Matrix and Graph Orders Derived from Locally Constrained Graph Homomorphisms

• Published in: Lecture notes in computer science pp. 340 - 351
• Main Authors: Fiala, Jiří, Paulusma, Daniël, Telle, Jan Arne
• Format: Book Chapter Conference Proceeding
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2005; Springer
• Series: Lecture Notes in Computer Science
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Computer science; control theory; systems; Exact sciences and technology; Local Constraint; Partial Order; Regular Graph; Surjective Homomorphism; Theoretical computing; Universal Cover; Polynomial matrix; Computer theory; Refinement method; Partial type; Graph degree; Partial ordering; Matrix polynomial; Homomorphism; Cycle(graph); Ordered graph; Polynomial time
• ISBN: 9783540287025; 3540287027
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/11549345_30

We consider three types of locally constrained graph homomorphisms: bijective, injective and surjective. We show that the three orders imposed on graphs by existence of these three types of homomorphisms are partial orders. 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 the orders imposed on degree refinement matrices by our locally constrained graph homomorphisms are also partial orders. We provide several equivalent characterizations of degree (refinement) matrices, e.g. in terms of the dimension of the cycle space of a graph related to the matrix. 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.