Subprime and superprime graphs

Abstract

Let $G$ be a graph with at least four vertices. The graph $G$ is prime if all its modules are trivial. For example, for every integer $n\ge 4$, the path $P_n$ is prime. So the graph $G$ can be made prime by modifying the adjacency relation between some pairs of vertices, i.e., by adding some edges to $G$ and removing some other edges from it. (The first two authors proved that the graph $G$ can be made prime by at most $\left|V\left(G\right)\right|-1$ adjacency modifications.) Nevertheless, there are graphs that cannot be made prime by only adding or only removing edges. So let us say that $G$ is superprime (resp. subprime) if it can be made prime by removing (resp. adding) edges. Given that subprime graphs are the complements of superprime ones, we have chosen to focus solely on superprime graphs, which are graphs admitting a spanning prime subgraph. These graphs are connected. They were considered by D.P. Sumner, who proved that given a connected graph $G$ with at least four vertices, $G$ is superprime if it does not admit a stable module of size 2. This sufficient condition for $G$ to be superprime is not necessary. In this paper, we provide a necessary and sufficient condition for $G$ to be superprime. This condition involves neighborhood complexes that we associate with stable sets.