Metric and edge-metric dimensions of bobble-neighbourhood-corona graphs

Resolving set in a graphG =(V(G), E(G)) is an ordered subset W of V(G) such that every vertex in V(G) has distinct representation with respect to W. Resolving set of G of minimum cardinality is called basis of G.Cardinality of basis of G is called metric dimension of G, dim(G). An ordered set W is called edge resolving set of G if every edge in E(G) has distinct edge-representation with respect to W. Edge-resolving sets in a graph Gof minimum cardinality is called edge-metric basis of G. Cardinality of edge-basis of G is called as edge-metric dimension of G, edim(G). Neighbourhood corona of G and H, G*H, is agraph obtained by taking graph G and |V (G)| graph Hi with Hi i = 1,2,…, |7(G)| is copy of H, then all vertices in H¡ are connected with neighbouring vertex of vertex v¡ in V (G). In this paper, we determine and analyse metric and edge-metric dimension of bobble-neighbourhood-corona, that is metric and edge-metric dimension of neighbourhood-corona of G and H, G*H, with H is trivial graph K1, and G £ {Cn,Kn}.


Introduction
The metric dimension is ne of the areas of studies in graph theory. Metric dimension of graphs first was introduced by Slater in 1975, then Harary and Melter in 1976 [1]. Metric dimension of graphs that resulted fromgraph operations werealso discovered. For metric dimension of corona graphs, Iswadi et al. have found them in 2011 [2]. They found corona graph ⊙ , where was a trivial graph . One of variants ofcorona graphs is edge-corona graph. In 2017, Rinurwati et al. Have determined metric dimension of edge-corona graph  where is trivial graph [3]. Beside metric dimension, Rinurwati et al. have also found local metric dimension [3]. For local adjacency metric dimension, Rinurwati et al. have found it for some wheel related graphs with pendant points [4]. The other variant of corona graphs is neighbourhood-corona graph. The graph was introduced by Gopalapillai [5]. The neighbourhood corona graph in [5] was studied for spectrum of the graph.
Another study about graph dimension is edge-metric dimension. Edge-metric dimension of graphs were introduced by Kelenc

Main results
In this part, we determineand analyse value of metric and edge-metric dimensions of * where is trivial graph 1 .

Metric dimension of bobble-neighbourhood-corona graphs
In this section, here is the result of metric dimension of * 1 where is belong to { , } which n is odd. From (a) to (d), we know that there are some vertices of * 1 have same representation to . Thus = { 1 , 2 , … , −1 } is a basis for * 1 . By direct observation on * 1 , we will show that dim( * 1 ) = ⌈ 3 ⌉, for ≥ 7, with the following steps: I. Create a distance matrix of * 1 (the elements of the matrix are distance between two distinct vertices of * 1 ). The elements of each row of the matrix, in order from left to right represents 1st, 2nd, ..., nth coordinates of vertice representation ∈ * 1 . II.
Observe a single value on: 1) First column of the distance matrix of * 1 , then we obtain some vertices representations with single value from 1st coordinate. The vertices with single representations are collected into set A. 2) First two columns of the distance matrix of * 1 , then we obtain some vertices representations with single velue from first two coordinates. The vertices with single representations are collected into set B. So on with the same steps, the observation is continues until first ⌈ 3 ⌉ columns of the distance matrix. The vertices with single representations are collected into set M.

Edge metric dimension of
From 5 cases above, we can see clearly that there are no edge which have same representation to , so is a edge resolving set for * 1 . Otherwise, let is a vertices set with cardinality 2 − 3, then is not a edge resolving set for * 1 , because: From (a) and (b), we know that in a set with cardinality 2 − 3, there are edges of * 1 that have same representation to . Therefore, = { 1 , 2 , … , −1 , 1 , 2 , … , −1 } is a minimum edge resolving set for * 1 . ■ Proof. In this paper, we will proof the theorem for ≥ 11. 2) First two columns of the distance matrix of * 1 , then we obtain some edge representations with single value from first two coordinates. The edges with single representations are collected into set 2 . So on with the same steps, the observation is continues until first ⌈ 3 ⌉ columns of the distance matrix.

Choose
The edges with single representations are collected into set T.