On the Distance Pattern Distinguishing Number of a Graph

Let G = (V, E) be a connected simple graph and letM be a nonempty subset of V. TheM-distance pattern of a vertex u in G is the set of all distances from u to the vertices inM. If the distance patterns of all vertices in V are distinct, then the setM is a distance pattern distinguishing set of G. A graph G 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.


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 V with respect to a -tuple of vertices  = (V 1 , V 2 , . . ., V  ) as ((V, V 1 ), (V, V 2 ), . . ., (V, V  )), where (V, V  ) denotes the distance of the vertex V from the vertex V  .Thus, entries in the code of a vertex may vary from 0 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.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 (0, 1)-matrix that is defined as follows.
Also, and the -distance pattern neighborhood matrix is ) . (3)

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  ≥ 7.
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. (a) The trivial graph  1 is the only graph with distance pattern distinguishing number as the order of that graph.
(b) Path is the only graph with distance pattern distinguishing number one.
(c) There exists no graph with distance pattern distinguishing number 2.
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 1; otherwise, the vertices adjacent to  have the same distance pattern as {1}.Since  is not a path, it contains a vertex whose degree is at least three.Let V be such a vertex of  with the least (, V) and let  = {V 1 , V 2 , . . ., V  } be the set of all vertices adjacent to V.Then, the distance pattern of each of the vertices in  is any one of { − 1}, {}, or { + 1}, where  = (, V).None of the vertices in  may have the distance pattern {}, as it is the distance pattern of the vertex V. Therefore, since || ≥ 3, at least two vertices in  have the same distance pattern, a contradiction.
(c) Let two vertices  and V form a distance pattern distinguishing set of a graph .Then the distance pattern of  and V is the same and   () =   (V) = {0, (, V)}, which is a contradiction to the concept of distance pattern distinguishing set.Hence, (c) holds. (d a cycle on  vertices.By Theorem 6,   ;  ≤ 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.
, are given as follows.
Theorem 8 motivates one to raise the following problem of theoretical interest.
Problem 1.Given a natural number , other than 2, 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,  + 1 − 2  ≤  ≤ 2  − ( + 1).
Upper Bound.Let  ∈ .There is at most 2  − 1 choices for   () and 0 ∈   ().By Theorem 8, there is no distance pattern distinguishing set of cardinality 2 and therefore we exclude all 2-element sets, for which one of the two elements is zero, from the choices.Thus, the upper bound  ≤ 2  − ( + 1) 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 The inequality given in the lower bound of Theorem 9 can be strict.For example, by Theorem 8, the cycle  7 of an order  = 7 and diameter  = 3 has the distance pattern distinguishing number ( 7 ) = 3, but  + 1 − 2  = 0. On the other hand, the path  2 shows that the lower bound in Theorem 9 can be sharp since  2 has order  = 2 and diameter  = 1, while either end-vertex of  2 constitutes a distance pattern distinguishing set and so  + 1 − 2  = 1 and ( 2 ) = 1.
The upper bound in Theorem 9 can be attained for the path  3 of order  = 3 and diameter  = 2 for which the distance pattern distinguishing number ( 3 ) = 1.But the upper bound cannot be sharp for the paths   ,  ≥ 4.
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  3 .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.
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  4 by attaching a path of length two to an arbitrary vertex of  4 .Then, () = 4, 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 V be a specified vertex in .Partition the edges of  by the equivalence relation = V defined as follows: two edges  = V  if and only if there is a path in  including  and  that does not have V as an internal vertex.The subgraphs induced by the edges of the equivalent classes of  are called the bridges of  relative to V. For each vertex V ∈  of a tree  = (, ), the legs at V are the bridges which are paths.We use  V to denote the number of legs at V. Theorem 13.For a tree  ≇   ,  ≥ ∑ V∈: V >1 ( V − 1).
Proof.Let  ≇   be a tree with distance pattern distinguishing set .Consider any vertex V with  V > 1.Then at least  V − 1 legs of V contain vertices in .Otherwise, let  1 and  2 be two legs of V whose vertices are not the elements of .Then the neighbors of V in those legs have the same distance pattern with respect to , a contradiction.Therefore, for each vertex V ∈  at least  V − 1 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 th branch is a path of length .
We denote the vertices on the th branch of  successively from the vertex adjacent to V to the pendant vertex of the branch as Then, the rows corresponding to the vertices in the th branch together with the row corresponding to V form a submatrix of  *   as follows: where in the first row 1 appears at the ( + 2)th, ( + 3)th, . . ., (2)th positions and from the second row onwards the entry 1 at each position is shifted one position to the left.Rows corresponding to the vertices in the ( − 1)th branch form a submatrix of  *   as follows: where in the first row 1 appears at the first, ( + 1)th, ( + 2)th, . . ., (2 − 2)th positions.From the second row onwards the entry 1 at the first position is shifted one position to the right and the entry 1 at ( + 2)th, ( + 3)th, . . ., (2 − 1)th positions is shifted one position to the left.When 3 ≤  ≤ ( − 2), the rows corresponding to the vertices in the th branch form a submatrix of  *   are as follows: ( where in the first row 1 appears at the first, ( + 2)th, ( + 3)th, . . ., (2)th, (2 + 2)th, (2 + 3)th, . . ., ( + )th positions.
From the second row onwards the entry 1 at the first position is shifted one position to the right and the entry 1 at ( + 2)th, ( + 3)th, . . ., (2)th, (2 + 2)th, (2 + 3)th, . . ., ( + )th 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  − 1.
Proof.By Theorem 8,   is the only graph with  = 1.From Lemmas 15 and 16 we have that, given any positive integer  ≥ 3, there exists more than one class of graphs with  = .

Algorithm
Let  = (  ) denote the distance matrix of a graph  and let 0 ̸ =  ⊆  with || = .Let   be an  × ; 1 ≤  ≤  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  ×  : 1 ≤  ≤  and  ̸ = 2 submatrices   of .By Theorem 8, cardinality of a distance pattern distinguishing set is not equal to 2.
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 1.Now we calculate the same for  3 of Figure 2 using the above algorithm.

Definition 1 .
For an arbitrarily fixed vertex  in  and for any nonnegative integer , let   [] = {V ∈ () : (, V) = } and   [] = () − (C  ) whenever  exceeds the eccentricity () of  in the component C  to which  belongs.Thus, if  is connected, then   [] = 0 if and only if  > ().We can generalize this concept as, given an arbitrary nonempty subset  ⊆ () and for each  ∈ (),    () = {V ∈  : (, V) = }.Definition 2. Let  be a given connected simple graph, 0 ̸ =  ⊆ (), and  ∈ ().Then, the -distance pattern of  is the set   () = {(, V) : V ∈ }.Clearly,   () = { :    [] ̸ = 0}.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 ().