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Comparing universal covers in polynomial time. (English) Zbl 1142.68456
Hirsch, Edward A. (ed.) et al., Computer science – theory and applications. Third international computer science symposium in Russia, CSR 2008 Moscow, Russia, June 7–12, 2008. Proceedings. Berlin: Springer (ISBN 978-3-540-79708-1/pbk). Lecture Notes in Computer Science 5010, 158-167 (2008).
Summary: The universal cover \(T _{G }\) of a connected graph \(G\) is the unique (possible infinite) tree covering \(G\), i.e., that allows a locally bijective homomorphism from \(T _{G }\) to \(G\). Universal covers have major applications in the area of distributed computing. It is well-known that if a graph \(G\) covers a graph \(H\) then their universal covers are isomorphic, and that the latter can be tested in polynomial time by checking if \(G\) and \(H\) share the same degree refinement matrix. We extend this result to locally injective and locally surjective homomorphisms by following a very different approach. Using linear programming techniques we design two polynomial time algorithms that check if there exists a locally injective or a locally surjective homomorphism, respectively, from a universal cover \(T _{G }\) to a universal cover \(T _{H }\). This way we obtain two heuristics for testing the corresponding locally constrained graph homomorphisms. As a consequence, we have obtained a new polynomial time algorithm for testing (subgraph) isomorphism between universal covers, and for checking if there exists a role assignment (locally surjective homomorphism) from a given tree to an arbitrary fixed graph \(H\).
For the entire collection see [Zbl 1136.68005].

MSC:
68R10 Graph theory (including graph drawing) in computer science
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C85 Graph algorithms (graph-theoretic aspects)
90C59 Approximation methods and heuristics in mathematical programming
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