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An e partition X of the vertex set V (H) of a connected graph H is the collection of e number of ordered disjoint subsets of V (H), denoted as X = {X1,X2,: ::, Xe}. The representation of a vertex u is a distance vector r(ujX) = (d(u,X1), d(u,X2),: ::, d(u,Xe)), where d(u,Xi) is the distance of u from Xi. Any ordered e partition X is referred as resolving partition if representations of all the vertices are distinct. The smallest integer e is referred as the partition dimension of the graph. The advancement in the concept of partition dimension is fault-tolerant partition dimension where the representations are distinct at two places for each pair of vertices. In this paper, we compute the partition dimension and fault-tolerant partition dimension of some planar graphs. [ABSTRACT FROM AUTHOR]

The partition dimension of a graph is the minimum number of vertex partitions such that every vertex has different distances to the ordered partitions. Many resolving partitions for trees have all vertices not in an end-path in the same partition. This reduces the problem of the partition dimension of trees into finding the partition dimension of palms, the end-paths from a branch. In this paper, we construct a resolving partition for trees using resolving partitions of their palms. We also study some bounds for the partition dimension of palms and also find the partition dimension of regular palm and olive trees. [ABSTRACT FROM AUTHOR]

Partition dimension problems involve dividing a graph's vertex set into a minimum number of disjoint sets so that each vertex is different with respect to the representation from each disjoint set. As a result of the development of this method, a number of applications have arisen in a number of fields such as drug design, navigation of robots, pattern recognition, and image processing. In this paper, we have calculated the partition dimension of oxide and zigzag benzenoid networks, and the subdivision of benzenoid hydrocarbon and triangular benzene networks. [ABSTRACT FROM AUTHOR]

Let $ G = (V, E) $ be a simple, connected graph with vertex set $ V(G) $ and $ E(G) $ edge set of $ G $. For two vertices $ a $ and $ b $ in a graph $ G $, the distance $ d(a, b) $ from $ a $ to $ b $ is the length of shortest path $ a-b $ path in $ G $. A $ k $-ordered partition of vertices of $ G $ is represented as $ {R}{p} = \{{R}{p_1}, {R}{p_2}, \dots, {R}{p_k}\} $ and the representation $ r(a|{R}{p}) $ of a vertex $ a $ with respect to $ {R}{p} $ is the vector $ (d(a|{R}{p_1}), d(a|{R}{p_2}), \dots, d(a|{R}{p_k})) $. The partition is called a resolving partition of $ G $ if $ r(a|{R}{p}) \ne r(b|{R}{p}) $ for all distinct $ a, b\in V(G) $. The partition dimension of a graph, denoted by $ pd(G) $, is the cardinality of a minimum resolving partition of $ G $. Computing precise and constant values for the partition dimension poses a interesting problem; therefore, it is possible to compute an upper bound for the partition dimension within a general family of graphs. In this paper, we studied partition dimension of the some families of convex polytopes, specifically $ \mathbb{T}_n $, $ \mathbb{U}_n $, $ \mathbb{V}_n $, and $ \mathbb{A}_n $, and proved that these graphs have constant partition dimension.

Let G = (V, E) be a simple, connected graph with vertex set V(G) and E(G) edge set of G. For two vertices a and b in a graph G, the distance d(a, b) from a to b is the length of shortest path a-b path in G. A k-ordered partition of vertices of G is represented as Rp = {Rp1, Rp2,. . ., Rpk} and the representation r(a|Rp) of a vertex a with respect to Rp is the vector (d(a|Rp1), d(a|Rp2), . . ., d(a|Rpk)). The partition is called a resolving partition of G if r(a|Rp) ≠ r(b|Rp) for all distinct a, b ∈ V(G). The partition dimension of a graph, denoted by pd(G), is the cardinality of a minimum resolving partition of G. Computing precise and constant values for the partition dimension poses a interesting problem; therefore, it is possible to compute an upper bound for the partition dimension within a general family of graphs. In this paper, we studied partition dimension of the some families of convex polytopes, specifically Tn, Un, Vn, and An, and proved that these graphs have constant partition dimension. [ABSTRACT FROM AUTHOR]

