The Partition Dimension of a Path Graph

Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph and is a subset of writen . Suppose there is , then the distance between and is denoted in the form . There is an ordered set of -partitions of, writen then the representation of with respect tois the The set of partitions ofis called a resolving partition if the representation of each to is different. The minimum cardinality of the solving-partition to is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitions of path graph, with , and are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely pd (Pn)=2Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph and is a subset of writen . Suppose there is , then the distance between and is denoted in the form . There is an ordered set of -partitions of, writen then the representation of with respect tois the The set of partitions ofis called a resolving partition if the representation of each to is different. The minimum cardinality of the solving-partition to is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitionsof path graph, with , and are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely.

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On the joint product graphs with respect to dominant metric dimension

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921500105 ◽

2020 ◽

pp. 2150010

Author(s):

Liliek Susilowati ◽

Imroatus Sa’adah ◽

Utami Dyah Purwati

Keyword(s):

Graph Theory ◽

Connected Graph ◽

Ordered Set ◽

Dominating Set ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality ◽

Joint Product ◽

Product Graphs

Some concepts in graph theory are resolving set, dominating set, and dominant metric dimension. A resolving set of a connected graph [Formula: see text] is the ordered set [Formula: see text] such that every pair of two vertices [Formula: see text] has the different representation with respect to [Formula: see text]. A Dominating set of [Formula: see text] is the subset [Formula: see text] such that for every vertex [Formula: see text] in [Formula: see text] is adjacent to at least one vertex in [Formula: see text]. A dominant resolving set of [Formula: see text] is an ordered set [Formula: see text] such that [Formula: see text] is a resolving set and a dominating set of [Formula: see text]. The minimum cardinality of a dominant resolving set is called a dominant metric dimension of [Formula: see text], denoted by [Formula: see text]. In this paper, we determine the dominant metric dimension of the joint product graphs.

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On fault-tolerant partition dimension of graphs

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-201390 ◽

2021 ◽

Vol 40 (1) ◽

pp. 1129-1135

Author(s):

Kamran Azhar ◽

Sohail Zafar ◽

Agha Kashif ◽

Zohaib Zahid

Keyword(s):

Connected Graph ◽

Intelligent Systems ◽

Fault Tolerant ◽

Robot Navigation ◽

Natural Extension ◽

Resolving Set ◽

Minimum Cardinality ◽

Sensor Networking ◽

Partition Dimension

Fault-tolerant resolving partition is natural extension of resolving partitions which have many applications in different areas of computer sciences for example sensor networking, intelligent systems, optimization and robot navigation. For a nontrivial connected graph G (V (G) , E (G)), the partition representation of vertex v with respect to an ordered partition Π = {Si : 1 ≤ i ≤ k} of V (G) is the k-vector r ( v | Π ) = ( d ( v , S i ) ) i = 1 k , where, d (v, Si) = min {d (v, x) |x ∈ Si}, for i ∈ {1, 2, …, k}. A partition Π is said to be fault-tolerant partition resolving set of G if r (u|Π) and r (v|Π) differ by at least two places for all u ≠ v ∈ V (G). A fault-tolerant partition resolving set of minimum cardinality is called the fault-tolerant partition basis of G and its cardinality the fault-tolerant partition dimension of G denoted by P ( G ) . In this article, we will compute fault-tolerant partition dimension of families of tadpole and necklace graphs.

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On the local metric dimension of t-fold wheel, Pn o Km, and generalized fan

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2018.2.2.4 ◽

2018 ◽

Vol 2 (2) ◽

pp. 88

Author(s):

Rokhana Ayu Solekhah ◽

Tri Atmojo Kusmayadi

Keyword(s):

Connected Graph ◽

Ordered Set ◽

Metric Dimension ◽

Minimum Cardinality ◽

Wheel Graph ◽

Metric Set ◽

Metric Basis ◽

Fan Graph

Let G be a connected graph and let u, v ∈ V(G). For an ordered set W = {w1, w2, ..., wn} of n distinct vertices in G, the representation of a vertex v of G with respect to W is the n-vector r(v∣W) = (d(v, w1), d(v, w2), ..., d(v, wn)), where d(v, wi) is the distance between v and wi for 1 ≤ i ≤ n. The set W is a local metric set of G if r(u ∣ W) ≠ r(v ∣ W) for every pair u, v of adjacent vertices of G. The local metric set of G with minimum cardinality is called a local metric basis for G and its cardinality is called a local metric dimension, denoted by lmd(G). In this paper we determine the local metric dimension of a t-fold wheel graph, Pn ⊙ Km graph, and generalized fan graph.

