On the structure of compact graphs

A simple graph G is called a compact graph if G contains no isolated vertices and for each pair x, y of non-adjacent vertices of G, there is a vertex z with N(x) ∪ N(y) ⊆ N(z), where N(v) is the neighborhood of v, for every vertex v of G. In this paper, compact graphs with sufficient number of edges are studied. Also, it is proved that every regular compact graph is strongly regular. Some results about cycles in compact graphs are proved, too. Among other results, it is proved that if the ascending chain condition holds for the set of neighbors of a compact graph G, then the descending chain condition holds for the set of neighbors of G.