Metric Dimension Of Graph Research Articles

Bounds on the sum of domination number and metric dimension of graphs

Let [Formula: see text] be a graph with vertex set [Formula: see text]. The domination number, [Formula: see text], of [Formula: see text] is the minimum cardinality of a set [Formula: see text] such that every vertex not in [Formula: see text] is adjacent to a vertex in [Formula: see text]. The metric dimension, [Formula: see text], of [Formula: see text] is the minimum cardinality of a set of vertices such that every vertex of [Formula: see text] is uniquely determined by its vector of distances to the chosen vertices. For a tree [Formula: see text] of order at least two, we show that [Formula: see text], where [Formula: see text] denotes the number of exterior major vertices of [Formula: see text]; further, we characterize trees [Formula: see text] achieving equality. For a connected graph [Formula: see text] of order [Formula: see text], Bagheri Gh. et al. proved that [Formula: see text] and equality holds if and only if [Formula: see text] for [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the complete graph and [Formula: see text] denotes a complete bi-partite graph of order [Formula: see text]. We characterize graphs [Formula: see text] for which [Formula: see text] equals two and three, respectively. We also characterize graphs [Formula: see text] satisfying [Formula: see text] when [Formula: see text] is a tree, a unicyclic graph, or a complete multi-partite graph.

Uniquely identifying the edges of a graph: The edge metric dimension

Let G=(V,E) be a connected graph, let v∈V be a vertex and let e=uw∈E be an edge. The distance between the vertex v and the edge e is given by dG(e,v)=min{dG(u,v),dG(w,v)}. A vertex w∈V distinguishes two edges e1,e2∈E if dG(w,e1)≠dG(w,e2). A set S of vertices in a connected graph G is an edge metric generator for G if every two edges of G are distinguished by some vertex of S. The smallest cardinality of an edge metric generator for G is called the edge metric dimension and is denoted by edim(G). In this paper, we introduce the concept of edge metric dimension and initiate the study of its mathematical properties. We make a comparison between the edge metric dimension and the standard metric dimension of graphs while presenting some realization results concerning the edge metric dimension and the standard metric dimension of graphs. We prove that computing the edge metric dimension of connected graphs is NP-hard and give some approximation results. Moreover, we present some bounds and closed formulae for the edge metric dimension of several classes of graphs.

On the strong metric dimension of antiprism graph, king graph, and graph

Let G be a connected graph with a set of vertices V(G) and a set of edges E(G). The interval I[u, v] between u and v to be the collection of all vertices that belong to some shortest u-v path. A vertex s ∊ V(G) is said to be strongly resolved for vertices u, v ∊ V(G) if v ∊ I[u, s] or u ∊ I[v, s]. A vertex set S ⊆ V(G) is a strong resolving set for G if every two distinct vertices of G are strongly resolved by some vertices of S. The strong metric dimension of G, denoted by sdim(G), is defined as the smallest cardinality of a strong resolving set. In this paper, we determine the strong metric dimension of an antiprism An graph, a king Km,n graph, and a graph. We obtain the strong metric dimension of an antiprim graph An are n for n odd and n + 1 for n even. The strong metric dimension of King graph Km,n is m + n − 1. The strong metric dimension of graph are n for m = 1, n > 1 and mn − 1 for m > 2, n > 1.

On the strong metric dimension of generalized butterfly graph, starbarbell graph, and graph

Let G be a connected graph with vertex set V(G) and edge set E(G). For every pair of vertices , the interval I[u, v] between u and v to be the collection of all vertices that belong to some shortest u − v path. A vertex strongly resolves two vertices u and v if u belongs to a shortest v − s path or v belongs to a shortest u − s path. A vertex set S of G is a strong resolving set of G if every two distinct vertices of G are strongly resolved by some vertex of S. The strong metric basis of G is a strong resolving set with minimal cardinality. The strong metric dimension sdim(G) of a graph G is defined as the cardinality of strong metric basis. In this paper we determine the strong metric dimension of a generalized butterfly graph, starbarbell graph, and graph. We obtain the strong metric dimension of generalized butterfly graph is sdim(BFn) = 2n − 2. The strong metric dimension of starbarbell graph is . The strong metric dimension of graph are for m > 3 and n = 2, and for m > 3 and n > 2.

On the (adjacency) metric dimension of corona and strong product graphs and their local variants: Combinatorial and computational results

The metric dimension is quite a well-studied graph parameter. Recently, the adjacency metric dimension and the local metric dimension have been introduced. We combine these variants and introduce the local adjacency metric dimension. We show that the (local) metric dimension of the corona product of a graph of order n and some non-trivial graph H equals n times the (local) adjacency metric dimension of H. This strong relation also enables us to infer computational hardness results for computing the (local) metric dimension, based on according hardness results for (local) adjacency metric dimension that we also provide. We also study combinatorial properties of the strong product of graphs and emphasize the role of different types of twins play in determining in particular the adjacency metric dimension of a graph.

