Geodesic
Changing and Unchanging of the Domination Number of a Graph: Path Addition Numbers
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 2, pp. 365-379

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Given a graph G=(V, E) and two its distinct vertices u and v, the (u, v)-P_k-addition graph of G is the graph G_u,v,k−2 obtained from disjoint union of G and a path P_k : x_0, x_1,...,x_k−1, k ≥ 2, by identifying the vertices u and x_0, and identifying the vertices v and x_k−1. We prove that γ(G) − 1 ≤ γ(G_u,v,k) for all k ≥ 1, and γ(G_u,v,k) gt;γ(G) when k ≥ 5. We also provide necessary and sufficient conditions for the equality γ(G_u,v,k)=γ(G) to be valid for each pair u, v ∈ V(G). In addition, we establish sharp upper and lower bounds for the minimum, respectively maximum, k in a graph G over all pairs of vertices u and v in G such that the (u, v)-P_k-addition graph of G has a larger domination number than G, which we consider separately for adjacent and non-adjacent pairs of vertices.
Keywords: domination number, path addition 
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     author = {Samodivkin, Vladimir},
     title = {Changing and {Unchanging} of the {Domination} {Number} of a {Graph:} {Path} {Addition} {Numbers}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {365--379},
     publisher = {mathdoc},
     volume = {41},
     number = {2},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2021_41_2_a1/}
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Samodivkin, Vladimir. Changing and Unchanging of the Domination Number of a Graph: Path Addition Numbers. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 2, pp. 365-379. http://geodesic.mathdoc.fr/item/DMGT_2021_41_2_a1/