scholarly journals The Simultaneous Strong Resolving Graph and the Simultaneous Strong Metric Dimension of Graph Families

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 125
Author(s):  
Ismael González Yero
Keyword(s):  
Vertex Cover ◽  
Common Vertex ◽  
Metric Dimension ◽  
Minimum Cardinality ◽  
New Approach ◽  
Strong Metric Dimension ◽  
Vertex Set ◽  
Graph Families ◽  
Vertex Sets ◽  
Np Hardness

We consider in this work a new approach to study the simultaneous strong metric dimension of graphs families, while introducing the simultaneous version of the strong resolving graph. In concordance, we consider here connected graphs G whose vertex sets are represented as V ( G ) , and the following terminology. Two vertices u , v ∈ V ( G ) are strongly resolved by a vertex w ∈ V ( G ) , if there is a shortest w − v path containing u or a shortest w − u containing v. A set A of vertices of the graph G is said to be a strong metric generator for G if every two vertices of G are strongly resolved by some vertex of A. The smallest possible cardinality of any strong metric generator (SSMG) for the graph G is taken as the strong metric dimension of the graph G. Given a family F of graphs defined over a common vertex set V, a set S ⊂ V is an SSMG for F , if such set S is a strong metric generator for every graph G ∈ F . The simultaneous strong metric dimension of F is the minimum cardinality of any strong metric generator for F , and is denoted by Sd s ( F ) . The notion of simultaneous strong resolving graph of a graph family F is introduced in this work, and its usefulness in the study of Sd s ( F ) is described. That is, it is proved that computing Sd s ( F ) is equivalent to computing the vertex cover number of the simultaneous strong resolving graph of F . Several consequences (computational and combinatorial) of such relationship are then deduced. Among them, we remark for instance that we have proved the NP-hardness of computing the simultaneous strong metric dimension of families of paths, which is an improvement (with respect to the increasing difficulty of the problem) on the results known from the literature.

scholarly journals Computing Minimal Doubly Resolving Sets and the Strong Metric Dimension of the Layer Sun Graph and the Line Graph of the Layer Sun Graph

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Jia-Bao Liu ◽  
Ali Zafari
Keyword(s):  
Connected Graph ◽  
Line Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Strong Metric Dimension ◽  
Vertex Set

Let G be a finite, connected graph of order of, at least, 2 with vertex set VG and edge set EG. A set S of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of S. The minimal doubly resolving set of vertices of graph G is a doubly resolving set with minimum cardinality and is denoted by ψG. In this paper, first, we construct a class of graphs of order 2n+Σr=1k−2nmr, denoted by LSGn,m,k, and call these graphs as the layer Sun graphs with parameters n, m, and k. Moreover, we compute minimal doubly resolving sets and the strong metric dimension of the layer Sun graph LSGn,m,k and the line graph of the layer Sun graph LSGn,m,k.

scholarly journals Fault-Tolerant Metric Dimension of Circulant Graphs

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 124
Author(s):  
Laxman Saha ◽  
Rupen Lama ◽  
Kalishankar Tiwary ◽  
Kinkar Chandra Das ◽  
Yilun Shang
Keyword(s):  
Connected Graph ◽  
Fault Tolerant ◽  
Metric Dimension ◽  
Resolving Set ◽  
Circulant Graphs ◽  
Minimum Cardinality ◽  
Vertex Set

Let G be a connected graph with vertex set V(G) and d(u,v) be the distance between the vertices u and v. A set of vertices S={s1,s2,…,sk}⊂V(G) is called a resolving set for G if, for any two distinct vertices u,v∈V(G), there is a vertex si∈S such that d(u,si)≠d(v,si). A resolving set S for G is fault-tolerant if S\{x} is also a resolving set, for each x in S, and the fault-tolerant metric dimension of G, denoted by β′(G), is the minimum cardinality of such a set. The paper of Basak et al. on fault-tolerant metric dimension of circulant graphs Cn(1,2,3) has determined the exact value of β′(Cn(1,2,3)). In this article, we extend the results of Basak et al. to the graph Cn(1,2,3,4) and obtain the exact value of β′(Cn(1,2,3,4)) for all n≥22.

scholarly journals Edge metric dimension of some classes of circulant graphs

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.

