Abstract

Let be a connected simple graph and let be a nonempty subset of . The -distance pattern of a vertex in is the set of all distances from to the vertices in . If the distance patterns of all vertices in are distinct, then the set is a distance pattern distinguishing set of . A graph with a distance pattern distinguishing set is called a distance pattern distinguishing graph. Minimum number of vertices in a distance pattern distinguishing set is called distance pattern distinguishing number of a graph. This paper initiates a study on the problem of finding distance pattern distinguishing number of a graph and gives bounds for distance pattern distinguishing number. Further, this paper provides an algorithm to determine whether a graph is a distance pattern distinguishing graph or not and hence to determine the distance pattern distinguishing number of that graph.

1. Introduction

One of the basic problems in graph theory is to select a minimum set of vertices in such a way that each vertex in the graph is uniquely determined by the distances to the chosen vertices. The vertices in that set uniquely determine the positions of the remaining vertices of the graph. Slater [1] defined the code of a vertex with respect to a -tuple of vertices as , where denotes the distance of the vertex from the vertex . Thus, entries in the code of a vertex may vary from to diameter of . If the codes of the vertices are to be distinct, then the subset is called resolving set of that graph. A resolving set of minimum cardinality is called a metric basis and is called the metric dimension of .

In 2006, Dr. B. D. Acharya introduced a new concept which is distance pattern distinguishing set of a graph. A detailed study of this concept has been done in [2, 3]. A distance pattern distinguishing set identifies the automorphism group of a graph and each vertex in the graph is uniquely identified by its graph properties and its relationship to the vertices of the distance pattern distinguishing set. However, a distance pattern distinguishing set of a graph (if it exists) need not be unique. Hence, determination of the minimum cardinality of a distance pattern distinguishing set in is an interesting problem to be investigated. This paper focuses on the problem of determining the minimum cardinality, , of a distance pattern distinguishing set in a graph and gives an algorithm to determine whether a graph has a distance pattern distinguishing set and also to determine , if it exists. In this paper we consider finite, simple, and connected graphs.

Definition 1. For an arbitrarily fixed vertex in and for any nonnegative integer , let and whenever exceeds the eccentricity of in the component to which belongs. Thus, if is connected, then if and only if . We can generalize this concept as, given an arbitrary nonempty subset and for each , .

Definition 2. Let be a given connected simple graph, , and . Then, the -distance pattern of is the set . Clearly, . If is an injective function, then the set is said to be a distance pattern distinguishing set of . A graph with distance pattern distinguishing set is called a distance pattern distinguishing graph. The number of vertices in a minimum distance pattern distinguishing set is called the distance pattern distinguishing number of and it is denoted by .
The expressibility of graphs and matrices in terms of each other is well known. Each of these two mathematical models has certain operational advantages. Definition 3 brings a matrix, related to the distance patterns of the vertices of a graph with respect to a subset of vertices and Lemma 4 characterizes the distance pattern distinguishing set of a graph in terms of -matrix that is defined as follows.

Definition 3. For an arbitrary nonempty subset , the matrix ; ; , where that denotes the diameter of is called the distance neighborhood pattern matrix of . Let be the matrix build from by replacing each nonzero entry by .

Lemma 4 (see [3]). In any graph , a nonempty is a distance pattern distinguishing set if and only if no two rows of are identical.

Example 5. Consider the graph given in Figure 1. Now, is a distance pattern distinguishing set of since Also, and the -distance pattern neighborhood matrix is

2. Distance Pattern Distinguishing Number of a Graph

Theorem 6 (see [3]). A cycle of order n admits a distance pattern distinguishing set if and only if .

Theorem 7 (see [2]). Let be any distance pattern distinguishing graph with a distance pattern distinguishing set . Then, the induced graph is disconnected.

The following theorem provides the distance pattern distinguishing number of some well-known classes of graphs.

Theorem 8. The trivial graph is the only graph with distance pattern distinguishing number as the order of that graph.
Path is the only graph with distance pattern distinguishing number one.
There exists no graph with distance pattern distinguishing number .
  , for all .

Proof. (a) Assume that is isomorphic to . Clearly, has the distance pattern distinguishing set , where and hence, .
Converse follows from the fact that if , then the distance patterns of diametrically opposite vertices are identical.
(b) It can be easily verified that one of the two pendant vertices of a path forms a distance pattern distinguishing set and hence distance pattern distinguishing number of a path is one. For the converse part, we assume that and that the distance pattern distinguishing set is a vertex of . First observe that the degree of is ; otherwise, the vertices adjacent to have the same distance pattern as . Since is not a path, it contains a vertex whose degree is at least three. Let be such a vertex of with the least and let be the set of all vertices adjacent to . Then, the distance pattern of each of the vertices in is any one of , , or , where . None of the vertices in may have the distance pattern , as it is the distance pattern of the vertex . Therefore, since , at least two vertices in have the same distance pattern, a contradiction.
(c) Let two vertices and form a distance pattern distinguishing set of a graph . Then the distance pattern of and is the same and , which is a contradiction to the concept of distance pattern distinguishing set. Hence, (c) holds.
(d) Let be a cycle on vertices. By Theorem 6, ; is not a distance pattern distinguishing graph. Also from (b) and (c), it follows that distance pattern distinguishing number of a cycle is not equal to one or two. Consider . Then, the rows representing the -distance neighborhood pattern of (taken in order) in , are given as follows.
Case 1. is an even integer and . Consider
Case 2. is an odd integer and . Consider In both of the cases it can be seen that none of the rows are identical and so forms a distance pattern distinguishing set of , which yields the desired result.

