On minimal prime extensions of a four-vertex graph in a prime graph Original Research Article

In a finite undirected graph image, a homogeneous set is a set image of at least two vertices such that every vertex in image is either adjacent to all vertices of U or nonadjacent to all of them. A graph is prime if it does not have a homogeneous set. We investigate the minimal prime extensions of a four-vertex subgraph in a prime graph. It turns out that the claw, the paw and the diamond (as well as their complements) have finitely many minimal prime extensions whereas the clique with four vertices has an infinite number of such extensions. This extends previous results of Hoàng and Reed on the image and of Olariu on a stable set of size three in a prime graph. The reducing pseudopath method of Zverovich is a general way to determine the minimal prime extensions of a non-prime graph; our paper, however, gives a short and direct proof for determining all prime extensions in the particular cases mentioned above.