scholarly journals Metric Dimension, Minimal Doubly Resolving Sets, and the Strong Metric Dimension for Jellyfish Graph and Cocktail Party Graph

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Jia-Bao Liu ◽  
Ali Zafari ◽  
Hassan Zarei
Keyword(s):  
Ordered Set ◽  
Undirected Graph ◽  
Metric Dimension ◽  
Np Hard ◽  
Cocktail Party ◽  
Resolving Set ◽  
Strong Metric Dimension ◽  
Vertex Set

Let Γ be a simple connected undirected graph with vertex set VΓ and edge set EΓ. The metric dimension of a graph Γ is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. For an ordered subset W=w1,w2,…,wk of vertices in a graph Γ and a vertex v of Γ, the metric representation of v with respect to W is the k-vector rvW=dv,w1,dv,w2,…,dv,wk. If every pair of distinct vertices of Γ have different metric representations, then the ordered set W is called a resolving set of Γ. It is known that the problem of computing this invariant is NP-hard. In this paper, we consider the problem of determining the cardinality ψΓ of minimal doubly resolving sets of Γ and the strong metric dimension for the jellyfish graph JFGn,m and the cocktail party graph CPk+1.

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

2018 ◽  
Vol 10 (05) ◽  
pp. 1850066 ◽  
Author(s):  
Cong X. Kang ◽  
Eunjeong Yi
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Domination Number ◽  
Unicyclic Graph ◽  
Metric Dimension ◽  
Minimum Cardinality ◽  
Dimension Formula ◽  
Vertex Set ◽  
Number Formula

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.

scholarly journals Further new results on strong resolving partitions for graphs

2020 ◽  
Vol 18 (1) ◽  
pp. 237-248 ◽  
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.

On enhanced power graphs of certain groups

2020 ◽  
pp. 2050099
Author(s):  
Sandeep Dalal ◽  
Jitender Kumar
Keyword(s):  
Cyclic Subgroup ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Independence Number ◽  
Metric Dimension ◽  
Matching Number ◽  
Graph Invariants ◽  
Power Graph ◽  
Strong Metric Dimension ◽  
Vertex Set

The enhanced power graph [Formula: see text] of a group [Formula: see text] is a simple undirected graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if both [Formula: see text] and [Formula: see text] belongs to same cyclic subgroup of [Formula: see text]. In this paper, we obtain various graph invariants viz. independence number, minimum degree and matching number of [Formula: see text], where [Formula: see text] is the dicyclic group or a class of groups of order [Formula: see text]. If [Formula: see text] is any of these groups, we prove that [Formula: see text] is perfect and then obtain its strong metric dimension.

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

2020 ◽  
Vol 12 (01) ◽  
pp. 2050010
Author(s):  
Eunjeong Yi
Keyword(s):  
Open Problem ◽  
Connected Graph ◽  
Domination Number ◽  
Metric Dimension ◽  
Unicyclic Graphs ◽  
Dimension Formula ◽  
Strong Metric Dimension ◽  
Vertex Set ◽  
Number Formula

Let [Formula: see text] be a connected graph of order at least two with vertex set [Formula: see text]. For [Formula: see text], let [Formula: see text] denote the length of an [Formula: see text] geodesic in [Formula: see text]. A function [Formula: see text] is called a dominating broadcast function of [Formula: see text] if, for each vertex [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], and the broadcast domination number, [Formula: see text], of [Formula: see text] is the minimum of [Formula: see text] over all dominating broadcast functions [Formula: see text] of [Formula: see text]. For [Formula: see text], let [Formula: see text] denote the set of vertices [Formula: see text] such that either [Formula: see text] lies on a [Formula: see text] geodesic or [Formula: see text] lies on a [Formula: see text] geodesic of [Formula: see text]. Let [Formula: see text] be a function and, for any [Formula: see text], let [Formula: see text]. We say that [Formula: see text] is a strong resolving function of [Formula: see text] if [Formula: see text] for every pair of distinct vertices [Formula: see text], and the strong metric dimension, [Formula: see text], of [Formula: see text] is the minimum of [Formula: see text] over all strong resolving functions [Formula: see text] of [Formula: see text]. For any connected graph [Formula: see text], we show that [Formula: see text]; we characterize [Formula: see text] satisfying [Formula: see text] equals two and three, respectively, and characterize unicyclic graphs achieving [Formula: see text]. For any tree [Formula: see text] of order at least three, we show that [Formula: see text], and characterize trees achieving equality. Moreover, for a tree [Formula: see text] of order [Formula: see text], we obtain the results that [Formula: see text] if [Formula: see text] is central, and that [Formula: see text] if [Formula: see text] is bicentral; in each case, we characterize trees achieving equality. We conclude this paper with some remarks and an open problem.