Theorem 8 motivates one to raise the following problem of theoretical interest.

Problem 1. Given a natural number , other than , does there exist a graph whose distance pattern distinguishing number is ? Is unique for that ?

The following three theorems establish sharp bounds for the distance pattern distinguishing number of a graph.

Theorem 9. Let be a graph of order with diameter and distance pattern distinguishing number . Then, .

Proof. Let be a graph with diameter and distance pattern distinguishing number . Then, for all , where is an injective function.
Upper Bound. Let . There is at most choices for and . By Theorem 8, there is no distance pattern distinguishing set of cardinality and therefore we exclude all 2-element sets, for which one of the two elements is zero, from the choices. Thus, the upper bound holds.
Lower Bound. For the lower bound, since is injective, each vertex in has distinct distance pattern of cardinality at most . Hence, has at most vertices. But , which implies that .

The inequality given in the lower bound of Theorem 9 can be strict. For example, by Theorem 8, the cycle of an order and diameter has the distance pattern distinguishing number , but . On the other hand, the path shows that the lower bound in Theorem 9 can be sharp since has order and diameter , while either end-vertex of constitutes a distance pattern distinguishing set and so and .

The upper bound in Theorem 9 can be attained for the path of order and diameter for which the distance pattern distinguishing number . But the upper bound cannot be sharp for the paths , .

It can be seen that every distance pattern distinguishing set of a graph is a resolving set of that graph. But not every resolving set is a distance pattern distinguishing set; the smallest counterexample is . Hence, the distance pattern distinguishing number of a graph may be the same as the metric dimension of that graph. Chartrand et al. obtained a sharp lower bound for the metric dimension of a graph in terms of maximum degree of [4]. By similar arguments, it can be shown that the same bound holds for distance pattern distinguishing number also. We exclude the proof in this case.

Theorem 10. If is a distance pattern distinguishing graph with diameter and maximum degree , then .

The lower bound in Theorem 10 can be attained for graphs . On the other hand, if , then the lower bound given in Theorem 10 cannot be sharp.

Remark 11. The upper bound for the metric dimension of a graph in terms of diameter of given by Chartrand et al. [4] is not valid for distance pattern distinguishing number of that graph. For example, let be a graph obtained from by attaching a path of length two to an arbitrary vertex of . Then, , which implies that the inequality does not hold for the distance pattern distinguishing number of .

The result that follows uses the following definitions and notations recalled from [5].

Definition 12. Let be a tree and let be a specified vertex in . Partition the edges of by the equivalence relation defined as follows: two edges if and only if there is a path in including and that does not have as an internal vertex. The subgraphs induced by the edges of the equivalent classes of are called the bridges of relative to . For each vertex of a tree , the legs at are the bridges which are paths. We use to denote the number of legs at .

Theorem 13. For a tree , .

Proof. Let be a tree with distance pattern distinguishing set . Consider any vertex with . Then at least legs of contain vertices in . Otherwise, let and be two legs of whose vertices are not the elements of . Then the neighbors of in those legs have the same distance pattern with respect to , a contradiction. Therefore, for each vertex at least are in . Since is not a path, the legs corresponding to distinct vertices are disjoint. Therefore, distance pattern distinguishing number is at least the sum stated above.

The following lemma gives a class of graphs attaining the lower bound in Theorem 13.

Definition 14. An olive tree is a rooted tree that consisted of branches, and the branch is a path of length .

Lemma 15. Olive tree with branches has distance pattern distinguishing number .

Proof. Since and , by Theorem 8, . Let be an olive tree with branches , where and let be its root vertex. Then, and for all , . Let be a distance pattern distinguishing set in . Then by Theorem 13, contains vertices from at least branches and . Now, we prove the lemma by showing that pendant vertices from the branches, , form a distance pattern distinguishing set.
We denote the vertices on the branch of successively from the vertex adjacent to to the pendant vertex of the branch as . Let . Then, the rows corresponding to the vertices in the th branch together with the row corresponding to form a submatrix of as follows: where in the first row appears at the positions and from the second row onwards the entry at each position is shifted one position to the left. Rows corresponding to the vertices in the branch form a submatrix of as follows: where in the first row appears at the first, positions. From the second row onwards the entry at the first position is shifted one position to the right and the entry at positions is shifted one position to the left. When , the rows corresponding to the vertices in the branch form a submatrix of are as follows: where in the first row appears at the first, positions. From the second row onwards the entry at the first position is shifted one position to the right and the entry at positions is shifted one position to the left.
Thus, all the rows in are nonidentical and hence, is a distance pattern distinguishing set. Hence, an olive tree with branches has distance pattern distinguishing number .

