a set $wsubseteq v(g)$ is called a resolving set for $g$, if for each two distinct vertices $u,vin v(g)$ there exists $win w$ such that $d(u,w)neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. the minimum cardinality of a resolving set for $g$ is called the metric dimension of $g$, and denoted by $dim(g)$. in this paper, it is proved that in a connected graph $...

A set $Wsubseteq V(G)$ is called a resolving set for $G$, if for each two distinct vertices $u,vin V(G)$ there exists $win W$ such that $d(u,w)neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The minimum cardinality of a resolving set for $G$ is called the metric dimension of $G$, and denoted by $dim(G)$. In this paper, it is proved that in a connected graph $...

A set $Wsubset V (G)$ is called a resolving set, if for every two distinct vertices $u, v in V (G)$ there exists $win W$ such that $d(u,w) not = d(v,w)$, where $d(x, y)$ is the distance between the vertices $x$ and $y$. A resolving set for $G$ with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a uniquely dimensional graph. In this paper, we establish...

Given a graph , a set is a resolving set if for each pair of distinct vertices there is a vertex such that . A resolving set containing a minimum number of vertices is called a minimum resolving set or a basis for . The cardinality of a minimum resolving set is called the resolving number or dimension of and is denoted by . A resolving set is said to be a star resolving set if it induces a star...

In a simple connected undirected graph G, an ordered set R of vertices is called resolving if for every pair distinct u and v, there vertex w∈R such that d(u,w)≠d(v,w). A F the G fault-tolerant each v∈F, F∖{v} also G. this article, we determine optimal fault-resolving r-th power any path Pn when n≥r(r−1)+2. For other values n, give bounds size set. We have presented algorithm to construct Pmr f...

For a given undirected graph G, an ordered subset S = {s1, s2, . . . , sk} ⊆ V of vertices is a resolving set for the graph if the vertices of the graph are distinguishable by their vector of distances to the vertices in S. While a superset of any resolving set is always a resolving set, a proper subset of a resolving set is not necessarily a resolving set, and we are interested in determining ...

A subset S of vertices in a graph G is said to be an independent set of G if each edge in the graph has at most one endpoint in S and a set W ( V is said to be a resolving set of G, if the vertices in G have distinct representations with respect to W. A resolving set W is said to be an independent resolving set, or an ir-set, if it is both resolving and independent. The minimum cardinality of W...

A subset S of vertices in a graph G is said to be an independent set of G if each edge in the graph has at most one endpoint in S and a set W ( V is said to be a resolving set of G, if the vertices in G have distinct representations with respect to W. A resolving set W is said to be an independent resolving set, or an ir-set, if it is both resolving and independent. The minimum cardinality of W...

A subset S of vertices in a graph G is said to be an independent set of G if each edge in the graph has at most one endpoint in S and a set W ( V is said to be a resolving set of G, if the vertices in G have distinct representations with respect to W. A resolving set W is said to be an independent resolving set, or an ir-set, if it is both resolving and independent. The minimum cardinality of W...