Geodesic
Graph domination in distance two
Discussiones Mathematicae. Graph Theory, Tome 25 (2005) no. 1-2, pp. 121-128

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Let G = (V,E) be a graph, and k ≥ 1 an integer. A subgraph D is said to be k-dominating in G if every vertex of G-D is at distance at most k from some vertex of D. For a given class of graphs, Domₖ is the set of those graphs G in which every connected induced subgraph H has some k-dominating induced subgraph D ∈ which is also connected. In our notation, Dom coincides with Dom₁. In this paper we prove that Dom Dom _u = Dom₂ _u holds for _u = all connected graphs without induced P_u (u ≥ 2). (In particular, ₂ = K₁ and ₃ = all complete graphs.) Some negative examples are also given.
Keywords: graph, dominating set, connected domination, distance domination, forbidden induced subgraph 
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Bacsó, Gábor; Tálos, Attila; Tuza, Zsolt. Graph domination in distance two. Discussiones Mathematicae. Graph Theory, Tome 25 (2005) no. 1-2, pp. 121-128. http://geodesic.mathdoc.fr/item/DMGT_2005_25_1-2_a12/