The local metric dimension of edge corona and corona product of cycle graph and path graph

Let G be a nontrivial connected graph with vertex set V ( G ). For an ordered set W = { w 1 , w 2 , …, w n } of n distinct vertices in G , the representation of a vertex v ∈ V ( G ) with respect to W is an ordered value of distance between v and every vertex of W . The set W is a local metric set of G if the representations of every pair of adjacent vertices with respect to W are different. The local metric set with minimum cardinality is called local metric basis and its cardinality is the local metric dimension of G and denoted by dim l ( G ). The edge corona product of cycle graph and path graph denoted by C m ʘ P n , this graph is obtained from a cycle graph C m and m copies of path graph P n , and then joining two end-vertices of i th edge of C m to every vertex in the i th copy of P n , where 1 ≤ i ≤ m . The corona product of cycle graph and path graph denoted by C m ◊ P n , this graph is obtained from a cycle graph C m and m copies of path graph P n , and then joining by an edge each vertex from the i th copy of P n with i th vertex of C m . In this paper, we determine the local metric dimension of edge corona and corona product of cycle graph and path graph for positive integer m ≥ 3 and n ≥ 1. We obtain the local metric dimension of edge corona product of cycle and path graphs is dim l ( C m ◊ P n )=2 for n = 1 and dim l ( C m ⋄ P n ) = m ⌈ n − 1 4 ⌉ for   n ≥ 2 for n ≥ 2. The local metric dimension of corona product of cycle and path graphs is dim l ( C m ◊ P n )=1 for n = 1 and even positive integer m ≥ 3, dim l ( C m ʘ P n )=2 for n = 1 and odd positive integer m ≥ 3, and dim l ( C m ⊙ P n ) = m ⌈ n − 1 4 ⌉ for   n ≥ 2 for n ≥ 2.