Total Partial Domination in Graphs Under Some Binary Operations

Abstract

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.

On Connected Partial Domination in Graphs

“…In 2019, R. Macapodi, R. Isla and S. Canoy [7] characterized the partial dominating sets in the join, corona, lexicographic and Cartesian products of graphs and determined the exact values or sharp bounds of the corresponding partial domination number of the said graphs. In the same year, R. Macapodi and R. Isla [10] published another paper where they characterized the total partial dominating sets in the join, corona, lexicographic product and Cartesian product of graphs. They also determined the exact values or sharp bounds of the corresponding total partial domination number of these graphs.…”

Independent partial domination

“…The theory of domination is one of the profusely researched areas in graph theory. Recently a new domination parameter called partial domination number was introduced simultaneously in [3], [4] and [6], and studied in [12,13,14,15]. We extend the concept of partial domination to independent domination in graphs.…”