Geodesic
Distance 2-Domination in Prisms of Graphs
Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 2, pp. 383-397

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A set of vertices D of a graph G is a distance 2-dominating set of G if the distance between each vertex u ∊ (V (G) − D) and D is at most two. Let γ2(G) denote the size of a smallest distance 2-dominating set of G. For any permutation π of the vertex set of G, the prism of G with respect to π is the graph πG obtained from G and a copy G′ of G by joining u ∊ V(G) with v′ ∊ V(G′) if and only if v′ = π(u). If γ2(πG) = γ2(G) for any permutation π of V(G), then G is called a universal γ2-fixer. In this work we characterize the cycles and paths that are universal γ2-fixers.
Keywords: distance 2 dominating set, prisms of graphs, universal fixer 
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Hurtado, Ferran; Mora, Mercè; Rivera-Campo, Eduardo; Zuazua, Rita. Distance 2-Domination in Prisms of Graphs. Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 2, pp. 383-397. http://geodesic.mathdoc.fr/item/DMGT_2017_37_2_a5/