***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St000301 ----------------------------------------------------------------------------- Collection: Graphs ----------------------------------------------------------------------------- Description: The number of facets of the stable set polytope of a graph. The stable set polytope of a graph $G$ is the convex hull of the characteristic vectors of stable (or independent) sets of vertices of $G$ inside $\mathbb{R}^{V(G)}$. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def stable_set_polytope(G): V = G.vertices() Vpos = { v:i for i,v in enumerate(V) } poly_vertices = [] for S in Subsets(V): if G.is_independent_set(S): vert = [0]*len(V) for s in S: vert[Vpos[s]] = 1 poly_vertices.append(vert) return Polyhedron(vertices=poly_vertices) def statistic(G): return stable_set_polytope(G).n_facets() ----------------------------------------------------------------------------- Statistic values: ([],1) => 2 ([],2) => 4 ([(0,1)],2) => 3 ([],3) => 6 ([(1,2)],3) => 5 ([(0,2),(1,2)],3) => 5 ([(0,1),(0,2),(1,2)],3) => 4 ([],4) => 8 ([(2,3)],4) => 7 ([(1,3),(2,3)],4) => 7 ([(0,3),(1,3),(2,3)],4) => 7 ([(0,3),(1,2)],4) => 6 ([(0,3),(1,2),(2,3)],4) => 7 ([(1,2),(1,3),(2,3)],4) => 6 ([(0,3),(1,2),(1,3),(2,3)],4) => 6 ([(0,2),(0,3),(1,2),(1,3)],4) => 8 ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 6 ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 5 ([],5) => 10 ([(3,4)],5) => 9 ([(2,4),(3,4)],5) => 9 ([(1,4),(2,4),(3,4)],5) => 9 ([(0,4),(1,4),(2,4),(3,4)],5) => 9 ([(1,4),(2,3)],5) => 8 ([(1,4),(2,3),(3,4)],5) => 9 ([(0,1),(2,4),(3,4)],5) => 8 ([(2,3),(2,4),(3,4)],5) => 8 ([(0,4),(1,4),(2,3),(3,4)],5) => 9 ([(1,4),(2,3),(2,4),(3,4)],5) => 8 ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 8 ([(1,3),(1,4),(2,3),(2,4)],5) => 10 ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 10 ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 8 ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 8 ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 8 ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 11 ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 8 ([(0,4),(1,3),(2,3),(2,4)],5) => 9 ([(0,1),(2,3),(2,4),(3,4)],5) => 7 ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 8 ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 7 ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 11 ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 9 ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 8 ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 8 ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 7 ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 7 ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 7 ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 9 ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 9 ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 7 ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6 ([],6) => 12 ([(4,5)],6) => 11 ([(3,5),(4,5)],6) => 11 ([(2,5),(3,5),(4,5)],6) => 11 ([(1,5),(2,5),(3,5),(4,5)],6) => 11 ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 11 ([(2,5),(3,4)],6) => 10 ([(2,5),(3,4),(4,5)],6) => 11 ([(1,2),(3,5),(4,5)],6) => 10 ([(3,4),(3,5),(4,5)],6) => 10 ([(1,5),(2,5),(3,4),(4,5)],6) => 11 ([(0,1),(2,5),(3,5),(4,5)],6) => 10 ([(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 11 ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(2,4),(2,5),(3,4),(3,5)],6) => 12 ([(0,5),(1,5),(2,4),(3,4)],6) => 10 ([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 12 ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 11 ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 10 ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 11 ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 12 ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 10 ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 13 ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 12 ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 13 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 14 ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,5),(1,4),(2,3)],6) => 9 ([(1,5),(2,4),(3,4),(3,5)],6) => 11 ([(0,1),(2,5),(3,4),(4,5)],6) => 10 ([(1,2),(3,4),(3,5),(4,5)],6) => 9 ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 11 ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 10 ([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 10 ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 9 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 9 ([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 13 ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 12 ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 11 ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 13 ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 10 ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 10 ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 11 ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 11 ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => 11 ([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => 9 ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 12 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 10 ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 10 ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => 11 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 10 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 10 ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 15 ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 12 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 10 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 11 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 11 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 12 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 13 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 11 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 11 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 11 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 11 ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 12 ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 13 ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 11 ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 12 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 10 ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 11 ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 11 ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 10 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 10 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 10 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 14 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 11 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 12 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 11 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 11 ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 11 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 11 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 10 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 15 ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 12 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 12 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => 8 ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => 10 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => 9 ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 9 ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6) => 10 ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 10 ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 10 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6) => 13 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 12 ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 12 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 12 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 11 ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 11 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 11 ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 10 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9 ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8 ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8 ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8 ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 12 ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 10 ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 10 ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 14 ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8 ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 7 ----------------------------------------------------------------------------- Created: Nov 26, 2015 at 12:26 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Nov 26, 2015 at 12:26 by Christian Stump