Homomorphism-Homogeneous Graphs

We answer two open questions posed by Cameron and Nesetril concerning homomorphism-homogeneous graphs. In particular we show, by giving a characterization of these graphs, that extendability to monomorphism or to homomorphism leads to the same class of graphs when defining homomorphism homogeneity. Further, we show that there are homomorphism-homogeneous graphs that do not contain the Rado graph as a spanning subgraph 'answering the second open question. We also treat the case of homomorphism-homogeneous graphs with loops allowed, showing that the corresponding decision problem is co-NP complete. Finally, we extend the list of considered morphism-types and show that the graphs for which monomorphisms can be extended to epimorphisms are complements of homomorphism-homogeneous graphs. (C) 2010 Wiley Periodicals. Inc. J Graph Theory 65: 253-262, 2010