A big amount of vanadium is used in the industry. It is used as a vanadium carbide stabiliser in steel production. Only $ 0.5\% $ 0.5 % vanadium in an iron alloy will double the tensile strength. Vanadium salts are used in leather, ink, dye manufacturing and yellow ceramic glaze. On the other hand, vanadium pentoxide is used as a catalyst, in the chemical industry, for the production of sulphuric acid and in the petroleum refinery industry. Resolvability parameter is the most discussed topic in graph theory. This helps to reorganise a network in a unique way, sometimes in terms of atoms (metric dimension) and sometimes bound (edge metric dimension). We studied resolvability parameters of vanadium carbide network in this article i.e. metric, edge metric and partition dimension. Among these metrics resolvability defined by parameters taking a subset from the set of vertices, if it has a unique representation with all the vertices, then it is called a resolving set. The number of elements in the smallest resolving set is called the metric dimension. While the number of elements of the minimal subset of vertices taking from vertex set is called the edge metric dimension, if the representations are unique of all edges and the set is called the edge resolving set. This study can have an effect in many ways in the research, particularly, with the help of some key points such as network structure analysis, material science via material properties, graph theory applications, predictive modelling, material testing and characterisation, as well as in industrial applications. [ABSTRACT FROM AUTHOR]

Chemical graph theory is an extension of mathematical chemistry that explores chemical phenomena and entities using the conceptual frameworks of graph theory. In particular, chemical graphs are used to represent molecular structures in chemical graph theory. Edges and vertices in this chemical graph substitute for bonds and atoms, respectively. The primary data types used throughout cheminformatics to depict chemical structures are chemical graphs. The basis for (quantitative) structure-property and structure-activity predictions, a central area of cheminformatics, is laid by the computable properties of graphs. The physical characteristics of molecules are thus reflected in these graphs, which can subsequently be reduced to graph-theoretical indices or descriptors. The resolving set Z , which distinguish every pair of distinct vertices in a connected simple graph, is one of the most well-studied distance-based graph descriptors. In this manuscript, we consider the most significant variation of a metric dimension known as partition dimension and determine it for the molecular complex graph of one-pentagonal carbon nanocone (PCN). We demonstrate that all of the vertices and edges in PCN can be uniquely identified only by considering three distinct non-neighboring partition sets (minimal requirement) from PCN. [ABSTRACT FROM AUTHOR]

The notion of partition dimension was initially introduced in the field of graph theory, primarily to examine distances between vertices. The local partition dimension extends this idea by incorporating specific conditions into how vertices are represented. In graph theory, it is customary to represent the partition dimension of a graph as $pd(G)$ . Network localization, on the other hand, is the process of precisely determining the position of nodes within a network concerning a selected subset of nodes, called the locating set. The smallest size of its locating set is used to represent the locating number of a network. The generalized hexagonal cellular network provides an innovative framework for network planning and analysis. In our study, we investigate the partition dimension of a generalized hexagonal cellular network and provide a rigorous proof of its exact partition dimension. Hence, our approach ensures the distinct recognition of each node within a generalized hexagonal cellular network. Additionally, we explore the utilization of the metric dimension in flood relief camping by the National Disaster Management Authority (NDMA) Pakistan during floods in 2022. NDMA established relief camps and relief centers with unique codes to rescue humans and animals.

In the aircraft sector, honeycomb composite materials are frequently employed. Recent research has demonstrated the benefits of honeycomb structures in applications involving nanohole arrays in anodized alumina, micro-porous arrays in polymer thin films, activated carbon honeycombs, and photonic band gap honeycomb structures. The resolvability parameter is the area of graph theory that is most commonly explored. This results in an original network reconfiguration. Occasionally in terms of atoms (metric dimension), and sometimes in terms of bounds (edge metric dimension). In this article, we examined the honeycomb rhombic torus’s metric, edge metric, mixed metric, and partition dimension. The application of edge metric dimension is also discussed in it.

