The Fractional Local Metric Dimension of Comb Product Graphs

For the connected graph G with vertex set V(G) and edge set E(G), the local resolving neighborhood Rl{u, v} of two adjacent vertices u, v is defined by Rl{u, v} = {x ∈ V(G): d(x, u) ≠ d(x, v)}. A local resolving0function fl of G is a real valued function fl: V(G) → [0,1] such0that fl(Rl{u, v}) ≥ 1 for every two adjacent vertices u, v ∈ V(G). The fractional local metric0dimension of graph G denoted dimfl(G), is defined by dimfl(G) = min⁡{|fl|: fl ⁡is⁡a⁡⁡local⁡resolving⁡function⁡of⁡G}. One of the operation in graph is the comb product graphs. The comb product graphs of G and H is denoted by G ⊳ H. The purpose of this research is to determine the fractional local metric0dimension of G ⊳ H, for graph G is a connected0graph and graph H is a complete graph (Kn). The result of G ⊳ Kn is dimfl(G ⊳ Kn) = |V(G)|. dimfl(Kn−1).


Introduction:
The first authers to discuss the minimum0resolving set and the metric0dimension problems is (1,2). They assumed that the graph used is a connected graph, simple graph and a finite graph. In (3), graph is defined as a finite and nonempty set of ( ) whose elements are called vertices and sets ( ) (maybe empty) whose elements are called edges which are non-ordered pairs of two different elements of ( ).
Let and be two vertices in , ( , ) is the distance0between two0vertices to of , defines as the shortest path between to . For an ordered subset = { 1 , 2 , … , } ⊆ ( ) and ∈ ( ), the representation of with respect to is an ordered k-tuple ( | ) = ( ( , 1 ), ( , 2 ), … , ( , )), where ( , ) is the distance0between two0vertices to . The set is called a resolving set for if each vertex in has a different representation of . A resolving set that has a minimal cardinality is called a basis of . The number of vertex on the basis of graph is called0dimension of and denoted by ( ). In (4) introduced the local metric0dimension of0graph, they defined the local0resolving set and the local0metric dimension of a0graph. In (5,6) studied the commutative0characterization0of graph operations with0respect to the local metric0dimension and metric dimension, respectively. The development of the metric0dimension is the fractional0metric0dimension. The fractional0metric dimensions were first examined by (7) they defined the concept of the fractional0metric dimension of involving resolving0set, resolving function and fractional metric dimension. Then their research was continued by (8). Furthermore, in (9) also found characterization where is a connected graphs. Meanwhile, the fractional0metric0dimension of trees and uncyclic graphs can0be seen in (10).
Furthermore, research about the fractional metric dimensions of a product graph has been investigated by (11,8), and in (12) who studied the fractional metric0dimensions on permutation.
The latest development of fractional metric dimension of graphs was conducted by (13). In (14,15) found the fractional0metric dimension of comb product graph. Figure 1 shows examples of comb product graphs In (14) discussed fractional metric dimension of comb product graph. In this paper discussing the fractional local metric dimension of comb product graphs of and , for is an arbitrary graph and graph is a complete graph.

Results:
In this research, we investigate the fractional local metric0dimension of comb product graphs where are some special graphs. We first recall some fractional local metric0dimension of a special graphs. ( ) = 1 3. for the wheel graph ( ), then The fractional local metric dimension of comb product graph for some special graphs, is presented as below.
( −1 ) where is a connected graph. This research can be continued for graph and graph is arbitrary graph, and for further research Cartesian product can be used.