Minuscule posets from neighbourly graph sequences

We begin by associating to any sequence of vertices in a simple graph X, here always assumed connected, a partially ordered set called a heap. This terminology was introduced by Viennot ([11]) and used extensively by Stembridge in the context of fully commutative elements of Coxeter groups (see [8]), but our context is more general and graph-theoretic. The heap of a sequence of vertices is that partially ordered set whose total linear orders correspond to all possible sequences obtained from the original one by switching adjacent elements which are not neighbours in X. Furthermore sequences which are equivalent under such interchanges (of adjacent elements which are not neighbours in the graph) give rise to identical heaps. A heap will be called neighbourly if the associated sequences have the property that between any two successive occurrences of a vertex x there occurs at least two occurrences of a neighbour of x. Heaps arising from maximal neighbourly sequences which in fact have exactly two neighbours between any two occurrences of a vertex x are classified. In our main result, we prove that any graph X having such a maximal neighbourly heap which is in fact two-neighbourly must be one of the Dynkin Coxeter diagrams An, Dn, or E6, E7, and that the corresponding heaps are exactly the minuscule posets defined and studied by Proctor in [4].