Lemma 16. Let with be a complete graph with . Let be the graph obtained from by attaching path to for all . Then, .

Proof. Let be a complete graph with . Let be the graph obtained from by attaching path to for all . Then, at least paths contain vertices in . Otherwise, let and be two paths whose vertices are not in . Then, , a contradiction. Therefore, at least vertices are in . Choose . Then, the rows corresponding to the vertices in the path ; in are of the form where in the first row the entry appears at the columns. From the second row onwards the entry in the first position shifts one position to the right and the entry in positions shifts one position to the left. The rows corresponding to the vertices of the path in are of the form where in the first row the entry appears at the columns. From the second row onwards the entry in the first position shifts one position to the right and the entry in positions shifts one position to the left. It is easy to see that the rows in are nonidentical and therefore, is a distance pattern distinguishing set.

Theorem 17. Given any positive integer , there exists a graph with . Furthermore, is not unique except for .

Proof. By Theorem 8, is the only graph with . From Lemmas 15 and 16 we have that, given any positive integer , there exists more than one class of graphs with .

3. Algorithm

Let denote the distance matrix of a graph and let with . Let be an submatrix of whose columns correspond to the vertices in . Then each row of (considered as a set) gives the distance patterns of the corresponding vertices in . Thus, we can check whether is a distance pattern distinguishing set or not. We design the following algorithm to determine whether a graph is a distance pattern distinguishing graph or not and to determine the distance pattern distinguishing number of that graph.

Algorithm 18.
Preprocess. Apply Floyd-Warshall algorithm to compute the distance matrix of .
Step 1 (input: distance matrix). Input is the distance matrix of the graph of order .
Step 2 (selection of columns of to find all the distance pattern distinguishing sets of ). Select all and submatrices of . By Theorem 8, cardinality of a distance pattern distinguishing set is not equal to .
Step 3 (formation of distance patterns of vertices in from the rows of ). Make distance patterns of the vertices in by considering each row of as sets.

Step 4 (identify the distance pattern distinguishing sets). If all the distance patterns are distinct, then a set of vertices corresponding to the columns in form the distance pattern distinguishing set. Otherwise, is not a distance pattern distinguishing set.

Step 5 (find distance pattern distinguishing number of ). If there is no distance pattern distinguishing set for , then distance pattern distinguishing number of is zero. Otherwise, distance pattern distinguishing number of is the minimum cardinality of distance pattern distinguishing sets.

The following examples illustrate the correctness and efficiency of the algorithm given above.

Example 19. By Theorem 8, the distance pattern distinguishing number of a path is . Now we calculate the same for of Figure 2 using the above algorithm.
Step 1 (input: distance matrix of ). Consider
Step 2 (selection of all and submatrices of ). Consider
Step 3 (formation of distance patterns of vertices of from the rows of above submatrices). Possible distance patterns of the vertices of are , , , and .
Step 4 (identify the distance pattern distinguishing sets from Step 3). From Step 3, and give distinct distance patterns for the vertices of and hence either the vertices or forms a distance pattern distinguishing set.
Step 5 (find distance pattern distinguishing number of ). From Step 4, distance pattern distinguishing number of is .

Example 20. By Theorem 6, the cycle is not a distance pattern distinguishing graph.
Step 1 (input: distance matrix of ). Consider
Step 2 (selection of all and submatrices of ). Consider
Step 3 (formation of distance patterns of vertices of from the rows of above submatrices). Possible distance patterns of the vertices of are
Step 4 (identify the distance pattern distinguishing sets from Step 3). From Step 3, it can be seen that at least one set is repeated in all the above given distance patterns. Hence, has no distance pattern distinguishing sets.

Remark 21. An R program has been developed based on this algorithm and calculated the distance pattern distinguishing number of several classes of graphs. The same program tested whether the input graph is distance pattern distinguishing graph or not.

Appendix

An R program for finding the distance pattern distinguishing number of a graph when the user supplies the distance matrix of that graph has been developed. See Algorithm 1.

dpd function ( )
{
     # dimension of
     # outer loop for times
  {
  
       # inner loop for sub matrices
  
{
    # sub matrix of columns
 {
    # checking distinct rows
   {
   
   
    &&
  {
   
  }
   }
 }
{
}
{
    # find dpd set
(“is a distance pattern distinguishing set”)
}
}
}
 {
(“Given graph has no distance pattern distinguishing sets”)
 }
(“distance pattern distinguishing number is”)   # find dpd number
}

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work reported in this note is a part of the research work done under Project no. SR/S4/MS:287/05 funded by the Department of Science & Technology (DST). The authors would like to thank Sajin Gopi for helping us to establish an R program, for determining the distance pattern distinguishing number of a graph, from the algorithm given in Section 3.