Abstract
In Bauslaugh (1995) we defined and explored the notion of homomorphic compactness for infinite digraphs. In this paper we show that there are exactly 2 N 0 compact digraphs, up to homomorphic equivalence. We then define the notion of finite equivalence for infinite digraphs. We show that for almost any infinite digraph G, the class of digraphs which are finitely equivalent to G (modulo homomorphic equivalence) is either a proper class or consists of a single homomorphic equivalence class. For undirected graphs we show that this is true in all cases. We also examine some basic properties of a natural partial order we may impose on these classes of digraphs.
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