k-Fair domination in the join, corona, composition, and Cartesian product of graphs

On k-Fair Total Domination in Graphs

“…The following characterizations of k-fair dominating sets in the join, corona, and lexicographic product of two nontrivial, connected graphs are found in Maravilla et al [5]. Theorem 5.…”

“…Theorem 5. [5] Let G and H be nontrivial connected graphs of orders m and n, respectively, and k a positive integer with 1 ≤ k ≤ max{m, n}. Then S ⊆ V (G + H) is a kf d-set in G + H if and only if one of the following holds:…”

“…In 2014, Maravilla et al [5] characterized the k-fair dominating sets in the join, corona, lexicographic product, and Cartesian product of graphs and determined the bounds or exact values of the k-fair domination numbers of these graphs. Two variants of k-fair domination, namely connected k-fair domination and neighborhood connected k-fair domination, were studied by Bent-Usman et al [1,6] in 2018 and 2019, respectively.…”

On Semitotal k-Fair and Independent k-Fair Domination in Graphs

“…Lemma 1. [6] Let G be a connected graph of order n ≥ 1 and let k be a positive integer such that k ≤ n. Then:…”

“…Theorem 8. [6] Let G and H be nontrivial connected graphs of orders m and n, respectively, and k a positive integer with 1 ≤ k ≤ max{m, n}. Then S ⊆ V (G + H) is a kf d-set in G + H if and only if one of the following holds:…”