Editing to Prime Graphs

Abstract

Given a graph G, a subset M of V(G) is a module of G if for each $$v\in V(G)\setminus M$$ and $$x,y\in M$$ , $$vx \in E(G)$$ if and only if $$vy \in E(G)$$ . The trivial modules of G are $$\varnothing ,\{u\}(u\in V(G))$$ and V(G). Let G be a graph with at least four vertices. The graph G is prime if all its modules are trivial. We are interested in the minimum number $$\delta (G)$$ of edge modifications (edge deletions and/or additions) that are needed to transform G into a prime graph. We prove that $$\delta (G) \le v(G)-1$$ , where $$v(G) = |V(G)|$$ , and that the upper bound is uniquely reached by the complete graphs and their complements. We then characterize the graphs G such that $$\delta (G) = v(G)-2$$ or $$v(G)-3$$ , respectively. Lastly, we look for the $$\delta $$ -critical graphs, i.e. the graphs G for which $$\delta (H) > \delta (G)$$ for every graph H obtained from G by deleting one vertex. We prove that the $$\delta $$ -critical graphs are precisely the half graphs and their complements.