GUARDING RECTANGULAR PARTITIONS

2009 ◽  
Vol 19 (06) ◽  
pp. 579-594 ◽  
Author(s):  
YEFIM DINITZ ◽  
MATTHEW J. KATZ ◽  
ROI KRAKOVSKI
Keyword(s):  
Planar Graphs ◽  
Vertex Cover ◽  
Plane Graph ◽  
Common Point ◽  
Distinct Vertex ◽  
Np Hard ◽  
Worst Case ◽  
Minimum Cardinality ◽  
Graph Theoretic ◽  
Np Hardness

A rectangular partition is a partition of a rectangle into non-overlapping rectangles, such that no four rectangles meet at a common point. A vertex guard is a guard located at a vertex of the partition (i.e., at a corner of a rectangle); it guards the rectangles that meet at this vertex. An edge guard is a guard that patrols along an edge of the partition, and is thus equivalent to two adjacent vertex guards. We consider the problem of finding a minimum-cardinality guarding set for the rectangles of the partition. For vertex guards, we prove that guarding a given subset of the rectangles is NP-hard. For edge guards, we prove that guarding all rectangles, where guards are restricted to a given subset of the edges, is NP-hard. For both results we show a reduction from vertex cover in non-bipartite 3-connected cubic planar graphs of girth greater than three. For the second NP-hardness result, we obtain a graph-theoretic result which establishes a connection between the set of faces of a plane graph of vertex degree at most three and a vertex cover for this graph. More precisely, we prove that one can assign to each internal face a distinct vertex of the cover, which lies on the face's boundary. We show that the vertices of a rectangular partition can be colored red, green, or black, such that each rectangle has all three colors on its boundary. We conjecture that the above is also true for four colors. Finally, we obtain a worst-case upper bound on the number of edge guards that are sufficient for guarding rectangular partitions with some restrictions on their structure.

scholarly journals FAULT-TOLERANT METRIC DIMENSION OF CIRCULANT GRAPHS

2019 ◽  
pp. 781
Author(s):  
Narjes Seyedi ◽  
Hamid Reza Maimani
Keyword(s):  
Fault Tolerant ◽  
Additive Group ◽  
Metric Dimension ◽  
Circulant Graph ◽  
Resolving Set ◽  
Circulant Graphs ◽  
Minimum Cardinality ◽  
Vertex Set

A set $W$ of vertices in a graph $G$ is called a resolving setfor $G$ if for every pair of distinct vertices $u$ and $v$ of $G$ there exists a vertex $w \in W$ such that the distance between $u$ and $w$ is different from the distance between $v$ and $w$. The cardinality of a minimum resolving set is called the metric dimension of $G$, denoted by $\beta(G)$. A resolving set $W'$ for $G$ is fault-tolerant if $W'\setminus \left\lbrace w\right\rbrace $ for each $w$ in $W'$, is also a resolving set and the fault-tolerant metric dimension of $G$ is the minimum cardinality of such a set, denoted by $\beta'(G)$. The circulant graph is a graph with vertex set $\mathbb{Z}_{n}$, an additive group of integers modulo $n$, and two vertices labeled $i$ and $j$ adjacent if and only if $i -j \left( mod \ n \right)  \in C$, where $C \in \mathbb{Z}_{n}$ has the property that $C=-C$ and $0 \notin C$. The circulant graph is denoted by $X_{n,\bigtriangleup}$ where $\bigtriangleup = \vert C\vert$. In this paper, we study the fault-tolerant metric dimension of a family of circulant graphs $X_{n,3}$ with connection set $C=\lbrace 1,\dfrac{n}{2},n-1\rbrace$ and circulant graphs $X_{n,4}$ with connection set $C=\lbrace \pm 1,\pm 2\rbrace$.

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