Locally constrained graph homomorphisms and equitable partitions

  • Authors:

  • Jiří Fiala;Daniël Paulusma;Jan Arne Telle

  • Affiliations:

  • Charles University, Faculty of Mathematics and Physics, DIMATIA and Institute for Theoretical Computer Science (ITI), Malostranské nám. 2/25, 11800, Prague, Czech Republic11Supported by ...;Department of Computer Science, Durham University, Science Laboratories, South Road, Durham DH1 3EY, England, United Kingdom;Department of Informatics, University of Bergen, N-5020 Bergen, Norway

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2008

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Abstract

We explore the connection between locally constrained graph homomorphisms and degree matrices arising from an equitable partition of a graph. We provide several equivalent characterizations of degree matrices. As a consequence we can efficiently check whether a given matrix M is a degree matrix of some graph and also compute the size of a smallest graph for which it is a degree matrix in polynomial time. We extend the well-known connection between degree refinement matrices of graphs and locally bijective graph homomorphisms to locally injective and locally surjective homomorphisms by showing that these latter types of homomorphisms also impose a quasiorder on degree matrices and a partial order on degree refinement matrices. Computing the degree refinement matrix of a graph is easy, and an algorithm deciding the comparability of two matrices in one of these partial orders could be used as a heuristic for deciding whether a graph G allows a homomorphism of the given type to H. For local surjectivity and injectivity we show that the problem of matrix comparability belongs to the complexity class NP.