Permutations avoiding connected graphs - Sauer, Norbert; Zaguia, Imed (Contributions to discrete mathematics, 2017)
There is a permutation of the vertices of a tree for which no proper subtree on at least two vertices is mapped to a subtree, if and only if twice the number of its endpoints is less than or equal to the number of points of the tree; Theorem 4.1. The following more general result follows: Let G = (V (G), E(G)) be a simple graph and let C(G) be the set of subsets A  V (G) which induce a connected subgraph of G containing at least two vertices and let Π(G) be the set of permutations of V (G) which do not map an element of C(G) to an element of C(G). In the case where G has n vertices and at most n − 1 edges we give a necessary and suﬃcient condition on G so that Π(G) [SPECIAL CHARACTER]= ∅. AMS subject classiﬁcation (2000): 05C70