# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2018-2019): c. 84%

# Discussiones Mathematicae Graph Theory

## CONNECTED ODD DOMINATING SETS IN GRAPHS

 Yair Caro Department of Mathematics University of Haifa - Oranim Tivon - 36006, Israel e-mail: yairc@macam.ac.il William F. Klostermeyer Department of Computer and Information Sciences University of North Florida Jacksonville, FL 32224, USA e-mail: klostermeyer@hotmail.com Raphael Yuster Department of Mathematics University of Haifa - Oranim Tivon - 36006, Israel e-mail: raphy@research.haifa.ac.il

## Abstract

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.

## 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. 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