e)$-minor-free graphs, all facets are generated by inequalities associated to triangles and non-interleaved cycles: - Triangle inequalities: $x_e - x_f - x_g \le 0$ for each triangle $\{e,f,g\}$; - 3-cycle cut-inequality: $x_e + x_f + x_g \le 2$; - Non-interleaved cycles: $\sum_{e\in E(C)} x_e \le 2$.

This yields a complete linear description for these classes.

5. Decomposition Theorems and Extension Complexity

By Wagner's theorem, every maximal $(K_5\setminus e)$-minor-free graph can be decomposed into a bounded number of base graphs (wheels $W_n$, prism, $K_2$, $K_3$, $K_{3,3}$) by a sequence of 1-sums and 2-sums. Leveraging Balas' union-of-polytopes theorem and the glueing lemma for 0/1-polytopes,

$\operatorname{xc}(P_{\rm bond}(G)) \leq O(n)$

where $\operatorname{xc}$ denotes the extension complexity, and $n=|V(G)|$ (Kolman et al., 16 Jan 2026). For each constant-size base, a constant-size EF is immediate; for the wheel $W_n$, a linear-size EF is constructed by dynamic programming.

Summing the extension complexities of components yields a linear bound for all $(K_5\setminus e)$-minor-free graphs. This provides a tight polyhedral description and efficient solvability for networks lacking $K_5$ minors minus one edge.

6. Algorithms for MaxBond and Computational Aspects

For $(K_5\setminus e)$-minor-free graphs, the MaxBond problem is solvable in $O(n)$ time (Chimani et al., 2020, Kolman et al., 16 Jan 2026). Algorithmic workflow:

• Decompose $G$ into 2- and 3-connected components via the SPQR-tree in linear time.
• Base graphs are handled as follows: for wheels $W_n$, all bonds cut off a contiguous rim path; by adapting Kadane’s algorithm, the best such bond can be found in two rim passes; other bases are constant-size.
• Combine solutions for larger graphs via the 2-sum/1-sum rules, solving recursively and aggregating results with edge-weight adjustments.
• For MaxBond, at each step, solve two subproblems for each subgraph $H$: the maximum bond not using the virtual edge $e$ ($\omega_H^-$), and the maximum bond using $e$ ($\omega_H^+$). Replace $H$ by $e$ of weight $\omega_H^+$, and recurse.

This recursive pruning and aggregation yields the global optimum.

7. Open Problems and Directions

Open questions remain on full characterization of facets arising from interleaved cycles in general 3-connected graphs, and on unifying the handling of 2-sums and their effect on the facial structure of the bond polytope (Chimani et al., 2020). The search for a “glue theorem” analogous to the cut-polytope case, detailing the facial structure under clique-sum operations, remains ongoing.

Additionally, although the focus thus far is on $(K_5\setminus e)$-minor-free and bounded-treewidth graphs, the extension complexity and polyhedral combinatorics of the bond polytope in general classes of graphs (e.g., non-planar, high-genus, expander families) present substantial directions for further study.

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References:

• "On the Bond Polytope" (Chimani et al., 2020)
• "Bond Polytope under Vertex- and Edge-sums" (Kolman et al., 16 Jan 2026)