The zero forcing number for cubic Cayley graphs of dihedral groups

The zero forcing number Z ( Γ ) of a graph Γ is the minimum cardinality of a zero forcing set Z ⊆ V ( Γ ) . In this paper, we study the zero forcing number for cubic connected Cayley graphs Γ = Cay ( D 2 n , S ) , where D 2 n = 〈 a , b | a n = b 2 = 1 , b a b = a − 1 〉 is the dihedral group of order 2 n . We determine Z ( Γ ) exactly when Γ is isomorphic to the complete graph K 4 , a ladder graph L ( 2 n ) , or a Möbius ladder M ( 2 n ) . For the remaining cases, namely those of type III, we give exact values in two normalized prime cases and establish explicit lower and upper bounds in general. These estimates are expressed in terms of the girth of Γ and arithmetic parameters associated with the differences among the exponents in the generating set.