Two-edge-connected (not necessarily spanning) subgraphs and polyhedra

Abstract

Given a graph $G$, we study the $2$-edge-connected subgraph polytope $\mathrm{TECSP}(G)$, which is given by the convex hull of the incidence vectors of all $2$-edge-connected subgraphs of $G$. We describe the lattice points of this polytope by linear inequalities which provides an ILP-algorithm for finding a $2$-edge-connected subgraph of maximum weight. Furthermore, we characterize when these inequalities define facets of $\mathrm{TECSP}(G)$. We also consider further types of supporting hyperplanes of $\mathrm{TECSP}(G)$ and study when they are facet-defining. Finally, we investigate the efficiency of our considered inequalities practically on some classes of graphs.

Figures (12)

• Figure 1: The structure of a coparallel class $C=\{e_1,\ldots,e_k\}$ of a graph $G$. Every grey filled cycle represents a $2$-edge-connected subgraph of $G$.
• Figure 2: The structure of the coparallel classes of $G-C$ for $C=\{e_1,\ldots,e_k\} \in \mathrm{CP}(G)$. The coparallel class $C'=\{f_1,\ldots,f_\ell,\ldots,f_m\}$ is completely contained in $G_1$
• Figure 3: Example for an asymmetric cut inequality with a non-minimal cut $\delta(S) \supsetneq \delta(S_1)$, where $S=S_1 \dot{\cup} S_2$.
• Figure 4: Example for $G[S]$ containing a bridge $f$ that is not contained in a coparallel class with an edge of $\delta(S)$.
• Figure 5: The graph used in the initial motivating example for connectivity cut inequalitites.

Theorems & Definitions (23)

• Definition 2.1
• Theorem 2.6
• proof
• Remark 2.7
• Theorem 2.8: BARAHONA198640
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.4