(untitled) (European Journal of Pure and Applied Mathematics, 2021)
Let G = (V (G), E(G)) be a simple non-empty graph. For an integer k â‰¥ 1, a k-fair total dominating set (kf td-set) is a total dominating set S âŠ† V (G) such that |NG(u) âˆ© S| = k for every u âˆˆ V (G)§. The k-fair total domination number of G, denoted by Î³kf td(G), is the minimum cardinality of a kf td-set. A k-fair total dominating set of cardinality Î³kf td(G) is called a minimum k-fair total dominating set or a Î³kf td-set. We investigate the notion of k-fair total domination in this paper. We also characterize the k-fair total dominating sets in the join, corona, lexicographic product and Cartesian product of graphs and determine the exact values or sharp bounds of their corresponding k-fair total domination number.