Efficient Partition in Join and Corona Product of Graphs

Abstract

As a generalization of efficient domination, -efficient domination is defined in terms of partitions of . A set  is a -efficient dominating set if there is a partition of  which is a collection of -neighborhoods of vertices of , where ’s vary between  and . The minimum cardinality of a -efficient dominating set is the -efficient domination number of , denoted by . The study explores the existence of an efficient dominating set and an exact -efficient dominating set for the join and corona product of graphs. Additionally, -efficient domination number for join of graph, corona product of cycles, wheel, and complete multipartite graphs with arbitrary graph is obtained.