In graph theory, a commonly used concept is the partition dimension or partition metric basis is to uniquely identify the node set of a structure by dividing it into smaller subsets, known as partition resolving sets. These subsets can then be used to define the partition dimension or partition metric of the graph. This concept is useful in the analysis and understanding of the structure and properties of graphs. This article describes a partition dimension of the line graph of the honeycomb network, the Aztec diamond network, and the extended Aztec diamond network. [ABSTRACT FROM AUTHOR]

Apart from many branches in mathematical sciences, the main share of applications of graph theory nowadays is in chemistry. The graph-theoretical ideas have been used by many chemists for more than two centuries but over the last 30 years, the great significance of graph theory in chemistry has been extensively acknowledged. In a chemical graph, the vertices can symbolize atoms, orbitals, bonds, collection of atoms, molecules, or group of molecules and the edges symbolize the connection between chemical substances and are used to describe reactions, chemical bonds, kinetic models, reaction mechanisms, or other interactions and alterations of the chemical substances. Benzene is the archetypal aromatic molecule, showing exceptional chemical stability as compared with other unsaturated hydrocarbons. Fullerenes are a unique family of carbon-based cage molecules having remarkable properties. Fullerene has potential applications in medicinal chemistry due to its size, hydrophobicity, three-dimensionality, and electronic configurations. In this paper, we will calculate the fault-tolerant partition dimension of classes of fullerene graphs, n-linear benzene graph and partition dimension of polyphenyl chains. [ABSTRACT FROM AUTHOR]

For every graph G, the dominating partition dimension of G is either the same as its partition dimension or one higher than its partition dimension. In this paper, we consider some general connections among these three graph parameters: partition dimension, locating-chromatic number, and dominating partition dimension. We will show that βp(G)≤ηp(G)≤χL(G) for any graph G with at least 3 vertices. Therefore, we will derive properties for which graphs G have ηp(G)=βp(G) or ηp(G)=βp(G)+1. [ABSTRACT FROM AUTHOR]

Carbon nanotubes have been actively studied for their applications in filtering of polycyclic aromatic hydrocarbons, sensors, electronics, stretchable devices, and energy storage. The large structures of nanotubes are studied using their induced subgraphs. The partition dimension focusses on the unique distance-based representation of nodes with respect to partition sets such that each node is exclusively identified. The partition and 2 − partition dimension of induced subgraphs of certain nanotubes have been computed in this article. [ABSTRACT FROM AUTHOR]

The characterizations of all graphs of order n with partition dimension 2; n - 2; n - 1 or n have been completely studied. Recently, all graphs of order n ≥ 11 and diameter two with partition dimension n-3 have been characterized. In this paper, we continue characterizing all graphs on n vertices with partition dimension n-3 and diameter either 3 or 4. This completes the characterization of all graphs of order n ≥ 11 with partition dimension n - 3. [ABSTRACT FROM AUTHOR]

Let Λ = {B1, B2, . . ., Bl} be an ordered l-partition of a connected graph G(V (G), E(G)). The partition representation of vertex x with respect to Λ is the l-vector, r(x|Λ) = (d(x, B1), d(x, B2), . . ., d(x, Bl)), where d(x, B) = min{d(x, y)|y ∈ B} is the distance between x and B. If the l - vectors r(x|Λ), for all x ∈ V (G) are distinct then l - partition is called a resolving partition. The least value of l for which there is a resolving l - partition is known as the partition dimension of G symbolized as pd(G). In this paper, the partition dimension of circulant graphs Cn(1, 2, 3, 4) is computed for n ≥ 8 as, pd(Cn(1, 2, 3, 4)) = { if 8 ≤ n ≤ 9; { 6, if n = 10; {5, if n ≥ 11. [ABSTRACT FROM AUTHOR]

