Lifting of Multicuts

For every simple, undirected graph G = (V,E), a one-to-one relation exists between the decompositions and the multicuts of G. A decomposition of G is a partition Π of V such that, for every U ∈ Π, the subgraph of G induced by U is connected. A multicut of G is a subset M ⊆ E of edges such that, for every (chordless) cycle C ⊆ E of G, |M ∩ C| 6 = 1. The multicut induced by a decomposition is the set of edges that straddle distinct components. The characteristic function x ∈ {0, 1} of a multicut M = x−1(1) of G makes explicit, for every pair {v, w} ∈ E of neighboring nodes, whether v and w are in distinct components. In order to make explicit also for non-neighboring nodes, specifically, for all {v, w} ∈ E′ with E ⊆ E′ ⊆ ( V 2 ) , whether v and w are in distinct components, we define a lifting of the multicuts of G to multicuts of G′ = (V,E′). We show that, if G is connected, the convex hull of the characteristic functions of those multicuts of G′ that are lifted from G is an |E′|-dimensional polytope in R ′ . We establish some properties of some facets of this polytope.