The partition dimension of circulant graphs (original) (raw)

On the Partition Dimension of Circulant Graphs

For a vertex v of a connected graph G(V, E) and a subset S of V , the distance between v and S is defined by d(v, S) = min{d(v, x) : x ∈ S}. For an ordered k -partition Π = {S 1 , S 2 . . . S k } of V , the representation of v with respect to

On the k-partition dimension of graphs

Theoretical Computer Science, 2020

As a generalization of the concept of the partition dimension of a graph, this article introduces the notion of the k-partition dimension. Given a nontrivial connected graph G = (V, E), a partition Π of V is said to be a k-partition generator of G if any pair of different vertices u, v ∈ V is distinguished by at least k vertex sets of Π, i.e., there exist at least k vertex sets S 1 ,. .. , S k ∈ Π such that d(u, S i) = d(v, S i) for every i ∈ {1,. .. , k}. A k-partition generator of G with minimum cardinality among all their k-partition generators is called a k-partition basis of G and its cardinality the k-partition dimension of G. A nontrivial connected graph G is k-partition dimensional if k is the largest integer such that G has a k-partition basis. We give a necessary and sufficient condition for a graph to be r-partition dimensional and we obtain several results on the k-partition dimension for k ∈ {1,. .. , r}.

The partition dimension of a graph

For a vertex v of a connected graph G and a subset S of V (G), the distance between v and S is d(v, S) = min{d(v, x)|x ∈ S}. For an ordered k-partition Π = {S 1 , S 2 , · · · , S k } of V (G), the representation of v with respect to Π is the k-vector r(v|Π) = (d(v, S 1), d(v, S 2), · · · , d(v, S k)). The k-partition Π is a resolving partition if the k-vectors r(v|Π), v ∈ V (G), are distinct. The minimum k for which there is a resolving k-partition of V (G) is the partition dimension pd(G) of G. It is shown that the partition dimension of a graph G is bounded above by 1 more than its metric dimension. An upper bound for the partition dimension of a bipartite graph G is given in terms of the cardinalities of its partite sets, and it is shown that the bound is attained if and only if G is a complete bipartite graph. Graphs of order n having partition dimension 2, n, or n − 1 are characterized.

Resolvability in circulant graphs

Acta Mathematica Sinica, English Series, 2012

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v ∈ V (G) there is a vertex w ∈ W such that d (u, w) = d(v, w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim(G). For a vertex u of G and a subset S of V (G), the distance between u and S is the number min s∈S d (u, s)

On Partition Dimension of Some Cycle-Related Graphs

Mathematical Problems in Engineering

Let G be a simple connected graph. Suppose Δ = Δ 1 , Δ 2 , … , Δ l an l -partition of V G . A partition representation of a vertex α w . r . t Δ is the l − vector d α , Δ 1 , d α , Δ 2 , … , d α , Δ l , denoted by r α | Δ . Any partition Δ is referred as resolving partition if ∀ α i ≠ α j ∈ V G such that r α i | Δ ≠ r α j | Δ . The smallest integer l is referred as the partition dimension pd G of G if the l -partition Δ is a resolving partition. In this article, we discuss the partition dimension of kayak paddle graph, cycle graph with chord, and a graph generated by chain of cycles. It has been shown that the partition dimension of the said families of graphs is constant.

On the metric dimension of circulant graphs

Applied Mathematics Letters, 2012

Let G = (V , E) be a connected graph and d(x, y) be the distance between the vertices x and y in V (G). A subset of vertices W = {w 1 , w 2 ,. .. , w k } is called a resolving set or locating set for G if for every two distinct vertices x, y ∈ V (G), there is a vertex w i ∈ W such that d(x, w i) ̸ = d(y, w i) for i = 1, 2,. .. , k. A resolving set containing the minimum number of vertices is called a metric basis for G and the number of vertices in a metric basis is its metric dimension, denoted by dim(G). Let F be a family of connected graphs G n : F = (G n) n≥1 depending on n as follows: the order |V (G)| = ϕ(n) and lim n→∞ ϕ(n) = ∞. If there exists a constant C > 0 such that dim(G n) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension. The metric dimension of a class of circulant graphs C n (1, 2) has been determined by Javaid and Rahim (2008) [13]. In this paper, we extend this study to an infinite class of circulant graphs C n (1, 2, 3). We prove that the circulant graphs C n (1, 2, 3) have metric dimension equal to 4 for n ≡ 2, 3, 4, 5 (mod 6). For n ≡ 0 (mod 6) only 5 vertices appropriately chosen suffice to resolve all the vertices of C n (1, 2, 3), thus implying that dim(C n (1, 2, 3)) ≤ 5 except n ≡ 1 (mod 6) when dim(C n (1, 2, 3)) ≤ 6.