In November 2019, there was the first case of COVID-19 (Coronavirus) recorded, and up to 3rd of April 2020, 1,116,643 confirmed positive cases, and around 59,158 dying were recorded. Novel antiviral structures of the SARS-COV-2 virus is discussed in terms of the metric basis of their molecular graph. These structures are named arbidol, chloroquine, hydroxy-chloroquine, thalidomide, and theaflavin. Partition dimension or partition metric basis is a concept in which the whole vertex set of a structure is uniquely identified by developing proper subsets of the entire vertex set and named as partition resolving set. By this concept of vertex-metric resolvability of COVID-19 antiviral drug structures are uniquely identified and helps to study the structural properties of structure.

Fault-tolerance of a system measures its working capability in the presence of faulty components in the system. The fault-tolerant partition dimension of a network computes the least number of subcomponents of network required to distinctively identify each node in the presence of faults, having promising applications in telecommunication, robot navigation and geographical routing protocols. In this paper, certain triangular mesh networks including, triangular ladder ( $Tl_{s}$ ), triangular mesh ( $T_{s}$ ), reflection triangular mesh ( $rl(T_{s})$ ), tower triangular mesh ( $Tr_{s}$ ) and reflection tower triangular mesh ( $rl(Tr_{s})$ ) networks are discussed for their partition and fault-tolerant partition resolvability. In this regard, it is shown that the partition dimension of these networks is 3, whereas their fault-tolerant partition dimension is 4.

The concept of metric-related parameters permeates all of graph theory and plays an important role in diverse networks, such as social networks, computer networks, biological networks and neural networks. The graph parameters include an incredible tool for analyzing the abstract structures of networks. An important metric-related parameter is the partition dimension of a graph holding auspicious applications in telecommunication, robot navigation and geographical routing protocols. A fault-tolerant resolving partition is a preference for the concept of a partition dimension. A system is fault-tolerant if failure of any single unit in the originally used chain is replaced by another chain of units not containing the faulty unit. Due to the optimal fault tolerance, cycle-related graphs have applications in network analysis, periodic scheduling and surface reconstruction. In this paper, it is shown that the partition dimension (PD) and fault-tolerant partition dimension (FTPD) of cycle-related graphs, including kayak paddle and flower graphs, are constant and free from the order of these graphs. More explicitly, the FTPD of kayak paddle and flower graphs is four, whereas the PD of flower graphs is three. Finally, an application of these parameters in a scenario of installing water reservoirs in a locality has also been furnished in order to verify our findings. [ABSTRACT FROM AUTHOR]

In this paper, we compute the partition dimension and local metric dimension of the extended annihilating-ideal graph 피픸픾(R) associated to a commutative ring R which is denoted by dimP(피픸픾 (R)) and diml(피픸픾 (R)) respectively. In addition, we characterize diml (피픸픾 (R)) for direct product of rings and the ring of integers ℤn. [ABSTRACT FROM AUTHOR]

The symmetry of an interconnection network plays a key role in defining the functioning of a system involving multiprocessors where thousands of processor-memory pairs known as processing nodes are connected. Addressing the processing nodes helps to create efficient routing and broadcasting algorithms for the multiprocessor interconnection networks. Oxide interconnection networks are extracted from the silicate networks having applications in multiprocessor systems due to their symmetry, smaller diameter, connectivity and simplicity of structure, and a constant number of links per node with the increasing size of the network can avoid overloading of nodes. The fault tolerant partition basis assigns unique addresses to each processing node in terms of distances (hops) from the other subnets in the network which work in the presence of faults. In this manuscript, the partition and fault tolerant partition resolvability of oxide interconnection networks have been studied which include single oxide chain networks ( S O X C N ), rhombus oxide networks ( R H O X N ) and regular triangulene oxide networks ( R T O X N ). Further, an application of fault tolerant partition basis in case of region-based routing in the networks is included. [ABSTRACT FROM AUTHOR]