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Edge metric dimension of some classes of circulant graphs

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2020-0032 ◽

2020 ◽

Vol 28 (3) ◽

pp. 15-37

Author(s):

Muhammad Ahsan ◽

Zohaib Zahid ◽

Sohail Zafar

Keyword(s):

Connected Graph ◽

Ordered Set ◽

Metric Dimension ◽

Circulant Graph ◽

Circulant Graphs ◽

Minimum Cardinality ◽

Vertex Set ◽

Metric Basis

AbstractLet 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 e1 and e2, if d(e1, x) ≠ d(e2, x). Let WE = {w1, w2, . . ., 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, w1), d(e, w2), . . ., 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 {v1, v2, . . ., vn} and edge set {vivi+1 : 1 ≤ i ≤ n−1}∪{vnv1}∪{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.

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Further new results on strong resolving partitions for graphs

Open Mathematics ◽

10.1515/math-2020-0142 ◽

2020 ◽

Vol 18 (1) ◽

pp. 237-248 ◽

Cited By ~ 3

Author(s):

Dorota Kuziak ◽

Ismael G. Yero

Keyword(s):

Connected Graph ◽

Metric Dimension ◽

Minimum Cardinality ◽

Vertex Partition ◽

Strong Metric Dimension ◽

Partition Dimension

Abstract A set W of vertices of a connected graph G strongly resolves two different vertices x, y ∉ W if either d G (x, W) = d G (x, y) + d G (y, W) or d G (y, W) = d G (y, x) + d G (x, W), where d G (x, W) = min{d(x,w): w ∈ W} and d(x,w) represents the length of a shortest x − w path. An ordered vertex partition Π = {U 1, U 2,…,U k } of a graph G is a strong resolving partition for G, if every two different vertices of G belonging to the same set of the partition are strongly resolved by some other set of Π. The minimum cardinality of any strong resolving partition for G is the strong partition dimension of G. In this article, we obtain several bounds and closed formulae for the strong partition dimension of some families of graphs and give some realization results relating the strong partition dimension, the strong metric dimension and the order of graphs.

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Non-Isolated Resolving Number of Graph with Pendant Edges

Journal of Interconnection Networks ◽

10.1142/s0219265919500038 ◽

2019 ◽

Vol 19 (02) ◽

pp. 1950003

Author(s):

RIDHO ALFARISI ◽

DAFIK ◽

ARIKA INDAH KRISTIANA ◽

IKA HESTI AGUSTIN

Keyword(s):

Connected Graph ◽

Ordered Set ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality ◽

Isolated Vertex

We consider V, E are respectively vertex and edge sets of a simple, nontrivial and connected graph G. For an ordered set W = {w1, w2, w3, …, wk} of vertices and a vertex v ∈ G, the ordered r(v|W) = (d(v, w1), d(v, w2), …, d(v, wk)) of k-vector is representations of v with respect to W, where d(v, w) is the distance between the vertices v and w. The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W. The metric dimension, denoted by dim(G) is min of |W|. Furthermore, the resolving set W of graph G is called non-isolated resolving set if there is no ∀v ∈ W induced by non-isolated vertex. While a non-isolated resolving number, denoted by nr(G), is the minimum cardinality of non-isolated resolving set in graph. In this paper, we study the non isolated resolving number of graph with any pendant edges.

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THE LINEAR GEODETIC NUMBER OF A GRAPH

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830911001279 ◽

2011 ◽

Vol 03 (03) ◽

pp. 357-368 ◽

Cited By ~ 1

Author(s):

A. P. SANTHAKUMARAN ◽

T. JEBARAJ ◽

S. V. ULLAS CHANDRAN

Keyword(s):

Connected Graph ◽

Ordered Set ◽

Minimum Cardinality ◽

Connected Graphs ◽

Geodetic Number ◽

Geodetic Set ◽

Positive Integers

For a connected graph G of order n, an ordered set S = {u1, u2, …, uk} of vertices in G is a linear geodetic set of G if for each vertex x in G, there exists an index i, 1 ≤ i < k such that x lies on a ui - ui + 1 geodesic on G, and a linear geodetic set of minimum cardinality is the linear geodetic number gl(G). The linear geodetic numbers of certain standard graphs are obtained. It is shown that if G is a graph of order n and diameter d, then gl(G) ≤ n - d + 1 and this bound is sharp. For positive integers r, d and k ≥ 2 with r < d ≤ 2r, there exists a connected graph G with rad G = r, diam G = d and gl(G) = k. Also, for integers n, d and k with 2 ≤ d < n, 2 ≤ k ≤ n - d + 1, there exists a connected graph G of order n, diameter d and gl(G) = k. We characterize connected graphs G of order n with gl(G) = n and gl(G) = n - 1. It is shown that for each pair a, b of integers with 3 ≤ a ≤ b, there is a connected graph G with g(G) = a and gl(G) = b. We also discuss how the linear geodetic number of a graph is affected by adding a pendent edge to the graph.

