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Let G = (V (G), E(G)) be a simple graph and let α ∈ (0, 1]. A set S ⊆ V (G) isan α-partial dominating set in G if |N[S]| ≥ α |V (G)|. The smallest cardinality of an α-partialdominating set in G is called the α-partial domination number of G, denoted by ∂α(G). An α-partial dominating set S ⊆ V (G) is a total α-partial dominating set in G if every vertex in S isadjacent to some vertex in S. The total α-partial domination number of G, denoted by ∂T α(G), isthe smallest cardinality of a total α-partial dominating set in G. In this paper, we characterize thetotal partial dominating sets in the join, corona, lexicographic and Cartesian products of graphsand determine the exact values or sharp bounds of the corresponding total partial dominationnumber of these graphs.

In this paper, we characterize the fair total dominating sets in the join and corona of graphs and determine the corresponding fair total domination numbers. We also characterize some fair total dominating sets in the composition of graphs and give sharp upper bounds for the corresponding fair total domination numbers.

In this paper, we characterize the fair dominating sets in the join, corona and composition of graphs. We also determine the bounds or exact values of the fair domination numbers of these graphs.