Abstract
We determine the distance k-domination number for the total graph, shadow graph, and middle graph of path Pn.
Highlights
We begin with finite, connected, and undirected graphs, G =
A dominating set D of a graph G is a set of vertices of G such that every vertex of V(G) − D is adjacent to some vertex of D
The open k-neighbourhood set Nk(V) of vertex V ∈ V(G) is the set of all vertices of G which are different from V and at distance at most k from V in G, that is, Nk(V) = {u ∈ V(G)/d(u, V) ≤ k}
Summary
(V(G), E(G)) without loops or multiple edges. A dominating set D of a graph G is a set of vertices of G such that every vertex of V(G) − D is adjacent to some vertex of D. The open k-neighbourhood set Nk(V) of vertex V ∈ V(G) is the set of all vertices of G which are different from V and at distance at most k from V in G, that is, Nk(V) = {u ∈ V(G)/d(u, V) ≤ k}. The total graph T(G) of a graph G is the graph whose vertex set is V(G) ∪ E(G) and two vertices are adjacent whenever they are either adjacent or incident in G. The middle graph M(G) of a graph G is the graph whose vertex set is V(G) ∪ E(G) and in which two vertices are adjacent whenever either they are adjacent edges of G or one is a vertex of G and the other is an edge incident with it
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