Local 2-geodesic transitivity and clique graphs

A 2-geodesic in a graph is a vertex triple (u,v,w) such that v is adjacent to both u and w and u,w are not adjacent. We study non-complete graphs Γ in which, for each vertex u, all 2-geodesics with initial vertex u are equivalent under the subgroup of graph automorphisms fixing u. We call such graphs locally 2-geodesic transitive, and show that the subgraph [Γ(u)] induced on the set of vertices of Γ adjacent to u is either (i) a connected graph of diameter 2, or (ii) a union mKr of m⩾2 copies of a complete graph Kr with r⩾1. This suggests studying locally 2-geodesic transitive graphs according to the structure of the subgraphs [Γ(u)]. We investigate the family F(m,r) of connected graphs Γ such that [Γ(u)]≅mKr for each vertex u, and for fixed m⩾2, r⩾1. We show that each Γ∈F(m,r) is the point graph of a partial linear space S of order (m,r+1) which has no triangles (and 2-geodesic transitivity of Γ corresponds to natural strong symmetry properties of S). Conversely, each S with these properties has point graph in F(m,r), and a natural duality on partial linear spaces induces a bijection F(m,r)↦F(r+1,m−1).