In this paper we define the property of homomorphic compactness for digraphs. We prove that if a digraph H is homomorphically compact then H has a core, although the converse does not hold. We also examine a weakened compactness condition and show that when this condition is assumed, compactness is equivalent to containing a core. We use this result to prove that if a digraph H of size kappa is not compact, then there is a digraph G of size at most kappa(+) such that H is not compact with respect to G. We then give examples of some sufficient conditions for compactness. (C) 1996 Academic Press, Inc. [References: 21]
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