DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

DOI 10.7151/dmgt

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2017: 0.601

SCImago Journal Rank (SJR) 2017: 0.633

Rejection Rate (2017-2018): c. 84%

Discussiones Mathematicae Graph Theory

PDF

Discussiones Mathematicae Graph Theory 21(1) (2001) 119-136
DOI: 10.7151/dmgt.1137

ODD AND RESIDUE DOMINATION NUMBERS OF A GRAPH

Yair Caro

Department of Mathematics
University of Haifa - Oranim
Tivon - 36006, Israel

e-mail: yairc@oranim.macam98.ac.il

William F. Klostermeyer

Deptartment of Computer and Information Sciences
University of North Florida
Jacksonville, FL 32224, USA

e-mail: klostermeyer@hotmail.com

John L. Goldwasser

Department of Mathematics
West Virginia University
Morgantown, WV 26506

e-mail: jgoldwas@math.wvu.edu

Abstract

Let G = (V, E) be a simple, undirected graph. A set of vertices D is called an odd dominating set if |N[v] ∩D| ≡ 1 ( mod   2) for every vertex v ∈ V(G). The minimum cardinality of an odd dominating set is called the odd domination number of G, denoted by γ1(G). In this paper, several algorithmic and structural results are presented on this parameter for grids, complements of powers of cycles, and other graph classes as well as for more general forms of ``residue'' domination.

Keywords: dominating set, odd dominating set, parity domination.

2000 Mathematics Subject Classification: 05C35, 05C69, 05C85.

References

[1] A. Amin, L. Clark, and P. Slater, Parity Dimension for Graphs, Discrete Math. 187 (1998) 1-17, doi: 10.1016/S0012-365X(97)00242-2.
[2] A. Amin and P. Slater, Neighborhood Domination with Parity Restriction in Graphs, Congr. Numer. 91 (1992) 19-30.
[3] A. Amin and P. Slater, All Parity Realizable Trees, J. Comb. Math. Comb. Comput. 20 (1996) 53-63.
[4] Y. Caro, Simple Proofs to Three Parity Theorems, Ars Combin. 42 (1996) 175-180.
[5] Y. Caro and W. Klostermeyer, The Odd Domination Number of a Graph, J. Comb. Math. Comb. Comput. (2000), to appear.
[6] E. Cockayne, E. Hare, S. Hedetniemi and T. Wimer, Bounds for the Domination Number of Grid Graphs, Congr. Numer. 47 (1985) 217-228.
[7] M. Garey and D. Johnson, Computers and Intractability (W.H. Freeman, San Francisco, 1979).
[8] J. Goldwasser, W. Klostermeyer, G. Trapp and C.-Q. Zhang, Setting Switches on a Grid, Technical Report TR-95-20, Dept. of Statistics and Computer Science (West Virginia University, 1995).
[9] J. Goldwasser, W. Klostermeyer and G. Trapp, Characterizing Switch-Setting Problems, Linear and Multilinear Algebra 43 (1997) 121-135, doi: 10.1080/03081089708818520.
[10] J. Goldwasser and W. Klostermeyer, Maximization Versions of ``Lights Out'' Games in Grids and Graphs, Congr. Numer. 126 (1997) 99-111.
[11] J. Goldwasser, W. Klostermeyer and H. Ware, Fibonacci Polynomials and Parity Domination in Grid Graphs, Graphs and Combinatorics (2000), to appear.
[12] M. Halldorsson, J. Kratochvil and J. Telle, Mod-2 Independence and Domination in Graphs, in: Proceedings Workshop on Graph-Theoretic Concepts in Computer Science '99, Ascona, Switzerland (Springer-Verlag, Lecture Notes in Computer Science, 1999).
[13] T. Haynes, S. Hedetniemi and P. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).
[14] R. Johnson and C. Johnson, Matrix Analysis (Cambridge University Press, 1990).
[15] M. Jacobson and L. Kinch, On the Domination Number of Products of Graphs, Ars Combin. 18 (1984) 33-44.
[16] W. Klostermeyer and E. Eschen, Perfect Codes and Independent Dominating Sets, Congr. Numer. (2000), to appear.
[17] K. Sutner, Linear Cellular Automata and the Garden-of-Eden, The Mathematical Intelligencer 11 (2) (1989) 49-53.

Received 30 October 2000
Revised 17 January 2001