Power Domination in Some Cubic Graphs and Its Relations with Other Graph Parameters

Yeoh, Randy Xuin (2024) Power Domination in Some Cubic Graphs and Its Relations with Other Graph Parameters. Final Year Project (Bachelor), Tunku Abdul Rahman University of Management and Technology.

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Abstract

Graph Theory refers to the mathematical structures that are used to specify pairwise relationships between objects. In a graph, vertices and their connections to one another by edges make up a graph's essential elements. G = (V, E) denotes a graph in mathematics, where V stands for a vertex and E for an edge. A power dominating set is a subset S ⊆ V such that every vertex in V is either in S or adjacent to a vertex in S. The power domination number of a graph G, denoted by γp(G), is the size of the smallest power dominating set in S. Mathematically, γp(G) is the minimum cardinality of S that satisfies the power domination number condition. In this project, we investigate the power domination number for some of the cubic graphs, which are the Durer graph, Mobius-Kantor graph, Frucht Graph, Wagner graph, Franklin graph, Bidakis Cube and Tietze graph. Additionally, we will investigate the relationship between power domination number with its graph parameters i.e. its number of vertices and its chromatic number. Keywords: Power domination number, Durer graph, Mobius-Kantor graph, Frucht Graph, Wagner graph, Franklin graph, Bidakis Cube, Tietze graph, chromatic number.