The edge nucleus of a point-determining graph

Abstract In this paper we investigate the edge nucleus E 0 ( G ) of a point-determining graph G . We observe several relationships between E 0 ( G ) and the nucleus G 0 = { v ∈ V ( G )∣ G − v is point determining} and use these relationships to prove several properties of E 0 ( G ). In particular, we show that there are only a finite number of graphs with a given edge nucleus and we determine those graphs G for which | E 0 ( G )| ≤ 2. We also show that an n -clique of a point-determining graph G contains at least n −2 edges of E 0 ( G ) and if G is totally point determining, then every odd cycle of G meets E 0 ( G ).