### Permutations avoiding connected graphs

#### Abstract

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) = ∅.

AMS subject classiﬁcation (2000): 05C70

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