The local metric dimension of generalized broken fan graph and edge corona product of star graph and path graph

Let G be a connected nontrivial graph. The distance between two vertices u and v in G is the length of the shortest u - v path, denoted by d ( u, v ). For an ordered set W = { w 1 , w 2 , …, w n } of n vertices on G , the representation of a vertex v with respect to W is ordered pair r ( v|W ) = ( d ( v, w 1 ) , d ( v, w 2 ) , …, d ( v, w n )). W is local metric set of G if r ( u|W ) ≠ r ( v|W ) for every pair of adjacent vertices u and v in G . The local metric set W with minimum cardinality is called local metric basis and its cardinality is the local metric dimension of G , denoted by dim l ( G ). In this paper, we determine the local metric dimension of generalized broken fan graph and edge corona product K 1 ,m ◊ P n graph. We obtained the local metric dimension of generalized broken fan graph is dim l ( B F ( a 1 , a 2 , … , a n ) ) = ∑ i = 1 n ⌈ n − 1 4 ⌉ . The local metric dimension of edge corona graph K 1 ,m ◊ P n is dim l ( K 1. m ⋄ P n ) = ⌊ n + 10 4 ⌋ for m = 1 and 1 ≤ n ≤ 5, dim l ( K 1. m ⋄ P n ) = ⌊ n + 6 4 ⌋ for m = 1 and n ≥ 6, dim l ( K 1. m ⋄ P n ) = ⌊ n + 3 4 ⌋ m for m ≥ 2 and 1 ≤ n ≤ 5, and dim l ( K 1. m ⋄ P n ) = ⌊ n − 1 4 ⌋ m for m ≥ 2 and n ≥ 6.