On the connected (sub)partition polytope

Let k be a positive integer and let G be a graph with n vertices. A connected k-subpartition of G is a collection of k pairwise disjoint sets (a.k.a. classes) of vertices in G such that each set induces a connected subgraph. The connected k-partition polytope of G, denoted by P(G,k), is defined as the convex hull of the incidence vectors of all connected k-subpartitions of G. Many applications arising in off-shore oil-drilling, forest planning, image processing, cluster analysis, political districting, police patrolling, and biology are modeled in terms of finding connected (sub)partitions of a graph. This study focus on the facial structure of P(G,k) and the computational complexity of the corresponding separation problems. We first propose a set of valid inequalities having non-null coefficients associated with a single class that extends and generalizes the ones in the literature of related problems, show sufficient conditions for these inequalities to be facet-defining, and design a polynomial-time separation algorithm for them. We also devise two sets of inequalities that consider multiple classes, prove when they define facets, and study the computational complexity of associated separation problems.