On seed graphs with more than two components Original Research Article

The closed neighbourhood NG[x] of a vertex x in a graph G is the subgraph of G induced by x and all neighbours of x. The seed of a vertex x∈G is the subgraph of G induced by all vertices of G⧹NG[x] and we denote it by SG(x). A graph F is a seed graph if there exists a graph G such that SG(x)≅F for each x∈G. In this paper seed graphs with more than two components are studied. It is shown that if all components are of equal order, size or regularity then they are all isomorphic to a complete graph. In the general case it is shown how the structure of any component Fi of a seed graph F depends on the structure of all components ‘smaller’ than Fi in the sense of ‘smaller order’, ‘smaller size’ or ‘smaller degree’ in the case of regular components.