Strong resolving graphs: The realization and the characterization problems

The strong resolving graph GSR of a connected graph G was introduced in Oellermann and Peters-Fransen (2007) as a tool to study the strong metric dimension of G. Basically, it was shown that the problem of finding the strong metric dimension of G can be transformed to the problem of finding the vertex cover number of GSR. Since then, several articles on the strong metric dimension of graphs which are using this tool have been published. However, the tool itself has remained unnoticed as a properly structure. In this paper, we survey the state of knowledge on the strong resolving graphs, and also derive some new results regarding its properties.

Bounds on the Metric Dimension of Graphs

Bounds on the Metric Dimension of Graphs

Mixed metric dimension of graphs

Let G=(V,E) be a connected graph. A vertex w ∈ V distinguishes two elements (vertices or edges) x, y ∈ E ∪ V if dG(w, x) ≠ dG(w, y). A set S of vertices in a connected graph G is a mixed metric generator for G if every two distinct elements (vertices or edges) of G are distinguished by some vertex of S. The smallest cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by dimm(G). In this paper we consider the structure of mixed metric generators and characterize graphs for which the mixed metric dimension equals the trivial lower and upper bounds. We also give results about the mixed metric dimension of some families of graphs and present an upper bound with respect to the girth of a graph. Finally, we prove that the problem of determining the mixed metric dimension of a graph is NP-hard in the general case.

On (local) metric dimension of graphs with m-pendant points

ABSTRACTAn ordered set of vertices S is called as a (local) resolving set of a connected graph G = (VG, EG) if for any two adjacent vertices s ≠ t ∈ VG have distinct representation with respect to S, that is r(s | S) ≠ r(t | S). A representation of a vertex in G is a vector of distances to vertices in S. The minimum (local) resolving set for G is called as a (local) basis of G. A (local) metric dimension for G denoted by dim(G), is the cardinality of vertices in a basis for G, and its local variant by diml(G).Given two graphs, G with vertices s1, s2, …, sp and edges e1, e2, …, eq, and H. An edge-corona of G and H, G⋄H is defined as a graph obtained by taking a copy of G and q copies of H and for each edge ej = sish of G joining edges between the two end-vertices si, sh of ej and each vertex of j-copy of H.In this paper, we determine and compare between the metric dimension of graphs with m-pendant points, G⋄mK1, and its local variant for any connected graph G. We give an upper bound of the dimensions.

On optimal approximability results for computing the strong metric dimension

The strong metric dimension of a graph was first introduced by Sebö and Tannier (2004) as an alternative to the (weak) metric dimension of graphs previously introduced independently by Slater (1975) and by Harary and Melter (1976), and has since been investigated in several research papers. However, the exact worst-case computational complexity of computing the strong metric dimension has remained open beyond being NP-complete. In this communication, we show that the problem of computing the strong metric dimension of a graph of n nodes admits a polynomial-time 2-approximation, admits a O∗(20.287n)-time exact computation algorithm, admits a O(1.2738k+nk)-time exact computation algorithm if the strong metric dimension is at most k, does not admit a polynomial time (2−ε)-approximation algorithm assuming the unique games conjecture is true, does not admit a polynomial time (105−21−ε)-approximation algorithm assuming P≠NP, does not admit a O∗(2o(n))-time exact computation algorithm assuming the exponential time hypothesis is true, and does not admit a O∗(no(k))-time exact computation algorithm if the strong metric dimension is at most k assuming the exponential time hypothesis is true.

Computing the metric dimension of a graph from primary subgraphs

Let $G$ be a connected graph. Given an ordered set $W = \{w_1, w_2,\dots w_k\}\subseteq V(G)$ and a vertex $u\in V(G)$, the representation of $u$ with respect to $W$ is the ordered $k$-tuple $(d(u,w_1), d(u,w_2),\dots,$ $d(u,w_k))$, where $d(u,w_i)$ denotes the distance between $u$ and $w_i$. The set $W$ is a metric generator for $G$ if every two different vertices of $G$ have distinct representations. A minimum cardinality metric generator is called a \emph{metric basis} of $G$ and its cardinality is called the \emph{metric dimension} of G. It is well known that the problem of finding the metric dimension of a graph is NP-Hard. In this paper we obtain closed formulae for the metric dimension of graphs with cut vertices. The main results are applied to specific constructions including rooted product graphs, corona product graphs, block graphs and chains of graphs.

