On counting homomorphisms to directed acyclic graphs

Authors:
Martin Dyer;Leslie Ann Goldberg;Mike Paterson
Affiliations:
School of Computing, University of Leeds, Leeds, UK;Dept. of Computer Science, University of Warwick, Coventry, UK;Dept. of Computer Science, University of Warwick, Coventry, UK
Venue:
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
Year:
2006

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Abstract

We give a dichotomy theorem for the problem of counting homomorphisms to directed acyclic graphs. H is a fixed directed acyclic graph. The problem is, given an input digraph G, to determine how many homomorphisms there are from G to H. We give a graph-theoretic classification, showing that for some digraphs H, the problem is in P and for the rest of the digraphs H the problem is #P-complete. An interesting feature of the dichotomy, absent from related dichotomy results, is the rich supply of tractable graphs H with complex structure.