Exact Resolving Domination Numbers of Wheel-Derived Graphs With an ILP Formulation

Let G=V,E be a simple, connected graph, and let da,g denote the shortest distance between vertices a and g. An ordered subset R of V, with R=k and k∈N, is known as a resolving set if, for m≠c,∃t∈R such that dm,t≠dc,t, ∀m, c ∈ V–R. The distance code representation for b∈V−R w.r.t. R is given by codeb∣R=db,t1,…,db,tk, ∀ ti ∈ R. A dominating set D satisfies the condition that ∀w∈V−D is adjacent to at least one vertex within D. A resolving dominating set (RDS) R′⊆V guarantees both conditions, and the smallest cardinality among all RDS is termed as resolving domination number, denoted by γresG. We determine γres for several wheel-derived graph families, including water wheel graphs, closed water wheel graphs, sunflower graphs, and flower graphs, and obtain exact values for these graph classes. In addition, we develop an exact integer linear programming (ILP) formulation for the minimum RDS problem on arbitrary simple connected graphs. The formulation employs binary decision variables and linear domination and resolvability constraints, providing a general, exact optimization model that complements the theoretical results established for the wheel-derived graph families.