Homomorphism-homogeneous graphs

We answer two open questions posed by Cameron and Nesetril concerning homomorphismhomogeneous 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.