A reflexive graph is a simple undirected graph where a loop has been added at each vertex. If G and H are reflexive graphs and U ⊆ V ( H ) , then a vertex map f : U → V ( G ) is called nonexpansive if for every two vertices x , y ∈ U , the distance between f ( x ) and f ( y ) in G is at most that between x and y in H . A reflexive graph G is said to have the extension property (EP) if for every reflexive graph H , every U ⊆ V ( H ) and every nonexpansive vertex map f : U → V ( G ) , there is a graph homomorphism φ f : H → G that agrees with f on U . Characterizations of EP-graphs are well known in the mathematics and computer science literature. In this article we determine when exactly, for a given “sink”-vertex s ∈ V ( G ) , we can obtain such an extension φ f ; s that maps each vertex of H closest to the vertex s among all such existing homomorphisms φ f . A reflexive graph G satisfying this is then said to have the sink extension property (SEP). We then characterize the reflexive graphs with the unique sink extension property (USEP), where each such sink extensions φ f ; s is unique.