On the extension of vertex maps to graph homomorphisms Original Research Article

A reflexive graph is a simple undirected graph where a loop has been added at each vertex. If image and image are reflexive graphs and image, then a vertex map image is called nonexpansive if for every two vertices image, the distance between image and image in image is at most that between image and image in image. A reflexive graph image is said to have the extension property (EP) if for every reflexive graph image, every image and every nonexpansive vertex map image, there is a graph homomorphism image that agrees with image on image. 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 image, we can obtain such an extension image that maps each vertex of image closest to the vertex image among all such existing homomorphisms image. A reflexive graph image 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 image is unique.