Efficient domination and bondage numbers in lexicographic product of trees

The vertex subset 𝑆 in 𝐺 is called the efficient dominating set, if every vertex not in 𝑆 is adjacent exactly one vertex in 𝑆 and it is also required that none of the vertices in 𝑆 are adjacent to each other. The efficient domination number 𝛾𝑒(𝐺) of 𝐺 is the minimum number of vertices of an efficient dominating set in 𝐺. If the efficient domination number of the graph obtained by removing an edge subset 𝐷 is greater than the efficient domination number of the original graph and if 𝐷 is the set with the fewest such edges, then the cardinality of 𝐷 is called the efficient bondage number 𝑏𝑒(𝐺) of 𝐺. This study deals with the concept of efficient domination and efficient bondage numbers of a graph. It characterizes this phenomenon in the lexicographic product of some trees such as 𝑃𝑛, 𝐸𝑛𝑡, 𝐶𝑛,𝑡 with path graph 𝑃𝑚, 𝑚=2,3 and determine the exact values of efficient domination and efficient bondage numbers of these graphs.