Graphs of Neighborhood Metric Dimension Two

Statistics, Optimization & Information Computing, 2020

All graphs in this paper are nontrivial and connected simple graphs. For a set W = {s1,s2,...,sk} of verticesof G, the multiset representation of a vertex v of G with respect to W is r(v|W) = {d(v,s1),d(v,s2),...,d(v,sk)} whered(v,si) is the distance between of v and si. If the representation r(v|W)̸= r(u|W) for every pair of vertices u,v of a graph G, the W is called the resolving set of G, and the cardinality of a minimum resolving set is called the multiset dimension, denoted by md(G). A set W is a local resolving set of G if r(v|W) ̸= r(u|W) for every pair of adjacent vertices u,v of a graph G. The cardinality of a minimum local resolving set W is called local multiset dimension, denoted by µl(G). In our paper, we discuss the relationship between the multiset dimension and local multiset dimension of graphs and establish bounds of local multiset dimension for some families of graph.

2015

As a generalization of the concept of a metric basis, this article introduces the notion of $k$-metric basis in graphs. Given a connected graph $G=(V,E)$, a set $S\subseteq V$ is said to be a $k$-metric generator for $G$ if the elements of any pair of different vertices of $G$ are distinguished by at least $k$ elements of $S$, {\em i.e.}, for any two different vertices $u,v\in V$, there exist at least $k$ vertices $w_1,w_2,\ldots,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$ for every $i\in \{1,\ldots,k\}$. A metric generator of minimum cardinality is called a $k$-metric basis and its cardinality the $k$-metric dimension of $G$. A connected graph $G$ is \emph{$k$-metric dimensional} if $k$ is the largest integer such that there exists a $k$-metric basis for $G$. We give a necessary and sufficient condition for a graph to be $k$-metric dimensional and we obtain several results on the $k$-metric dimension.

2020

The metric representation of a vertex v with respect to an ordered subset W = {w1, w2, · · · , wn} ⊆ V (G) is an ordered k−tuple defined by r(v|W ) = (d(v, w1), d(v, w2), . . . , d(v, wn)), where d(u, v) denotes the distance between the vertices u and v. A subset W ⊆ V (G) is a resolving set if all vertices of G have distinct representations with respect to W . A resolving set of the largest order whose no proper subset resolves all vertices of G is called the upper basis of G and the cardinality of the upper basis is called the upper dimension of G. A vertex v having at least one pendent edge incident on it is called a star vertex and the number of pendent edges incident on a vertex v is called the star degree of v. We determine the upper dimension of certain families of graphs and characterize the cases in which upper dimension equals the metric dimension. For instance, it is shown that metric dimension equals upper dimension for the graphs defined by the Cartesian product of Kn a...

Discrete Applied Mathematics, 2015

A vertex v ∈ V is said to resolve two vertices x and y if dG(v, x) = dG (v, y). A set S ⊂ V is said to be a metric generator for G if any pair of vertices of G is resolved by some element of S. A minimum metric generator is called a metric basis, and its cardinality, dim(G), the metric dimension of G. A set S ⊆ V is said to be a simultaneous metric generator for a graph family G = {G1, G2, . . . , G k }, defined on a common (labeled) vertex set, if it is a metric generator for every graph of the family. A minimum cardinality simultaneous metric generator is called a simultaneous metric basis, and its cardinality the simultaneous metric dimension of G. We obtain sharp bounds for this invariants for general families of graphs and calculate closed formulae or tight bounds for the simultaneous metric dimension of several specific graph families. For a given graph G we describe a process for obtaining a lower bound on the maximum number of graphs in a family containing G that has simultaneous metric dimension equal to dim(G). It is shown that the problem of finding the simultaneous metric dimension of families of trees is N P -hard. Sharp upper bounds for the simultaneous metric dimension of trees are established. The problem of finding this invariant for families of trees that can be obtained from an initial tree by a sequence of successive edge-exchanges is considered. For such families of trees sharp upper and lower bounds for the simultaneous metric dimension are established.

