The Local Metric Dimension and Distance-Edge-Monitoring Number of Graph

Let G be a graph and u, v, w E V ( G ). The vertex w is said to resolve a pair u and v if and only if d(G)(w,u) not equal d(G) ( w,v ). A set W subset of V ( G ) is defined as a resolving set of G if for all u, v E V ( G ), the pair ( u, v ) is resolved by some w is an element of W . The minimum cardinality of a resolve set of G is defined as dim(G). A set R subset of V ( G ) is a local resolve set of G if for all u, v E V ( G ) such that uv E E ( G ), the pair ( u, v ) is resolved by some r E R . The minimum cardinality of a local resolve set of G is defined as dim`(G). An edge uv subset of E ( G ) is said to be monitored by x is an element of V ( G ) if d G ( x, u ) not equal d (G-uv )( x, u ) or d G ( x, v ) not equal d(G-uv )( x, v ). A set M subset of V ( G ) is a distance-edge-monitoring (DEM) set if for all e E subset of ( G ), e is monitored by some v E M . The minimum cardinality of a DEM set of G is defined as dem(G). In this paper, we obtained that 1 <= diml(G) G dem(G) G n - 1 for all non trivial graphs with order n , and the exact value of dim`(G) and dem(G) for G is an element of { T-n , K-n , B-n , NEn , Koch(n)}. Also, we obtained that if dim`(G) = 1 then deml(G) G [ n 2 ]. With respect to the relation between the defined graph invariants, it was proved a bound for (dem - dim)(n) for G E Gn = {G V (G) = n } and the exact values of (dim - diml)(n) for G E Gn, where (dem - dim)(n) (resp, (dim - diml)(n))is the maximum value of dem(G) - diml(G) (resp, dem(G) - diml(G)) over all graphs G