THE NON-ISOLATED RESOLVING NUMBER OF CARTESIAN PRODUCT OF ANY GRAPHS WITH A CERTAIN GRAPH

• Main Author: Mulia Hasibuan, Ismail
• Format: Dissertations
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/71452
• Institution: Institut Teknologi Bandung

Let G be a simple, connected, and finite graph. For an ordered subset W = fw1;w2; ;wkg of vertices in a graph G and a vertex v of G, the representation of v respect to W is the k-vector r(vjW) = (d(v;w1); d(v;w2); ; d(v;wk)) where d(v;wi) represents the distance between the vertices v and wi for every i in [1; k]. The set W is called a resolving set of G, if every vertex of G has a distinct representation. The resolving set W is called a non-isolated resolving set of G, if the induced subgraph hWi has no isolated vertices. The minimum cardinality of a nonisolated resolving set of G is called the non-isolated resolving number of G, denoted by nr(G). The non-isolated resolving number of a graph was first introduced by Chitra and Arumugam (2015) by determining the non-isolated resolving number of some classes of graphs. Then, several other researchers have determined the non-isolated resolving number for some graphs obtained by an operation, including a join operation, a shackle operation, a comb product, a corona product, and a Cartesian product. Little has been investigated about the non-isolated resolving number of graph by previous researchers. These include the non-isolated resolving number of Cartesian product of two paths of order n, and Cartesian product of a cycle of order n with a path of order 2. In this dissertation, we provide an upper bound and a lower bound of the non-isolated resolving number of Cartesian product of a graphs with a certain graph, namely a path, a star graph, a cycle, or a complete graph. Then, we also determine the non-isolated resolving number of Cartesian product of two certain graphs.