IJMSHE Volume 2 Issue 1 | E-ISSN: 3082-3021 | DOI: https://doi.org/10.70847/593139 International Journal of Multidisciplinary Studies in Higher Education Journal Homepage: https://ijmshe.com/index.php/pub/index Research Article Restrained Weakly Connected 2-Domination and the Lexicographic Product of Graphs Michael P. Tañalas 1 | Mae P. Militante 2 | Berlita Y. Disca 3 1-3 Mindanao State University-General Santos Article Info Article history: Received: 31 December 2024 Revised: 22 January 2025 Accepted: 08 February 2025 Keywords: Lexicographic product, Restrained weakly-connected 2-domination, Upper bound, Weakly connected A b s t r a c t Let be a nontrivial connected graph. A subset =  ( ),  ( ) ( ) is a restrained weakly connected 2-dominating set ⊆ ( ) (RWC2D-set) in if every vertex is adjacent to at least two  ∈ ( )\ vertices in and is adjacent to another vertex in and the subgraph   ( )∖ weakly induced by is connected. A subgraph is said to be weakly ⟨⟩   induced by a set if it includes all vertices in and all edges in the   original graph that are connected to at least one vertex in . The   minimum cardinality of a restrained weakly connected 2-dominating set, denoted by , is called the restrained weakly connected γ 2  ( ) 2-domination number. Leveraging the concept of weakly connected 2-domination by examining another parameter called restrained domination is the prime focus of this study. The newly defined parameter is explored, establishing improved upper bounds and providing conditions for graph to admit an RWC2D-set. Additionally, sufficient conditions  for the RWC2D-set on the lexicographic product of graphs were provided. It is shown that for any graph of order , .  ≥4 γ 2  ( )≤−2 Moreover, the upper bound for the restrained weakly connected 2-domination number of the lexicographic product of graphs is provided. Cite as: Tañalas, M., Militante, M., & Disca, B. (2025). Restrained Weakly Connected 2-Domination and the Lexicographic Product of Graphs. International Journal of Multidisciplinary Studies in Higher Education, 2(1), 138–149. https://doi.org/10.70847/593139 1 Corresponding Author: Michael P. Tañalas * Corresponding Email: michael.tanalas@msugensan.edu.ph 138