On the Partition Dimension of Cartesian Product Graphs

Applied Mathematics and Computation, 2010

Let G = (V, E) be a connected graph. The distance between two vertices u, v ∈ V , denoted by d (u, v), is the length of a shortest u − v path in G. The distance between a vertex v ∈ V and a subset P ⊂ V is defined as min{d(v, x) : x ∈ P }, and it is denoted by d(v, P ). An ordered partition {P 1 , P 2 , ..., P t } of vertices of a graph G, is a resolving partition of G, if all the distance vectors (d(v, P 1 ), d(v, P 2 ), ..., d(v, P t )) are different.

On the partition dimension of unicyclic graphs

2011

Given an ordered partition Π = {P 1 , P 2 , ..., P t } of the vertex set V of a connected graph G = (V, E), the partition representation of a vertex v ∈ V with respect to the partition Π is the vector r(v|Π) = (d(v, P 1 ), d(v, P 2 ), ..., d(v, P t )), where d(v, P i ) represents the distance between the vertex v and the set P i . A partition Π of V is a resolving partition if different vertices of G have different partition representations, i.e., for every pair of vertices u, v ∈ V , r(u|Π) = r(v|Π). The partition dimension of G is the minimum number of sets in any resolving partition for G. In this paper we obtain several tight bounds on the partition dimension of unicyclic graphs.

Edge metric dimension of some classes of circulant graphs

Analele Universitatii "Ovidius" Constanta - Seria Matematica

Let G = (V (G), E(G)) be a connected graph and x, y ∈ V (G), d(x, y) = min{ length of x − y path } and for e ∈ E(G), d(x, e) = min{d(x, a), d(x, b)}, where e = ab. A vertex x distinguishes two edges e 1 and e 2, if d(e 1, x) ≠ d(e 2, x). Let WE = {w 1, w 2, . . ., wk } be an ordered set in V (G) and let e ∈ E(G). The representation r(e | WE ) of e with respect to WE is the k-tuple (d(e, w 1), d(e, w 2), . . ., d(e, wk )). If distinct edges of G have distinct representation with respect to WE , then WE is called an edge metric generator for G. An edge metric generator of minimum cardinality is an edge metric basis for G, and its cardinality is called edge metric dimension of G, denoted by edim(G). The circulant graph Cn (1, m) has vertex set {v 1, v 2, . . ., vn } and edge set {vivi +1 : 1 ≤ i ≤ n−1}∪{vnv 1}∪{vivi+m : 1 ≤ i ≤ n−m}∪{vn−m+ivi : 1 ≤ i ≤ m}. In this paper, it is shown that the edge metric dimension of circulant graphs Cn (1, 2) and Cn (1, 3) is constant.

On the metric dimension of circulant and Harary graphs

Applied Mathematics and Computation, 2014

A metric generator is a set W of vertices of a graph GðV; EÞ such that for every pair of vertices u; v of G, there exists a vertex w 2 W with the condition that the length of a shortest path from u to w is different from the length of a shortest path from v to w. In this case the vertex w is said to resolve or distinguish the vertices u and v. The minimum cardinality of a metric generator for G is called the metric dimension. The metric dimension problem is to find a minimum metric generator in a graph G. In this paper, we make a significant advance on the metric dimension problem for circulant graphs Cðn; AEf1; 2; . . . ; jgÞ; 1 6 j 6 bn=2c; n P 3, and for Harary graphs.