N-2-LOCALLY DISCONNECTED GRAPHS

The edge-induced subgraph on the set of all edges of a graph G that ar e adjacent to a given vertex x is called the neighbourhood of the seco nd type of x in G and is denoted by N2(x, G) (an edge yz is said to be adjacent to x if y not-equal x not-equal z and y or z is adjacent to x). A graph G is N2-locally disconnected if N2(x, G) is disconnected f or every vertex x of G. The main aim of the present paper is to find t he maximum size of an N2-locally disconnected graph of a given order.