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## Greedy algorithms, H-colourings and a complexity-theoretic dichotomy.

### Abstract

Let H be a fixed undirected graph. An H-colouring of an undirected graph G is a homomorphism from G to H. If the vertices of G are partially ordered then there is a generic non-deterministic greedy algorithm which computes all lexicographically first maximal H-colourable subgraphs of G. We show that the complexity of deciding whether a given vertex of G is in a lexicographically first maximal H-colourable subgraph of G is NP-complete, if H is bipartite, and ${\bf \Sigma}_2^p$-complete, if H is non-bipartite. This result complements Hell and Nesetril's seminal dichotomy result that the standard H-colouring problem is in P, if H is bipartite, and NP-complete, if H is non-bipartite. Our proofs use the basic techniques established by Hell and Nesetril, combinatorially adapted to our scenario

Topics: Computational complexity, Constraint satisfaction, Graph algorithms, Dicotomies.
Publisher: Elsevier
Year: 2003
DOI identifier: 10.1016/S0304-3975(02)00329-8
OAI identifier: oai:dro.dur.ac.uk.OAI2:598
Journal:

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