Amas dans les graphes planaires

The problem examined in this paper comes from percolation theory. G = (X, E) is a simple geometric planar graph each vertex of which has a finite degree. We partition X in two subsets X₁, X₂ and we colour in blue each vertex and edge of the subgraph G_x generated by X₁ and in red each vertex and edge of G_x2. We obtain blue clusters (resp. red clusters) namely the components of G_x1 (resp. of G_x2). We want to characterize G so that for any such coloration, any finite cluster of one colour is surrounded by a cluster of the other colour. A necessary and sufficient condition is that every component of G is a maximal infinite planar graph and that every vertex x is surrounded by the cycle which connects the vertices adjacent to x.