Hyperbolicity and complement of graphs

If X is a geodesic metric space and x 1 , x 2 , x 3 ∈ X , a geodesic triangle T = { x 1 , x 2 , x 3 } is the union of the three geodesics [ x 1 x 2 ] , [ x 2 x 3 ] and [ x 3 x 1 ] in X . The space X is δ -hyperbolic (in the Gromov sense) if any side of T is contained in a δ -neighborhood of the union of the two other sides, for every geodesic triangle T in X . We denote by δ ( X ) the sharp hyperbolicity constant of X , i.e.  δ ( X ) ≔ inf { δ ≥ 0 : X  is  δ -hyperbolic } . The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. The main aim of this paper is to obtain information about the hyperbolicity constant of the complement graph G ¯ in terms of properties of the graph G . In particular, we prove that if diam ( V ( G ) ) ≥ 3 , then δ ( G ¯ ) ≤ 2 , and that the inequality is sharp. Furthermore, we find some Nordhaus–Gaddum type results on the hyperbolicity constant of a graph δ ( G ) .