## 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.

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.

Original language | English |
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Pages (from-to) | 225–239 |

Journal | Discussiones Mathematicae - Graph Theory |

Volume | 25 |

State | Published - 2005 |