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Let G be a connected graph. For a vertex v of G and a subset S of V(G), the distance between v and S is d(v, S) = min Given an ordered k-partition = of V(G), the representation of v with respect to is the k-vector If for each pair of distinct vertices then the k-partition is said to be a resolving partition. The partition dimension of G, denoted by pd(G), is determined by the minimum k for which there is a resolving partition of V(G). If each induced subgraph for Si, is connected in G, then the resolving partition = of V(G) is said to be connected. The connected partition dimension of G, denoted by cpd(G), is determined by the minimum k for which there is a connected resolving partition of V(G). In this paper, we compute the connected partition dimension of the truncated wheels TWn. It is shown that for any natural number the connected partition dimension of the truncated wheel TWn is 3 when n = 3 and when more...

The directed circulant graph Cn(1, 2, … , t) consists of the vertices v0 , v1 , … , vn-1 and edges directed from vi to vi+j for every i = 0, 1, … , n – 1, and j = 1, 2, … , t, where 1 ≤ t ≤ n – 1, the indices are taken modulo n. We present lower and upper bounds on the partition dimension of very general graphs 10.1080/09720529.2019.1694739Cn(1, 2, … , t). Our bounds yield exact values of the partition dimension of the graphs Cn(1, 2, … , t) for t = 2, 3, 4 and n ≥ 10. more...

The locating-chromatic number of a graph is combined two graph concept, coloring vertices and partition dimension of a graph. The locating-chromatic number, denoted by XLG, is the smallest k such that G has a locating k-coloring. In this paper, we discuss the locating- chromatic number for certain operation of generalized Petersen graphs sP(n,1). more...

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. more...

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. more...

Resolving partition is part of graph theory. Resolving partition is needed to obtain partition dimension. So far it has not been discussed regarding resolving partition of complete graphs. For this reason, this article explains the resolving partition of a complete graph.Given a connected graph and is a subset of write . Suppose there is then the distance between to is denoted in the form of . Then, suppose there is a partitioned set of , write , then we can obtain a representation of in the form of The partition set is called a resolving partition if the representation of each ∈ with is different. The minimum cardinality of the resolving -partition against is called the partition dimension of , denoted by pd (). To get the partition dimension of a complete graph, first we find the resolving partition og the graph, resolving partition orde After that, to assist in obtaining the partition dimensions of complete graph, we explained that if is is a connected graph then Based on these steps, finally obtained that the partition dimension of the complete graph, namely more...

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. more...

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. more...

In peer-to-peer (P2P) overlay networks, multimedia contents are in nature distributed to peers by downloading and caching. Here, a peer which transmits a multimedia content and a peer which receives the multimedia content are referred to as source and receiver peers, respectively. A peer is realized in a process of a computer and there are mobile and fixed types of computers. A peer on a mobile computer moves in the network. Furthermore, a peer maybe realized as a mobile agent. Thus, not only receiver peers but also source peers might move in the network. In this paper, we would like to discuss how source peers deliver multimedia contents to receiver peers in a streaming model so that enough quality of service (QoS) required is supported in change of QoS of network and peer, possibly according to the movements of the peers. In this paper, we discuss a multi-source streaming (MSS) protocol where a receiver peer can receive packets of a multimedia content from multiple source peers which can support enough QoS. If a current source peer is expected to support lower QoS than required, another source peer takes over the source peer and starts sending packets of the multimedia content. The receiver peer is required to receive packets of the multimedia content with enough QoS, e.g. no packet loss even if the source peer is being switched with a new source peer. We discuss how to switch source peers so as to support enough QoS to the moving receiver peer. We evaluate the MSS protocol in terms of the fault ratio, i.e. how frequently the receiver peer fails to receive packets with enough QoS and show the MSS protocol can reduce the fault ratio. more...

