ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2018: 0.741

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Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 25(3) (2005) 225-239
DOI: 10.7151/dmgt.1276


Yair Caro

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

William F. Klostermeyer

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

Raphael Yuster

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


An odd dominating set of a simple, undirected graph G = (V,E) is a set of vertices D ⊆ V such that |N[v] ∩D| ≡ 1 mod 2 for all vertices v ∈ V. It is known that every graph has an odd dominating set. In this paper we consider the concept of connected odd dominating sets. We prove that the problem of deciding if a graph has a connected odd dominating set is NP-complete. We also determine the existence or non-existence of such sets in several classes of graphs. Among other results, we prove there are only 15 grid graphs that have a connected odd dominating set.

Keywords: dominating set, odd dominating set.

2000 Mathematics Subject Classification: 05C35.


[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. Combin. Math. and Combin. 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. Combin. Math. Combin. Comput. 44 (2003) 65-84.
[6] Y. Caro, W. Klostermeyer and J. Goldwasser, Odd and residue domination numbers of a graph, Discuss. Math. Graph Theory 21 (2001) 119-136, doi: 10.7151/dmgt.1137.
[7] M. Conlon, M. Falidas, M. Forde, J. Kennedy, S. McIlwaine and J. Stern, Inversion numbers of graphs, Graph Theory Notes of New York XXXVII (1999) 43-49.
[8] R. Cowen, S. Hechler, J. Kennedy and A. Ryba, Inversion and neighborhood inversion in graphs, Graph Theory Notes of New York XXXVII (1999) 38-42.
[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 18 (2002) 271-283, doi: 10.1007/s003730200020.
[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) 101-109.
[13] T. Haynes, S. Hedetniemi and P. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).
[14] K. Sutner, Linear cellular automata and the Garden-of-Eden, The Mathematical Intelligencer 11 (2) (1989) 49-53, doi: 10.1007/BF03023823.

Received 7 May 2003
Revised 19 January 2005