A Study on Avoiding RFI in the Movement of Robots via Radio Resolving Number Problem

Let G = (V, E) be a connected graph with diameter d and order n. A non-empty subset X of V is called a resolving set if for each pair u, v ∈ V such that u ≠ v, then there is a vertex x in X satisfies d(x, u) ≠ d(x, v). The minimum cardinality of all such resolving sets is called the resolving number of G. Let X be a non-empty minimum cardinality resolving set of G. An injection σ: V (G) → N is said to be a radio resolving labelling if d (u, x)+|σ (u) − σ(x)| ≥ 1 + d, ∀ u ∈ V \X, x ∈ X, where d (u, x) is the distance between u and x. The radio resolving number of σ denoted rβ(σ) is the biggest number labelled under the mapping σ. The minimum taken over all rβ(σ) is called radio resolving number, denoted by rβ (G) . If rβ (G) = n, then G is called radio resolving graceful graph. In this research article, we introduce the concept of radio resolving number and present some results connecting the radio resolving number with resolving number and radio number. Further, we investigate the exact radio resolving number for complete graph, path, cycle, star graph, complete k−partite graph, and complete binary tree. © Palestine Polytechnic University-PPU 2023.