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On the Strong Partition Dimension of Graphs

The Electronic Journal of Combinatorics ◽

10.37236/3474 ◽

2014 ◽

Vol 21 (3) ◽

Cited By ~ 4

Author(s):

Ismael González Yero

Keyword(s):

Connected Graph ◽

Minimum Cardinality ◽

Single Vertex ◽

Mathematical Properties ◽

Vertex Partition ◽

Vertex Set ◽

Partition Dimension

We present a new style of metric generator in graphs. Specifically we introduce a metric generator based on a partition of the vertex set of a graph. The sets of the partition will work as the elements which will uniquely determine the position of each single vertex of the graph. A set $W$ of vertices of a connected graph $G$ strongly resolves two different vertices $x,y\notin W$ if either $d_G(x,W)=d_G(x,y)+d_G(y,W)$ or $d_G(y,W)=d_G(y,x)+d_G(x,W)$, where $d_G(x,W)=\min\left\{d(x,w)\;:\;w\in W\right\}$. An ordered vertex partition $\Pi=\left\{U_1,U_2,...,U_k\right\}$ of a graph $G$ is a strong resolving partition for $G$ if every two different vertices of $G$ belonging to the same set of the partition are strongly resolved by some set of $\Pi$. A strong resolving partition of minimum cardinality is called a strong partition basis and its cardinality the strong partition dimension. In this article we introduce the concepts of strong resolving partition and strong partition dimension and we begin with the study of its mathematical properties.

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Characterization of n-Vertex Graphs of Metric Dimension n − 3 by Metric Matrix

Mathematics ◽

10.3390/math7050479 ◽

2019 ◽

Vol 7 (5) ◽

pp. 479 ◽

Cited By ~ 1

Author(s):

Juan Wang ◽

Lianying Miao ◽

Yunlong Liu

Keyword(s):

Connected Graph ◽

Ordered Set ◽

Distance Matrix ◽

Metric Dimension ◽

Extremal Graphs ◽

Resolving Set ◽

Minimum Cardinality

Let G = ( V ( G ) , E ( G ) ) be a connected graph. An ordered set W ⊂ V ( G ) is a resolving set for G if every vertex of G is uniquely determined by its vector of distances to the vertices in W. The metric dimension of G is the minimum cardinality of a resolving set. In this paper, we characterize the graphs of metric dimension n − 3 by constructing a special distance matrix, called metric matrix. The metric matrix makes it so a class of graph and its twin graph are bijective and the class of graph is obtained from its twin graph, so it provides a basis for the extension of graphs with respect to metric dimension. Further, the metric matrix gives a new idea of the characterization of extremal graphs based on metric dimension.

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On The Local Metric Dimension of Line Graph of Special Graph

CAUCHY ◽

10.18860/ca.v4i3.3694 ◽

2016 ◽

Vol 4 (3) ◽

pp. 125

Author(s):

Marsidi Marsidi ◽

Dafik Dafik ◽

Ika Hesti Agustin ◽

Ridho Alfarisi

Keyword(s):

Connected Graph ◽

Ordered Set ◽

Line Graph ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality

Let G be a simple, nontrivial, and connected graph. is a representation of an ordered set of k distinct vertices in a nontrivial connected graph G. The metric code of a vertex v, where , the ordered of k-vector is representations of v with respect to W, where is the distance between the vertices v and wi for 1≤ i ≤k. Furthermore, the set W is called a local resolving set of G if for every pair u,v of adjacent vertices of G. The local metric dimension ldim(G) is minimum cardinality of W. The local metric dimension exists for every nontrivial connected graph G. In this paper, we study the local metric dimension of line graph of special graphs , namely path, cycle, generalized star, and wheel. The line graph L(G) of a graph G has a vertex for each edge of G, and two vertices in L(G) are adjacent if and only if the corresponding edges in G have a vertex in common.

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