Computing the TCCD-Number for the Strong Product of Graphs and Its Application in Urban Water Treatment: Addressing Water Scarcity Through Drainage Purification

A dominating set <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$S$</tex> of a graph <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex> is termed as a triple connected certified dominating set (TCCD-set) if, for every vertex <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$v\in S$</tex>, the condition <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\vert N(v)\cap(V-S)\vert \neq 1$</tex> holds, and the subgraph <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\langle S \rangle$</tex> is triple connected. The minimum size of a TCCD-set is known as the triple connected certified domination number (TCCD-number), denoted by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\gamma_{TCC}(G)$</tex>. This study investigates the behavior of the strong product of graphs with respect to the TCCD-number. The TCCD-set concept offers valuable insights into graph connectivity and its applications in enhancing network reliability. In addition to graph theory, this study examines practical applications, such as the establishment of advanced water treatment plants in strategically chosen locations within cities. These plants enable companies to efficiently purify water while utilizing sufficient resources. This model facilitates a reduction in the installation and maintenance costs of drainage water systems, with a primary focus on transforming drainage water into purified water.