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Homomorphisms to oriented paths. (English) Zbl 0819.05030
A homomorphism of a digraph $$G= (V, A)$$ to a digraph $$H= (V', A')$$ is a mapping $$f: V\to V'$$ of the vertices of $$G$$ to the vertices of $$H$$ (not necessarily onto) which preserves arcs, i.e., such that $$xy\in A$$ implies $$f(x) f(y)\in A'$$. If such a homomorphism exists, $$G$$ is said to be homomorphic to $$H$$ and the notation $$G\to H$$ is used. Otherwise the notation $$G\nrightarrow H$$ is used.
Given an oriented path $$P$$, the authors characterize those digraphs $$G$$ which are homomorphic to $$P$$. The characterization equates the nonexistence of a homomorphism $$G\to P$$ with the existence of a homomorphism $$W\to G$$, for some oriented path $$W$$ which is not homomorphic to $$P$$. This result complements the recent polynomial time algorithm of W. Gutjahr, E. Welzl and G. Woeginger to find such a homomorphism (if one exists) [Polynomial graph-colorings, Discrete Appl. Math. 35, No. 1, 29-45 (1992; Zbl 0761.05040)].
Say that $$H$$ has tree-duality if $$G\nrightarrow H$$ if and only if there is an oriented tree $$T$$ such that $$T\to G$$ and $$T\nrightarrow H$$. The main result in this paper is that oriented paths have tree-duality. In another recent paper with J. Nešetřil, the authors have proved that whenever $$H$$ has tree-duality then there is a polynomial algorithm to test for the existence of homomorphisms to $$H$$.

MSC:
 05C20 Directed graphs (digraphs), tournaments 05C38 Paths and cycles 05C75 Structural characterization of families of graphs 05C15 Coloring of graphs and hypergraphs
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References:
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