On the extension of vertex maps to graph homomorphisms

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 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 f;s that maps each vertex of H closest to the vertex s among all such existing homomorphisms 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 e...

• Type: Journal
• Year: 2006
• Where: DM
• Authors: Geir Agnarsson, Li Chen
• Keywords: DM 2006, Reflexive Graphs, Simple Undirected Graph, Vertex Map