Let $G=(V,E)$ be a simple graph which has the following property. If $(u,v)\in E$ and $(v,s)\in E$ then $(u,s)\in E$. What kind of graph is this? Does it have any name and special property?
Such a graph is a disjoint union of complete graphs. Indeed, it's easy to prove by induction on the number of vertices of any connected subgraph that any connected subgraph is complete.