Let  be a simple graph and let  A set  is an a-partial dominating set in G if  The smallest cardinality of an a-partial dominating set in G is called the a-partial domination number of G, denoted by  This paper extends the study on partial domination in graphs independently worked on by Case et al. [5] and Das [7] who published papers of the same title in May 2017 and July 2017, respectively. It also introduces the concept of total partial domination.

In this paper, we characterize the partial dominating sets in the join, corona, lexicographic and Cartesian products of graphs and determine the exact values or sharp bounds of the corresponding partial domination number of these graphs.

partial domination, total partial domination, join, corona, lexicographic product, Cartesian product.