DMGT

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

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.

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Received 7 May 2003
Revised 19 January 2005