Weak and strong domination on some graphs

¹ Arts and Science Faculty, Manisa Celal Bayar University, 45140 Manisa, Turkey
² Institute of Natural and Applied Sciences, Manisa Celal Bayar University, 45140 Manisa, Turkey

^* Corresponding author: derya.dogan@cbu.edu.tr

Received: 22 March 2020
Accepted: 24 March 2022

Abstract

Let G = (V(G), E(G)) be a graph and uvεE. A subset D ⊆ V of vertices is a dominating set if every vertex in V − D is adjacent to at least one vertex of D. The domination number is the minimum cardinality of a dominating set. Let u and v be elements of V. Then, u strongly dominates u and v weakly dominates u if (i)uvεE and (ii)deg(u) ≥ deg(v). A set D ⊆ V is a strong (weak) dominating set (sd-set)(wd-set) of G if every vertex in V − D is strongly dominated by at least one vertex in D. The strong (weak) domination number γ_s(γ_w) of G is the minimum cardinality of a sd-set (wd-set). In this paper, the strong and weak domination numbers of comet, double comet, double star and theta graphs are given. The theta graphs are important geometric graphs that have many applications, including wireless networking, motion planning, MST construction and real-time animation.