***************************************************************************** * 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: St001391 ----------------------------------------------------------------------------- Collection: Graphs ----------------------------------------------------------------------------- Description: The disjunction number of a graph. Let $V_n$ be the power set of $\{1,\dots,n\}$ and let $E_n=\{(a,b)| a,b\in V_n, a\neq b, a\cap b=\emptyset\}$. Then the disjunction number of a graph $G$ is the smallest integer $n$ such that $(V_n, E_n)$ has an induced subgraph isomorphic to $G$. ----------------------------------------------------------------------------- References: [1] van der Zypen, D. Disjunction number of a graph [[MathOverflow:331366]] ----------------------------------------------------------------------------- Code: def Dominics_graph(n): V = map(frozenset, powerset(range(n))) return Graph([V, lambda a, b: a != b and a.isdisjoint(b)]) def statistic(G): n = 0 while True: H = Dominics_graph(n) H.relabel() if H.subgraph_search(G, induced=True): return n n += 1 ----------------------------------------------------------------------------- Statistic values: ([],1) => 0 ([],2) => 2 ([(0,1)],2) => 1 ([],3) => 3 ([(1,2)],3) => 2 ([(0,2),(1,2)],3) => 2 ([(0,1),(0,2),(1,2)],3) => 2 ([],4) => 3 ([(2,3)],4) => 3 ([(1,3),(2,3)],4) => 3 ([(0,3),(1,3),(2,3)],4) => 3 ([(0,3),(1,2)],4) => 4 ([(0,3),(1,2),(2,3)],4) => 3 ([(1,2),(1,3),(2,3)],4) => 3 ([(0,3),(1,2),(1,3),(2,3)],4) => 2 ([(0,2),(0,3),(1,2),(1,3)],4) => 4 ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3 ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3 ([],5) => 4 ([(3,4)],5) => 3 ([(2,4),(3,4)],5) => 3 ([(1,4),(2,4),(3,4)],5) => 4 ([(0,4),(1,4),(2,4),(3,4)],5) => 3 ([(1,4),(2,3)],5) => 4 ([(1,4),(2,3),(3,4)],5) => 3 ([(0,1),(2,4),(3,4)],5) => 5 ([(2,3),(2,4),(3,4)],5) => 4 ([(0,4),(1,4),(2,3),(3,4)],5) => 4 ([(1,4),(2,3),(2,4),(3,4)],5) => 3 ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 3 ([(1,3),(1,4),(2,3),(2,4)],5) => 4 ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 4 ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4 ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 3 ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3 ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 5 ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4 ([(0,4),(1,3),(2,3),(2,4)],5) => 4 ([(0,1),(2,3),(2,4),(3,4)],5) => 6 ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 5 ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 4 ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 5 ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 4 ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3 ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 4 ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4 ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3 ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3 ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 4 ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 4 ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4 ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4 ([],6) => 4 ([(4,5)],6) => 4 ([(3,5),(4,5)],6) => 4 ([(2,5),(3,5),(4,5)],6) => 4 ([(1,5),(2,5),(3,5),(4,5)],6) => 4 ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 4 ([(2,5),(3,4)],6) => 4 ([(2,5),(3,4),(4,5)],6) => 3 ([(1,2),(3,5),(4,5)],6) => 5 ([(3,4),(3,5),(4,5)],6) => 4 ([(1,5),(2,5),(3,4),(4,5)],6) => 4 ([(0,1),(2,5),(3,5),(4,5)],6) => 5 ([(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 4 ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 3 ([(2,4),(2,5),(3,4),(3,5)],6) => 4 ([(0,5),(1,5),(2,4),(3,4)],6) => 5 ([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 5 ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 3 ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 4 ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 4 ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,4),(2,3)],6) => 4 ([(1,5),(2,4),(3,4),(3,5)],6) => 4 ([(0,1),(2,5),(3,4),(4,5)],6) => 5 ([(1,2),(3,4),(3,5),(4,5)],6) => 6 ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 4 ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 5 ([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 5 ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 5 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 4 ([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 5 ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 4 ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 5 ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 3 ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3 ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 5 ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => 6 ([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => 7 ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 5 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 6 ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 7 ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 6 ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => 6 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 6 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 5 ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 6 ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 5 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 4 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 4 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 5 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 4 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 4 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 5 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 4 ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 5 ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 4 ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 4 ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 6 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 5 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 5 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 4 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 4 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 6 ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 5 ([(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) => 5 ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => 9 ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => 6 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => 8 ([(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) => 7 ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 7 ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6) => 7 ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 6 ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 6 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6) => 5 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 7 ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 6 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 5 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 6 ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 5 ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(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) => 4 ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 5 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(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) => 4 ([(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) => 4 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(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) => 4 ([(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) => 6 ([(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) => 5 ([(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) => 5 ([(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) => 5 ----------------------------------------------------------------------------- Created: May 13, 2019 at 08:43 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: May 14, 2019 at 12:51 by Martin Rubey