For any (not necessary connected) graph $G(V,E)$ and $A\subseteq V(G)$, the distance of a vertex $x\in V(G)$ and $A$ is $d(x,A)=\min\{d(x,a): a\in A\}$. A subset of vertices $A$ resolves two vertices $x,y \in V(G)$ if $d(x,A)\neq d(y,A)$. For an ordered partition $\Lambda=\{A_1, A_2,\ldots, A_k\}$ of $V(G)$, if all $d(x,A_i)\infty$ for all $x\in V(G)$, then the representation of $x$ under $\Lambda$ is $r(x|\Lambda)=(d(x,A_1), d(x,A_2), \ldots, d(x,A_k))$. Such a partition $\Lambda$ is a resolving partition of $G$ if every two distinct vertices $x,y\in V(G)$ are resolved by $A_i$ for some $i\in [1,k]$. The smallest cardinality of a resolving partition $\Lambda$ is called a partition dimension of $G$ and denoted by $pd(G)$ or $pdd(G)$ for connected or disconnected $G$, respectively. If $G$ have no resolving partition, then $pdd(G)=\infty$. In this paper, we studied the partition dimension of disjoint union of complete bipartite graph, namely $tK_{m,n}$ where $t\geq 1$ and $m\geq n\geq 2$. We gave the necessary condition such that the partition dimension of $tK_{m,n}$ are finite. Furthermore, we also derived the necessary and sufficient conditions such that $pdd(tK_{m,n})$ is either equal to $m$ or $m+1$.

Little is known about the partition dimension of infinite graphs. Tomescu studied graphs where the set of vertices is the set of points of the integer lattice. We generalize these graphs and present several exact values, lower bounds and upper bounds on the partition dimension of infinite graphs. [ABSTRACT FROM AUTHOR]

The partition dimension is the most complicated resolving parameter of all the resolvability parameters. In this parameter, the vertex set of a graph is completely subdivided into subsets so that each vertex of a graph has a unique identification in terms of the partitioning of its selected subsets. In this paper, we consider Carbon nanocone structure (CNC) more specifically one pentagonal carbon nanocone CNCζ,5. It is also regarded as a useful structure in terms of various applications. To delve deeper into this structure, we provide a graphical representation of it and identified sharp bounds for the partition dimension. We conclude that the pentagonal carbon nanocone CNCζ,5 has partition dimension ≤4.

The production of low-cost, small in size, and high in efficiency objects is the topic of research in almost all scientific fields, especially of engineering. In this scenario, nanotechnology becomes of great importance. To achieve these tasks, one needs to study the different physical and chemical aspects of a chemical compound. In mathematical graph theory, chemical compounds are transformed in a unique mathematical representation and then examined under various parameters for these purposes. The Partition Dimension is also one of these parametric tools, which are used to identify each atom or vertex of a chemical structure under some chosen conditions. In the present paper, we use this tool to show the unique identification of each atom of a tri-hexagonal lattice of the $\alpha $ -boron nanotube.

Complex networks are not easy to decode and understand to work on it, similarly, the Möbius structure is also considered as a complex structure or geometry. But making a graph of every complex and huge structure either chemical or computer-related networks becomes easy. After making easy of its construction, recognition of each vertex (node or atom) is also not an easy task, in this context resolvability parameters plays an important role in controlling or accessing each vertex with respect to some chosen vertices called as resolving set or sometimes dividing entire cluster of vertices into further subparts (subsets) and then accessing each vertex with respect to build in subsets called as resolving partition set. In these parameters, each vertex has its own unique identification and is easy to access despite the small or huge structures. In this article, we provide a resolving partition of hexagonal Möbius ladder graph and discuss bounds of partition dimension of hexagonal Möbius ladder network.

