On (1, 2)-realizable graphs Original Research Article

A graph H is (1,2)-realizable if there exists a graph G in which each vertex has the first neighbourhood as well as the second neighbourhood isomorphic to H. We prove that if a (1,2)-realizable graph H has n vertices, n ⩾ 3, then there is a unique connected graph G which realizes it and that G has 2n + 2 vertices. We give a necessary and sufficient condition for (1,2)-realizability of H, and use it to analyze regular (1,2)-realizable graphs.