Abstract
We consider some problems concerning generalizations of embeddings of acyclic digraphs inton-dimensional dicubes. In particular, we define an injectioni from a digraphD into then-dimensional dicubeQ n to be animmersion if for any two elementsa andb inD there is a directed path inD froma tob iff there is a directed path inQ n fromi(a) toi(b). We further define the immersion to bestrong iff there is a way of choosing these paths so that paths inQ n corresponding to arcs inD have disjoint interiors, and we introduce a natural notion of “minimality” on the set of arcs of a digraph in terms of its paths. Our main theorem then becomes:Every (minimal) n-element acyclic digraph can be (strongly) immersed in Q n. We also present examples ofn-element digraphs which cannot be immersed inQ n−1 and examples of 9n-element non-minimal digraphs which cannot be strongly immersed inQ10n −1. We conclude with some applications.
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