WSEAS TRANSACTIONS ON MATHEMATICS

Let be a connected graph. is said to be unicyclic if it contains exactly one cycle, and bicyclic if the number of edges equals the number of vertices plus one. For a-ordered set = { 1 , 2 , … , } ⊂ V(G), the multiset representation of a vertex in with respect to is given as (|) = { (, 1), (, 2), … , (,)}, where (,) is the distance between and the ordered subset of together with their multiplicities. The set is called a local-resolving set of if for every ∈ (), (|) ≠ (|). The local-resolving set with minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of , denoted by (). If has no localresolving set, we write () = ∞ and say that has an infinite local multiset dimension. In this paper, we determine the local multiset dimension of the unicyclic and bicyclic graphs.

Heliyon, 2020

The be a connected graph with vertex set () and edge set (). A subset ⊆ () is called a dominating set of if for every vertex in () ⧵ , there exists at least one vertex in such that is adjacent to. An ordered set ⊆ () is called a resolving set of , if every pair of vertices and in () have distinct representation with respect to. An ordered set ⊆ () is called a dominant resolving set of , if is a resolving set and also a dominating set of. The minimum cardinality of dominant resolving set is called a dominant metric dimension of , denoted by (). In this paper, we investigate the dominant metric dimension of some particular class of graphs, the characterisation of graph with certain dominant metric dimension, and the dominant metric dimension of joint and comb products of graphs.

IAEME, 2019

Let G be a connected graph with vertex set V(G) and edge set E(G). The distance between vertices u and v in G is denoted by d(u, v), which serves as the shortest path length from u to v. Let 𝑊 = {𝑤1, 𝑤2, … , 𝑤𝑘 } ⊆ 𝑉(𝐺) be an ordered set, and v is a vertex in G. The representation of v with respect to W is an ordered set 𝑘 − 𝑡𝑢𝑝𝑙𝑒, 𝑟(𝑣|𝑊) = (𝑑(𝑣, 𝑤1), 𝑑(𝑣, 𝑤2), … , 𝑑(𝑣, 𝑤𝑘 )). The set W is called a resolving set for G if each vertex in G has a different representation with respect to W. A resolving set containing minimum cardinality is called a basis for G. The number of vertices in a basis of G is called metric dimension of G, which is denoted by𝑑𝑖𝑚(𝐺). The 𝑆 ⊆ 𝑉(𝐺) is a complement resolving set of G if there are two vertices𝑢, 𝑣 ∈ 𝑉(𝐺) ∖ 𝑆, such that𝑟(𝑢|𝑆) = 𝑟(𝑣|𝑆). A complement basis of G is the complement resolving set containing maximum cardinality. The number of vertices in a complement basis of G is called complement metric dimension of G, which is denoted by 𝑑̅̅𝑖̅𝑚̅̅(𝐺). In this paper, we examined complement metric dimension of particular graphs and their characteristics. Furthermore, we determined complement metric dimension of corona and comb products graphs

International journal of computer applications, 2016

The near common-neighborhood graph of a graph G, denoted by ncn(G), is the graph on the same vertices of G, two vertices being adjacent in ncn(G) if there is at least one vertex in G not adjacent to both of them. A graph is called near-common neighborhood graph if it is the near-common neighborhood of some graph. In this paper we introduce the near-common neighborhood of a graph, the near common neighborhood graph, near-completeness number of a graph, basic properties of these new graphs are obtained and interesting results are established.

Siam Journal on Discrete Mathematics, 2007

A set of vertices S resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric dimension of cartesian products G H. We prove that the metric dimension of G G is tied in a strong sense to the minimum order of a so-called doubly resolving set in G. Using bounds on the order of doubly resolving sets, we establish bounds on G H for many examples of G and H. One of our main results is a family of graphs G with bounded metric dimension for which the metric dimension of G G is unbounded.

Electronic Notes in Discrete Mathematics, 2014

Let G be a simple graph with vertex set {v 1 , v 2 ,. .. , v n }. The common neighborhood graph (congraph) of G, denoted by con(G), is a graph with vertex set {v 1 , v 2 ,. .. , v n }, in which two vertices are adjacent if and only if they have at least one common neighbor in the graph G. In this paper we compute the common neighborhood of some composite graphs. In continue we investigate the relation between hamiltonicity of graph G and con(G). Also we obtain a lower bound for the clique number of con(G) in terms of clique number of graph G. Finally we state that the total chromatic number of G is bounded by chromatic number of con(T (G)).