Relatively Prime Domination Number in Quadrilateral Snake Graphs

A set S⊆V is said to be relatively prime dominating set if it is a dominating set with at least two elements and for every pair of vertices u and v in S, (deg⁡(u),deg⁡〖(v))〗=1. The minimum cardinality of a relatively prime dominating set is called relatively prime domination number and it is denoted by γ_rpd (G). If there is no such pair exist, then γ_rpd (G)=0. For a finite undirected graph G(V,E) and a subset  V, the switching of G by  is defined as the graph  (V,  ) which is obtained from G by removing all edges between  and its complement V-  and adding as edges all non-edges between  and V- . This article delves into the discussion of the relatively prime domination number on quadrilateral snake graphs and their complements. The findings reveal that for quadrilateral snake graphs, the relatively prime domination number γ_rpd(G^v) equals either 2, 3 or 4. Similarly, for alternate quadrilateral snake graphs, the γ_rpd(G^v) is determined to be 2, 3 or 4. In the case of double quadrilateral snake graphs, the relatively prime domination number γ_rpd(G^v) is established as 2, 3, 4, 6 or 7, while for double alternate quadrilateral snake graphs, it is 2, 3, 4 or 5. Notably, the complements of quadrilateral, alternate quadrilateral, double quadrilateral, and double alternate quadrilateral snake graphs exhibit a relatively prime domination number of 2.