Vertices, edges, distances and metric dimension in graphs

Given a connected graph G=(V,E), a set of vertices S⊂V is an edge metric generator for G, if any two edges of G are identified by S by mean of distances to the vertices in S. Moreover, in a natural way, S is a mixed metric generator, if any two elements of G (vertices or edges) are identified by S by mean of distances. In this work we study the (edge and mixed) metric dimension of graphs.

Complexity of metric dimension on planar graphs

The metric dimension of a graph G is the size of a smallest subset L⊆V(G) such that for any x,y∈V(G) with x≠y there is a z∈L such that the graph distance between x and z differs from the graph distance between y and z. Even though this notion has been part of the literature for almost 40 years, prior to our work the computational complexity of determining the metric dimension of a graph was still very unclear. In this paper, we show tight complexity boundaries for the Metric Dimension problem. We achieve this by giving two complementary results. First, we show that the Metric Dimension problem on planar graphs of maximum degree 6 is NP-complete. Then, we give a polynomial-time algorithm for determining the metric dimension of outerplanar graphs.

On the fractional strong metric dimension of graphs

For any two vertices x and y of a graph G, let S{x,y} denote the set of vertices z such that either x lies on a y−z geodesic or y lies on an x−z geodesic. For a function g defined on V(G) and U⊆V(G), let g(U)=∑x∈Ug(x). A function g:V(G)→[0,1] is a strong resolving function of G if g(S{x,y})≥1, for every pair of distinct vertices x,y of G. The fractional strong metric dimension, sdimf(G), of a graph G is min{g(V(G)):g is a strong resolving function of G}. This paper furthers the study of fractional strong metric dimension initiated in COCOA 2013 (Lecture Notes in Comput. Sci.). First, we clarify or correct the proofs to two characterization theorems contained in two papers on fractional (strong) metric dimension. Next, results on fractional strong metric dimension analogous to the work of Feng, Lv, and Wang on fractional metric dimension are offered. We provide new upper and lower bounds on sdimf(G), partly in analogy with the work done by Feng et al. and partly by exploiting the particular nature of the strong metric dimension. Finally, motivated by the work of Arumugam, Mathew, and Shen, we describe a class of graphs G for which sdimf(G)=|V(G)|2.

The Local Metric Dimension of Strong Product Graphs

A vertex $$v\in V(G)$$v?V(G) is said to distinguish two vertices $$x,y\in V(G)$$x,y?V(G) of a nontrivial connected graph G if the distance from v to x is different from the distance from v to y. A set $$S\subset V(G)$$S?V(G) is a local metric generator for G if every two adjacent vertices of G are distinguished by some vertex of S. A local metric generator with the minimum cardinality is called a local metric basis for G and its cardinality, the local metric dimension of G. It is known that the problem of computing the local metric dimension of a graph is NP-Complete. In this paper we study the problem of finding exact values or bounds for the local metric dimension of strong product of graphs.

Identifying Codes in Vertex-Transitive Graphs and Strongly Regular Graphs

We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and $2\ln(|V|)+1$ where $V$ is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order $|V|^{\alpha}$ with $\alpha \in \{\frac{1}{4},\frac{1}{3},\frac{2}{5}\}$. These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs.

Metric Dimension and Uncertainty of Traversing Robots in a Network

Metric dimension in graph theory has many applications in the real world. It has been applied to the optimization problems in complex networks, analyzing electrical networks; show the business relations, robotics, control of production processes etc. This paper studies the metric dimension of graphs with respect to contraction and its bijection between them. Also an algorithm to avoid the overlapping between the robots in a network is introduced.

An Application of Gd-Metric Spaces and Metric Dimension of Graphs

An Application of Gd-Metric Spaces and Metric Dimension of Graphs

The metric dimension of strong product graphs

For an ordered subset S = {s1, s2, . . . sk} of vertices in a connected graph G, the metric representation of a vertex u with respect to the set S is the k-vector r(u|S) = (dG(v, s1), dG(v, s2), . . . , dG(v, sk)), where dG(x, y) represents the distance between the vertices x and y. The set S is a metric generator for G if every two different vertices of G have distinct metric representations with respect to S. A minimum metric generator is called a metric basis for G and its cardinality, dim(G), the metric dimension of G. It is well known that the problem of finding the metric dimension of a graph is NP-Hard. In this paper we obtain closed formulae and tight bounds for the metric dimension of strong product graphs.

The effect of vertex or edge deletion on the metric dimension of graphs

The effect of vertex or edge deletion on the metric dimension of graphs