On generating all maximal independent sets
Information Processing Letters
On the complexity of H-coloring
Journal of Combinatorial Theory Series B
&Dgr;P2-complete lexicographically first maximal subgraph problems
Theoretical Computer Science
List homomorphisms to reflexive graphs
Journal of Combinatorial Theory Series B
The complexity of counting graph homomorphisms
Proceedings of the ninth international conference on on Random structures and algorithms
A Generic Greedy Algorithm, Partially-Ordered Graphs and NP-Completeness
WG '01 Proceedings of the 27th International Workshop on Graph-Theoretic Concepts in Computer Science
Graphs and Hypergraphs
Dichotomies for classes of homomorphism problems involving unary functions
Theoretical Computer Science
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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 Σ2p-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.