Let $\Omega $ be a connected graph and for a given $l$ -ordered partition of vertices of a connected graph $\Omega $ is represented as $\beta =\{\beta _{1},\beta _{2}, {\dots },\beta _{l}\}$ . The representation of a vertex $\mu \in V(\Omega)$ is the vector $r(\mu |\beta)=(d(\mu,\beta _{1}),d(\mu,\beta _{2}), {\dots }, d(\mu,\beta _{l}))$ . The partition $\beta $ is a resolving partition for vertices of $\Omega $ if all vertices of $\Omega $ having the unique representation with respect to $\beta $ . The minimum number of $l$ in the resolving partition for $\Omega $ is known as the partition dimension of $\Omega $ and represented as $pd(\Omega)$ . Resolving partition and partition dimension have multipurpose applications in networking, optimization, computer, mastermind games and modeling of chemical structures. The problem of computing constant values of partition dimension is NP-hard so one can find sharp bound for the partition dimension of graph. In this article, we computed the upper bound for the convex polytopes $\mathbb {E}_{n},~\mathbb {S}_{n},~\mathbb {T}_{n},~\mathbb {G}_{n},~\mathbb {Q}_{n}$ and flower graph $f_{n\times 3}$ .

Fibonacenes are unbranched catacondensed benzenoid hydrocarbons in which all the non-terminal hexagons are angularly annelated. A hexagon is said to be angularly annelated if the hexagon is adjacent to exactly two other hexagons and possesses two adjacent vertices of degree 2. Fibonacenes possess remarkable properties related with Fibonacci numbers. Various graph properties of fibonacenes have been extensively studied, such as their saturation numbers, independence numbers and Wiener indices. In this paper, we show that the locating-chromatic number of any fibonacene graph is 4 and the partition dimension of such a graph is 3.

Resolving partition and partition dimension have multipurpose applications in computer, networking, optimization, mastermind games and modelling of chemical substances. The problem of finding exact values of partition dimension is hard so one can find bound for the partition dimension of a general family of graph. In the present article, we give the sharp upper bounds and lower bounds for the partition dimension of m-wheel, Wn,m for all n ≥ 4 and m ≥ 1. Presented data generalise some already available results.

The concept of metric-related parameters permeates all of graph theory and plays an important role in diverse networks, such as social networks, computer networks, biological networks and neural networks. The graph parameters include an incredible tool for analyzing the abstract structures of networks. An important metric-related parameter is the partition dimension of a graph holding auspicious applications in telecommunication, robot navigation and geographical routing protocols. A fault-tolerant resolving partition is a preference for the concept of a partition dimension. A system is fault-tolerant if failure of any single unit in the originally used chain is replaced by another chain of units not containing the faulty unit. Due to the optimal fault tolerance, cycle-related graphs have applications in network analysis, periodic scheduling and surface reconstruction. In this paper, it is shown that the partition dimension (PD) and fault-tolerant partition dimension (FTPD) of cycle-related graphs, including kayak paddle and flower graphs, are constant and free from the order of these graphs. More explicitly, the FTPD of kayak paddle and flower graphs is four, whereas the PD of flower graphs is three. Finally, an application of these parameters in a scenario of installing water reservoirs in a locality has also been furnished in order to verify our findings.

The partition dimension is a variant of metric dimension in graphs. It has arising applications in the fields of network designing, robot navigation, pattern recognition and image processing. Let G (V (G) , E (G)) be a connected graph and Γ = {P1, P2, ..., Pm} be an ordered m-partition of V (G). The partition representation of vertex v with respect to Γ is an m-vector r (v|Γ) = (d (v, P1) , d (v, P2) , ..., d (v, Pm)), where d (v, P) = min {d (v, x) |x ∈ P} is the distance between v and P. If the m-vectors r (v|Γ) differ in at least 2 positions for all v ∈ V (G), then the m-partition is called a 2-partition generator of G. A 2-partition generator of G with minimum cardinality is called a 2-partition basis of G and its cardinality is known as the 2-partition dimension of G. Circulant graphs outperform other network topologies due to their low message delay, high connectivity and survivability, therefore are widely used in telecommunication networks, computer networks, parallel processing systems and social networks. In this paper, we computed partition dimension of circulant graphs Cn (1, 2) for n ≡ 2 (mod 4), n ≥ 18 and hence corrected the result given by Salman et al. [Acta Math. Sin. Engl. Ser. 2012, 28, 1851-1864]. We further computed the 2-partition dimension of Cn (1, 2) for n ≥ 6. [ABSTRACT FROM AUTHOR]

