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

✍ Scribed by Antonio Puricella; Iain A. Stewart

Publisher: Elsevier Science
Year: 2003
Tongue: English
Weight: 566 KB
Volume: 290
Category: Article
ISSN: 0304-3975
DOI: 10.1016/s0304-3975(02)00329-8

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✦ Synopsis

Let H be a ÿxed 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 ÿrst maximal Hcolourable subgraphs of G. We show that the complexity of deciding whether a given vertex of G is in a lexicographically ÿrst maximal H -colourable subgraph of G is NP-complete, if H is bipartite, and p 2 -complete, if H is non-bipartite. This result complements Hell and NeÄ setÄ ril'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 NeÄ setÄ ril, combinatorially adapted to our scenario.

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