Let 𝐺 = (𝑉, 𝐸) be a connected graph and 𝑆 ⊆ 𝑉(𝐺). For a vertex v ∈ V(G) and an ordered k-partition Π = {𝑆1 , 𝑆2 , … , 𝑆𝑘 } of 𝑉(𝐺), the representation of v with respect to Π is the k-vector 𝑟(𝑣|𝛱 = (𝑑(𝑣, 𝑆1), 𝑑(𝑣, 𝑆2), . . . , 𝑑(𝑣, 𝑆𝑘)), where d(v,Si) denotes the distance between v and Si. The k-partition Π is said to be resolving if for every two vertices 𝑢, 𝑣  𝑉(𝐺), the representation 𝑟(𝑢|П)  𝑟(𝑣|Π). The minimum k for which there is a resolving k-partition of 𝑉(𝐺) is called the partition dimension of 𝐺, denoted by 𝑝𝑑(𝐺). The wheel graph 𝑊𝑛 𝑜𝑛 𝑛 + 1 vertices with 𝑉(𝑊𝑛) = {𝑣0, 𝑣1, . . . , 𝑣𝑛}. Let 𝑙2 ,𝑙2 ,… ,𝑙𝑛be non-negative integers, 𝑙𝑖 ≥ 1, for 𝑖  {0,1,2, . . . , 𝑛}. The thorn graph of the graph Wn, with parameters 𝑙0 ,𝑙1 ,… ,𝑙𝑛 is obtained by attaching li new vertices of degree one to the vertex vi of the graph Wn. The thorn graph is denoted by 𝑇ℎ(𝑊𝑛,𝑙0 ,𝑙1 ,… ,𝑙𝑛). In this paper we give the upper bounds for the partition dimension of 𝑊3 and 𝑊4 denoted by 𝑝𝑑(𝑇ℎ(𝑊3 ,𝑙0 ,𝑙1 ,𝑙2 ,𝑙3 )) and 𝑝𝑑(𝑇ℎ(𝑊4 ,𝑙0 ,𝑙1 ,𝑙2 ,𝑙3 ,𝑙4 )). Keywords : Partition Dimension, Resolving Partition, Thorn Graph, Wheel Graph. more...

The locating-chromatic number of a graph combines two graph concepts, namely coloring vertices and partition dimension of a graph. The locating-chromatic number is the smallest k such that G has a locating k-coloring, denoted by χL(G). This article proposes a procedure for obtaining a locating-chromatic number for an origami graph and its subdivision (one vertex on an outer edge) through two theorems with proofs. more...

In this paper, we determine and show the proof of the metric dimension of a wheel graph and the partition dimension of graph F = Ks + Kt. The solution for the metric dimension is divided into four cases. If n = 3 or 6, the metric dimension of wheel graph Wn is 3. If n = 4 or 5, the metric dimension of wheel graph Wn is 2. If n = 7 or 8, the metric dimension of wheel graph Wn+5k is 3 + 2k for k is a whole number. If n = 9, 10, or 11, the metric dimension of wheel graph Wn+5k is 4 + 2k for k is a whole number. The solution for the partition dimension is divided into two cases. If s ≥ t, the partition dimension of graph F is s + 1. If s < t, the partition dimension of graph F is t. more...

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 . more...

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. more...

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. more...

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., Mathematics; Partition dimension; Graph; Characterization; Diameter more...

Finding the partition dimension of graph is still an open problem in the graph theory. Therefore, several researchers investigate the problem in several operations of the graph. For example, the partition dimension of corona product, cartesian product, subdivision operation has been published by several researchers. Let G1, G2 be two connected graphs. We present the partition dimension of a graph G which is obtained from two graphs G1( G2 with a linking a vertex in G1 to one vertex in G2 (a bridge in graph G). This paper is devoted to find the upper bound of partition dimension of the connected two graphs by a bridge. more...

The locating-chromatic number was introduced by Chartrand in 2002. The locating-chromatic number of a graph is a combined concept between the coloring and partition dimension of a graph. The locating chromatic number of a graph G is defined as the cardinality of minimum color classes. In this paper, we discuss about the locating chromatic number of three types modified path with cycle having locating chromatic number four. more...

