  A coloring of the vertices of a connected graph is convex if each color class
induces a connected subgraph. We address the convex recoloring (CR) problem
defined as follows. Given a graph $G$ and a coloring of its vertices, recolor a
minimum number of vertices of $G$ so that the resulting coloring is convex.
This problem, known to be NP-hard even on paths, was firstly motivated by
applications on perfect phylogenies. In this work, we study CR on general
graphs from a polyhedral point of view. First, we introduce a full-dimensional
polytope based on the idea of connected subgraphs, and present a class of valid
inequalities with righthand side one that comprises all facet-defining
inequalities with binary coefficients when the input graph is a tree. Moreover,
we define a general class of inequalities with righthand side in $\{1, \ldots,
k\}$, where $k$ is the amount of colors used in the initial coloring, and show
sufficient conditions for validity and facetness of such inequalities. Finally,
we report on computational experiments for an application on mobile networks
that can be modeled by the polytope of CR on paths. We evaluate the potential
of the proposed inequalities to reduce the integrality gaps.
