Computing Vertex-Surjective Homomorphisms to Partially Reflexive Trees

A homomorphism from a graph G to a graph H is a vertex mapping f : VG → VH such that f(u) and f(v) form an edge in H whenever u and v form an edge in G. The H-Coloring problem is to test whether a graph G allows a homomorphism to a given graph H. A well-known result of Hell and Nesetřil determines the computational complexity of this problem for any fixed graph H. We study a natural variant of this problem, namely the SURJECTIVE H-COLORING problem, which is to test whether a graph G allows a homomorphism to a graph H that is (vertex-)surjective. We classify the computational complexity of this problem when H is any fixed partially reflexive tree. Thus we identify the first class of target graphs H for which the computational complexity of Surjective H-Coloring can be determined. For the polynomial-time solvable cases, we show a number of parameterized complexity results, especially on nowhere dense graph classes.

[1]  NARAYAN VIKAS,et al.  Computational Complexity of Compaction to Reflexive Cycles , 2002, SIAM J. Comput..

[2]  Robin Thomas,et al.  Deciding First-Order Properties for Sparse Graphs , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[3]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[4]  Narayan Vikas,et al.  Compaction, Retraction, and Constraint Satisfaction , 2004, SIAM J. Comput..

[5]  Jirí Fiala,et al.  A complete complexity classification of the role assignment problem , 2005, Theor. Comput. Sci..

[6]  Tomás Feder,et al.  The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory , 1999, SIAM J. Comput..

[7]  Maurizio Patrignani,et al.  The Complexity of the Matching-Cut Problem , 2001, WG.

[8]  P. Hell,et al.  Sparse pseudo-random graphs are Hamiltonian , 2003 .

[9]  Pavol Hell,et al.  List Homomorphisms to Reflexive Graphs , 1998, J. Comb. Theory, Ser. B.

[10]  Jaroslav Nesetril,et al.  On the complexity of H-coloring , 1990, J. Comb. Theory, Ser. B.

[11]  Stephan Kreutzer,et al.  Parameterized Complexity of First-Order Logic , 2009, Electron. Colloquium Comput. Complex..

[12]  Ge Xia,et al.  Strong computational lower bounds via parameterized complexity , 2006, J. Comput. Syst. Sci..

[13]  Celina M. H. de Figueiredo,et al.  Finding H-partitions efficiently , 2005, RAIRO Theor. Informatics Appl..

[14]  Jirí Fiala,et al.  Locally constrained graph homomorphisms - structure, complexity, and applications , 2008, Comput. Sci. Rev..

[15]  Simone Dantas,et al.  2k2-partition of Some Classes of Graphs , 2012, Discret. Appl. Math..

[16]  Russell Impagliazzo,et al.  Which Problems Have Strongly Exponential Complexity? , 2001, J. Comput. Syst. Sci..

[17]  Celina M. H. de Figueiredo,et al.  The external constraint 4 nonempty part sandwich problem , 2011, Discret. Appl. Math..

[18]  Jaroslav Nesetril,et al.  Graphs and homomorphisms , 2004, Oxford lecture series in mathematics and its applications.

[19]  Gustav Nordh,et al.  Retractions to Pseudoforests , 2010, SIAM J. Discret. Math..

[20]  Narayan Vikas,et al.  A complete and equal computational complexity classification of compaction and retraction to all graphs with at most four vertices and some general results , 2005, J. Comput. Syst. Sci..

[21]  Daniël Paulusma,et al.  Covering graphs with few complete bipartite subgraphs , 2009, Theor. Comput. Sci..