Metric dimension of Andrasfai graphs
Ali Behtoei, Shiroyeh Payrovi, S. Batool Pejman

TL;DR
This paper calculates the metric dimension of Andrasfai graphs, providing insights into their resolving set sizes and properties within graph theory.
Contribution
It offers the first explicit determination of the metric dimension for Andrasfai graphs and related graph classes.
Findings
Metric dimension of Andrasfai graphs determined
Results applicable to related graph classes
Enhances understanding of graph resolving sets
Abstract
In this paper we determine the metric dimension of Andrasfai graphs and some related graphs.
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Taxonomy
TopicsGraph Labeling and Dimension Problems
Metric dimension of Andrásfai graphs
S. Batool Pejman , Shiroyeh Payrovi , Ali Behtoei
*Department of Mathematics, Imam Khomeini International University, 34149-16818, Qazvin, Iran * [email protected]@ikiu.ac.irCorresponding author, [email protected]
Abstract
A set is called a resolving set, if for each pair of distinct vertices there exists such that , where is the distance between vertices and . The cardinality of a minimum resolving set for is called the metric dimension of and is denoted by . This parameter has many applications in different areas. The problem of finding metric dimension is NP-complete for general graphs but it is determined for trees and some other important families of graphs. In this paper, we determine the exact value of the metric dimension of graphs, their complements and . Also, we provide upper and lower bounds for .
Key words: Resolving set, Metric dimension, Andrásfai graph, Cayley graph.
**MSC code 2010: ** 05C12, 05C25.
1 Introduction
Throughout this paper all graphs are finite, simple and undirected. Let be a connected graph with vertex set and edge set . The distance between two vertices is the length of a shortest path between them and is denoted by , or for convenient. The neighborhood of is and the diameter of is . It is well known that almost all graphs have diameter 2. The notations and stand for the complement graph and the line graph of , respectively. For an ordered subset of vertices and a vertex , the -vector is called the metric representation of with respect to (the code of , for convenient). The set is called a resolving set for if distinct vertices of have distinct metric representations with respect to . The cardinality of a minimum resolving set is the metric dimension of and is denoted by . A graph with metric dimension is called -dimensional. These concepts were introduced by Slater in 1975 when he was working with U.S. Sonar and Coast Guard Loran stations and he described the usefulness of these concepts, [19]. Independently, Harary and Melter [11] discovered these concepts. They have applications in many areas including network discovery and verification [2], robot navigation [14], problems of pattern recognition and image processing [15], coin weighing problems [18], strategies for the Mastermind game [8], combinatorial search and optimization [18]. Determining the metric dimension of different families of graphs, operations and products, or characterizing -vertex graphs with a specified metric dimension are fascinating problems and atracts the attention of many researchers. The problem of finding metric dimension is NP-Complete for general graphs but the metric dimension of trees can be obtained using a polynomial time algorithm [14]. It is not hard to see that for each -vertex graph we have . Khuller [14] and Chartrand et al. [7] proved that if and only if is a path . Chartrand et al. [7] proved that for , if and only if is the complete graph . The metric dimension of each complete -partite graph with vertices is . They also provided a characterization of graphs of order with metric dimension , see [7]. Graphs of order with metric dimension are characterized in [13]. Béla Bollobás studied the metric dimension of random graphs [5]. Cceres studied this parameter for the Cartesian product of graphs [6]. Bailey and Cameron [1] have computed the exact value of the metric dimension for the diameter 2 Kneser and Johnson graphs. Fijav and Mohar studied this parameter for Paley graphs [10]. Salman studied this parameter for the Cayley graphs on cyclic groups [17]. In [16] and [9] the metric dimension of Cayley digraphs for the groups which are direct product of some cyclic groups is investigated. Imran studied the metric dimension of barycentric subdivision of Cayley graphs in [12]. Each cycle graph is a -dimensional graph . In [20] some properties of -dimensional graphs are obtained. All of 2-trees with metric dimension two are characterized in [3], -dimensional Cayley graphs on Abelian groups are characterized in [21] and -dimensional Cayley graphs on dihedral groups are characterized in [4]. For more results in this subject or related subject see [?].
Recall that the Cartesian product of two graphs and , denoted by , is a graph with vertex set , in which is adjacent to whenever and , or and . Note that the vertex set of can be arranged in rows and columns. Also if and are connected, then is connected. Let be a group and let be a subset of that is closed under taking inverse and does not contain the identity element. Recall that the Cayley graph is a simple graph whose vertex set is and two vertices and are adjacent in it when . For any integer , the Andrásfai graph is the Cayley graph where is the additive group of integers modulo and is the subset of consisting of the elements congruent to modulo . Note that is a path with two vertices, is isomorphic to the -cycle and is a . It is well known that is a reduced (twin free), circulant, vertex transitive, triangle-free and -regular graph whose diameter is two for .
In this paper, we determine the exact value of the metric dimension of graphs, their complements and . Also, we prove that .
2 Main Results
Note that if is a resolving set for , then for each the set is a larger resolving set for . Also, when is a graph with diameter 2, then is a resolving set for if and only if for each pair of distinct vertices there exist such that .
