Geodesic
Connected odd dominating sets in graphs
Discussiones Mathematicae. Graph Theory, Tome 25 (2005) no. 3, pp. 225-239.

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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 
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Caro, Yair; Klostermeyer, William; Yuster, Raphael. Connected odd dominating sets in graphs. Discussiones Mathematicae. Graph Theory, Tome 25 (2005) no. 3, pp. 225-239. http://geodesic.mathdoc.fr/item/DMGT_2005_25_3_a0/

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