All graphs of order n with partition dimension 2, n−2, n−1, or n have been characterized. However, finding all graphs on n vertices with partition dimension other than these above numbers is still open. In this paper, we characterize all graphs of order n≥11 and diameter 2 with partition dimension n−3.

The concept of minimal resolving partition and resolving set plays a pivotal role in diverse areas such as robot navigation, networking, optimization, mastermind games and coin weighing. It is hard to compute exact values of partition dimension for a graphic metric space, (G, dG) and networks. In this article, we give the sharp upper bounds and lower bounds for the partition dimension of generalized Möbius ladders, Mm, n, for all n≥3 and m≥2.

Let $G=(V(G),E(G))$ be a connected graph and $\Pi=\{S_{1},S_2,\dots,S_{k}\}$ be a $k$-partition of $V(G)$.The representation $r(v|\Pi)$ of a vertex $v$ with respect to $\Pi$ is the vector $(d(v,S_{1}),d(v,S_2),\dots,d(v,S_{k}))$, where $d(v,S_{i})=\min\{d(v,s_{i})\mid s_{i}\in S_{i}\}$.The partition $\Pi$ is called a resolving partition of $G$ if $r(u|\Pi)\neq r(v|\Pi)$ for all distinct $u,v\in V(G)$.The partition dimension of $G$, denoted by $pd(G)$, is the cardinality of a minimum resolving partition of $G$.In this paper, we calculate the partition dimension of two $(4,6)$-fullerene graphs. We also give conjectures on the partition dimension of two $(3,6)$-fullerene graphs.

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. [ABSTRACT FROM AUTHOR]

Finding the partition dimension of a graph is one of the interesting (and uncompletely solved) problems of graph theory. For instance, the values of the partition dimensions for most kind of trees are still unknown. Although for several classes of trees such as paths, stars, caterpillars, homogeneous firecrackers and others, we do know their partition dimensions. In this paper, we determine the partition dimension of a subdivision of a particular tree, namely homogeneous firecrackers. Let G be any graph. For any positive integer k and e ∈ E(G), a subdivision of a graph G, denoted by S(G(e; k)), is the graph obtained from G, by replacing an edge e with a (k + 1)-path. We show that the partition dimension of S(G(e; k)) is equal to the partition dimension of G, if G, is a homogeneous firecracker. [ABSTRACT FROM AUTHOR]

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 }. [ABSTRACT FROM AUTHOR]

For a graph G=(V,E), a partition Ω=\{O_1,O_2,…,O_k \} of the vertex set V is called a resolving partition if every pair of vertices u,v∈V(G) have distinct representations under Ω. The partition dimension of G is the minimum integer k such that G has a resolving k-partition. Many results in determining the partition dimension of graphs have been obtained. However, the known results are limited to connected graphs. In this study, the notion of the partition dimension of a graph is extended so that it can be applied to disconnected graphs as well. Some lower and upper bounds for the partition dimension of a disconnected graph are determined (if they are finite). In this paper, also the partition dimensions for some classes of disconnected graphs are given.

