DOI: 10.51354/mjen.1747369 ISSN: 1694-7398

Efficient domination and bondage numbers in lexicographic product of trees

BetΓΌl Atay
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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, 𝐸𝑛𝑑, 𝐢𝑛,𝑑 with path graph π‘ƒπ‘š, π‘š=2,3 and determine the exact values of efficient domination and efficient bondage numbers of these graphs.

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