On RIDS Analysis for Shade Tree Placement and Its Application to STGNN Multi-step Forecasting on RH and CO2 Concentration of Coffee Agroforestry

Let G(V,E) be a finite, simple, and connected graph, where |V | and |E| denote the number of vertices and edges, respectively. A subset D ⊆ V is called a dominating set if every vertex in V \ D is adjacent to at least one vertex in D. If no two vertices in D are adjacent, then D is referred to as an independent set. The independent domination number of G, denoted by γi(G), is the minimum size of an independent dominating set. For a given vertex v ∈ V , its metric representation with respect to an ordered set W = {w1,w2, . . . ,wk} is defined as the k-vector r(v|W){d(v|w1), d(v|w2), d(v|w3), . . . , d(v|wk)}, where d(v,w) is the shortest path distance between vertices v and w. A set W is called a resolving independent dominating set (RIDS) if it is an independent dominating set and every pair of distinct vertices in G has a unique metric representation relative toW. The smallest cardinality of such a set is known as the resolvingindependent domination number, denoted by γri(G). In this paper, we will obtain the lower and upper bound of γri(G) and determine the exact of value of the resolving independent domination number of some graph classes. Furthermore, to see the robust application of resolving independent domination, at the end of this paper we will illustrate the implementation of it on analyzing Spatial Temporal Graph Neural Network (STGNN) model for multi-step forecasting on relative humidity (RH) and carbon dioxide (CO2) concentration of coffee agroforestry.