A partition Π = { S 1 , ... , S k } of the vertex set of a connected graph G is called a resolving partition of G if for every pair of vertices u and v , d (u , S j) ≠ d (v , S j) , for some part S j. The partition dimension β p (G) is the minimum cardinality of a resolving partition of G. A resolving partition Π is called resolving dominating if for every vertex v of G , d (v , S j) = 1 , for some part S j of Π. The dominating partition dimension η p (G) is the minimum cardinality of a resolving dominating partition of G. In this paper we show, among other results, that β p (G) ≤ η p (G) ≤ β p (G) + 1. We also characterize all connected graphs of order n ≥ 7 satisfying any of the following conditions: η p (G) = n , η p (G) = n − 1 , η p (G) = n − 2 and β p (G) = n − 2. Finally, we present some tight Nordhaus–Gaddum bounds for both the partition dimension β p (G) and the dominating partition dimension η p (G). [ABSTRACT FROM AUTHOR]

The partition dimension problem is to decompose the vertex set of a graph into minimum number of vertex disjoint sets such that the ordered distances between every vertex of the graph and the disjoint sets of that decomposition have different values in the representation. This problem has emerging applications in areas such as structure-activity issues in drug design, navigation of robots, pattern recognition and image processing. In the present study, we provide an upper bound for the partition dimension of parallel composition of any graphs and derive an exact result for parallel composition of paths with different lengths. On the other side, we find the partition dimension of the series composition of cycles and complete graphs with different sizes and an upper bound is derived for series composition of any graphs. Further, this problem is applied to a graph which has been generated by parallel and series composition together on paths. [ABSTRACT FROM AUTHOR]

In this paper, we propose a method for constructing new graphs from a given graph G so that the resulting graphs have the partition dimension at most one larger than the partition dimension of the graph G. In particular, we employ this method to construct a family of graphs with partition dimension 3. [ABSTRACT FROM AUTHOR]

An ordered k -partition Π = { S 1 , S 2 , ... , S k } of V (G) is called a resolving partition if for every two distinct vertices u , v ∈ V (G) , there exists a set S i in Π such that the distance between u and S i is not equal to the distance between v and S i. The minimum k for which there is a resolving k -partition of V (G) is called the partition dimension of G. In this paper, we provide the tight bounds for the partition dimension of rooted product graphs. Further, partition dimension of particular class of rooted product graphs has been studied. [ABSTRACT FROM AUTHOR]

In this paper, the resolving parameters such as metric dimension and partition dimension for the nonzero component graph, associated to a finite vector space, are discussed. The exact values of these parameters are determined. It is derived that the notions of metric dimension and locating-domination number coincide in the graph. Independent sets, introduced by Boutin [Determining sets, resolving set, and the exchange property, Graphs Combin.25 (2009) 789–806], are studied in the graph. It is shown that the exchange property holds in the graph for minimal resolving sets with some exceptions. Consequently, a minimal resolving set of the graph is a basis for a matroid with the set V of nonzero vectors of the vector space as the ground set. The matroid intersection problem for two matroids with V as the ground set is also solved. [ABSTRACT FROM AUTHOR]

Let Π = {S1, S2, . . . , Sk} be an ordered partition of the vertex setV(G) of a graphG. The partition representation of a vertexv∈V(G) with respect to Π is thek-tupler(v|Π) = (d(v, S1), d(v, S2), . . . , d(v, Sk)), whered(v, S) is the distance betweenvand a setS. If for every pair of distinct verticesu, v∈V(G), we haver(u|Π) ≠r(v|Π), then Π is a resolving partition and the minimum cardinality of a resolving partition ofV(G) is called the partition dimension ofG. We study the partition dimension of circulant graphs, which are Cayley graphs of cyclic groups. Grigorious et al. [On the partition dimension of circulant graphs] proved thatpd(Cn(1,2, . . . , t)) ≥t+ 1 forn≥ 3. We disprove this statement by showing that ift≥ 4 is even, then there exists an infinite set of values ofn, such that. We also present exact values of the partition dimension of circulant graphs with 3 generators. [ABSTRACT FROM AUTHOR]