On the Bond Polytope

TL;DR

This paper initiates the structural study of bond polytopes, exploring their properties, relations to cut polytopes, and providing algorithms for maximum bond problems in specific graph classes.

Contribution

It offers the first structural analysis of bond polytopes, characterizes their facets, and develops a linear time algorithm for certain graph classes.

Findings

Complete linear description of bond polytopes for cycles.

Facet characterization for 3-connected planar (K5-e)-minor free graphs.

Linear time algorithm for maximum bond problem on (K5-e)-minor free graphs.

Abstract

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.