The full automorphism groups of general position graphs

Abstract

Let S be a non-empty finite set. A flag of S is a set f of non-empty proper subsets of S such that X⊆Y or Y⊆X for all X,Y∈f. The set {|X|:X∈f} is called the type of f. Two flags f and f′ are in general position with respect to S if X∩Y=∅ or X∪Y=S for all X∈f and Y∈f′. For a fixed type T, Klaus Metsch defined the general position graph Γ(S,T) whose vertices are the flags of S of type T with two vertices being adjacent when the corresponding flags are in general position. In this paper, we characterize the full automorphism groups of Γ(S,T) in the case that |T|=2. In particular, we solve an open problem proposed by Klaus Metsch.