A finiteness theorem for primal extensions Original Research Article

A set W⊆V(G)W⊆V(G) is called homogeneous in a graph G if 2⩽|W|⩽|V(G)|-12⩽|W|⩽|V(G)|-1, and N(x)⧹W=N(y)⧹WN(x)⧹W=N(y)⧹W for each x,y∈Wx,y∈W. A graph without homogeneous sets is called prime. A graph H is called a (primal) extension of a graph G if G is an induced subgraph of H, and H is a prime graph. An extension H of G is minimal if there are no extensions of G in the set ISub(H)⧹{H}ISub(H)⧹{H}. We denote by Ext(G)Ext(G) the set of all minimal extensions of a graph G.