All graphs in this paper are undirected and connected graphs. An ordered k-partition set [Formula: see text] where [Formula: see text], the representation of a vertex [Formula: see text] of [Formula: see text] with respect to [Formula: see text] is [Formula: see text] where [Formula: see text] is the distance between the vertex v and the set [Formula: see text] with [Formula: see text] for [Formula: see text]. The partition set [Formula: see text] is a local resolving partition of [Formula: see text] if [Formula: see text] for [Formula: see text] adjacent to [Formula: see text] of [Formula: see text]. The minimum local resolving partition [Formula: see text] is a local partition dimension of [Formula: see text], denoted by [Formula: see text]. In our paper, we found the sharp bounds of the local partition dimension of [Formula: see text] and determine the exact value of some special graph. more...

Determining a resolving partition of a graph is an interesting study in graph theory due to many applications like censor design, compound classification in chemistry, robotic navigation and internet network. Let and , the distance between an is . For an ordered partition of , the representation of with respect to is . The partition is called a resolving partition of if all representation of vertices are distinct. The partition dimension of graph is the smallest integer such that has a resolving partition with element.In this thesis, we determine the partition dimension of complete multipartite graph , which is limited by , with and . We found that , , and , . more...

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 G ( x , w ) : w ? W } and d G ( x , w ) is the length of a shortest x - w path. An ordered vertex partition ? = { U 1 , U 2 , ? , U k } of G is a strong resolving partition for G , if every two different vertices belonging to the same set of the partition are strongly resolved by some set of ? . The minimum cardinality of a strong resolving partition for G is the strong partition dimension of G . In this article we study the strong resolving partitions and the strong partition dimension of strong product graphs and Cartesian product graphs. more...

The concept of a graph partition dimension was introduced by Chartrand et al. (1998). Let Π = { L 1, L 2, L 3, · · ·, Lk } be a k -partition of V ( G ). The representation r ( v |Π) of a vertex v with respect to Π is the vector ( d ( v , L 1), d ( v , L 2), · · ·, d ( v , Lk) ). The partition Π is called a resolving partition of G if r ( w |Π) ≠ r ( v |Π) for all distinct w , v ∈ V ( G ). The partition dimension of a graph, denoted by pd ( G ), is the cardinality of a minimum resolving partition of G . This paper considers in finding partition dimensions of graphs obtained from a subdivision operation. In particular, we derive an upper bound of partition dimension of a subdivision of a complete graph Kn with n ≥ 9. Additionally for n ∈ [2,8], we obtain the exact values of the partition dimensions. more...

In 1998, Chartrand, E. Salehi and P. Zhang introduced the concept of graph partition dimension. This is a variant of graph metric dimension concept introduced independently by Slater in 1975 and Harary & Melter in 1976. In this paper, we determine the partition dimension of specific classes of trees, namely homogeneous caterpillars and homogeneous banana trees. In particular, we characterize all trees in these classes with partition dimension three. more...

For a vertex v of a connected graph G(V,E) and a subset S of V, the distance between a vertex v and S is defined by d(v,S)=min{d(v,x):x@?S}. For an ordered k-partition @p={S"1,S"2...S"k} of V, the partition representation of v with respect to @p is the k-vector r(v|@p)=(d(v,S"1),d(v,S"2)...d(v,S"k)). The k-partition @p is a resolving partition if the k-vectors r(v|@p), v@?V(G) are distinct. The minimum k for which there is a resolving k-partition of V is the partition dimension of G. Salman et al. [1] in their paper which appeared in Acta Mathematica Sinica, English Series proved that partition dimension of a class of circulant graph G(n,+/-{1,2}), for all even n>=6 is four. In this paper we prove that it is three. more...

Let be a connected graph G and -partition of end . The coordinat to is definition . If for every two vertices in t , then is a called k-resolving partition of . The minimum k such that is a k-resolving partition of is the partition dimension of and denoted by . In this paper, we show that the partition dimension for amlagamation of cycle graph for To proof this results, we was used mathematical induction method. more...