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On the extension of vertex maps to graph homomorphisms
AbstractA reflexive graph is a simple undirected graph where a loop has been added at each vertex. If G and H are reflexive graphs and UâV(H), then a vertex map f:UâV(G) is called nonexpansive if for every two vertices x,yâU, 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 UâV(H) and every nonexpansive vertex map f:UâV(G), there is a graph homomorphism Ïf:HâG 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 âsinkâ-vertex sâV(G), we can obtain such an extension Ïf;s that maps each vertex of H closest to the vertex s among all such existing homomorphisms Ïf. 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 Ïf;s is unique
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