Theorem 2.1
Let be an integer. Then, the metric dimension of the Andrásfai graph is .
**Proof. **
By investigation, it is easy to see that for . Hence, we assume that . Note that where . Since , contains the Hamiltonian cycle and we can consider a drawing of it in such a way that vertices are consecutively ordered clockwise around a cycle. Hereafter, all of vertex numbers will be cosidered in modulo . It is straightforward to check that the vertex is adjacent to every vertex in and each vertex has at least one non-adjacent vertex in . Consider the subset of with . We have and . Also, for each vertex we have if and only if (because in modulo , we have if and only if ). Therefore, , and . This means that two vertices and have unique codes among the vertices of . Since for each vertex there exists such that or , the code of is unique. Hence, is a resolving set for and this implies that .
In order to complete the proof, it is sufficient to show that for each resolving set of . Suppose on the contrary that there exists a resolving set of with . By including some additional vertices to (if it is necessary) we can assume that . If there exists a subset of four (clockwise) consecutive vertices such that , then for each vertex we have if and only if (because, if and only if ). This implies that two vertices and have the same metric representations with respect to , which contradics the resolvability of . Now, assume that where
[TABLE]
For each (and with the assumption that ) let
[TABLE]
Note that just when and that for each . Also, using previous facts we have
[TABLE]
For each let be the number of blocks with . Thus, using the fact we see that
[TABLE]
and
[TABLE]
Therefore,
[TABLE]
This implies that . Specially, . Now the Pigeonhole Principle implies that there exist two blocks of size 3 such that between them in at least one direction (clockwise or counterclockwise) only blocks of size 2 (if any exists) are located. Since is vertex transitive, whithout loss of generality and for convenient, we can assume that (i.e ) and is located (in clockwise direction) after blocks of size two (when they are ), i.e . Note that for the case two blocks and are consecutive. Therefore,
[TABLE]
Now consider two vertices and . Since , for each we have if and only if . Also, it is straightforward to check that
[TABLE]
This means that , which is a contradiction. Therefore, and this completes the proof.
- *
Note that is a 2-vertex path and hence, its complement is disconnected. is a 5-cycle and its metric dimension is 2. Also, for each the complement of is a connected -regular graph and its diameter is two.
Theorem 2.2
For each we have .
**Proof. **
Let be a non-empty ordered subset of and let be an arbitrary vertex. Assume that the metric representation of vertex with respect to in is and the metric representation of with respect to in is . Since both graphs and have diameter two, for each we have
[TABLE]
This means that for each vertex we have if and only if . Thus, there is a one-to-one correspondence between the vectors and the vectors by a switching on non-zero components. In the proof of Theorem 2.1 we see that is a minimum resolving set for and hence, . Therefore, is a minimum resolving set for and the result follows.
- *
In the following theorem we determine and .
Theorem 2.3
For each and we have
[TABLE]
Specially, the metric dimension of the prism generated by or its complement is .
**Proof. **
Assume that and . Hence,
[TABLE]
and the induced subgraph of on the set is isomorphic to for each . Using Corollary 3.2 in [6] and Theorem 2.1 we see that
[TABLE]
Let
[TABLE]
We want to show that is a resolving set for . For each and for each it is easy to see that
[TABLE]
which implies that
[TABLE]
Note that except the vertex whose metric representation with respect to is the all 1 vector , the metric representation of each vertex has at least one component equal to 2 and we have . Similarly, for each , except the vertex whose metric representation is , the metric representation of each vertex has at least one component equal to and . By the proof of Theorem 2.1, is a minimum resolving set for . Note that by Lemma 3.1 in [6] the projection of onto each copy of in (i.e the induced subgraph on each row) resolves the vertices of that copy (row). Therefore, each pair of distinct vertices and (with or ) have distinct metric representations with respect to . Hence, is a minimum resolving set for and . Using a similar argument we can show that .
- *
Let be an integer. Using Theorem 8.6 and Theorem 8.4 in [6] we see that
[TABLE]
and
[TABLE]
Proposition 2.1
If and , then .
**Proof. **
Corollary 3.2 in [6] using Theorem 2.1 implies that
[TABLE]
For the upper bound, assume that
[TABLE]
and let
[TABLE]
Using the structure of the Cartesian product of two graphs, for each and for each we have
[TABLE]
Note that (using the proof of Theorem 2.1 and Lemma 3.1 in [6]) the projection of onto each copy of in (i.e the induced subgraph on each row) resolves the vertices of that copy and the projection of onto each copy of in (each column) resolves its vertices. Also, for each and for each we have
[TABLE]
Thus, distinct vertices in have distinct metric representations with respect to (and hence, with respect to ) and distinct vertices in
[TABLE]
have distinct metric representations with respect to (and hence, with respect to ). These facts imply that for each and for each the metric representation of the vertex with respect to in is just equal to the code of and no other vertex (note that when is even and the vertex coincides with ). Since
[TABLE]
and
[TABLE]
for distinct vertices and we have
[TABLE]
Hence, is a resolving set for and .
- *
In the following theorems we investigate and .
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