STUDIES IN GRAPH THEORY - DISTANCE RELATED CONCEPTS IN GRAPHS. R. ANANTHA KUMAR (Reg. No ) DOCTOR OF PHILOSOPHY

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1 STUDIES IN GRAPH THEORY - DISTANCE RELATED CONCEPTS IN GRAPHS A THESIS Submitted by R. ANANTHA KUMAR (Reg. No ) In partial fulfillment for the award of the degree of DOCTOR OF PHILOSOPHY FACULTY OF SCIENCE AND HUMANITIES KALASALINGAM UNIVERSITY (KALASALINGAM ACADEMY OF RESEARCH AND EDUCATION) ANAND NAGAR, KRISHNANKOIL TAMIL NADU, INDIA. SEPTEMBER 2013

2 CERTIFICATE This is to certify that all the corrections/suggestions pointed out by the examiners have been incorporated in the thesis of Mr. R. ANANTHA KUMAR. Place: Krishnankoil Date: SIGNATURE Dr. S. ARUMUGAM SUPERVISOR Senior Professor (Research) National Centre for Advanced Research in Discrete Mathematics (n-cardmath) Kalasalingam University (Kalasalingam Academy of Research and Education) Anand Nagar, Krishnankoil , Tamil Nadu, INDIA.

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4 KALASALINGAM UNIVERSITY (Kalasalingam Academy of Research and Education) ANAND NAGAR, KRISHNANKOIL BONAFIDE CERTIFICATE Certified that this thesis titled STUDIES IN GRAPH THEORY - DISTANCE RELATED CONCEPTS IN GRAPHS is the bonafide work of Mr. R. ANANTHA KUMAR, who carried out the research under my supervision. Certified further that to the best of my knowledge the work reported herein does not form part of any other thesis or dissertation on the basis of which a degree or award was conferred on an earlier occasion of this or any other candidate. SIGNATURE Dr. S. ARUMUGAM SUPERVISOR Senior Professor (Research) National Centre for Advanced Research in Discrete Mathematics (n-cardmath) Kalasalingam University (Kalasalingam Academy of Research and Education) Anand Nagar, Krishnankoil , Tamil Nadu, INDIA.

5 ABSTRACT By a graph G = (V, E), we mean a finite undirected graph with neither loops nor multiple edges. The order and size of G are denoted by n = V and m = E respectively. For graph theoretic terminology we refer to Chartrand and Lesniak [7]. In Chapter 1, we collect some basic definitions and theorems on graphs which are needed for the subsequent chapters. The distance d(u, v) between two vertices u and v of a connected graph G is the length of a shortest u-v path in G. There are several distance related concepts and parameters such as eccentricity, radius, diameter, convexity and metric dimension which have been investigated by several authors in terms of theory and applications. An excellent treatment of various distances and distance related parameters are given in Buckley and Harary [6]. Let G = (V, E) be a graph. Let v V. The open neighborhood N(v) of a vertex v is the set of vertices adjacent to v. Thus N(v) = {w V : wv E}. The closed neighborhood of a vertex v, is the set N[v] = N(v) {v}. For a set S V, the open neighborhood N(S) is defined to be N(v). For any two disjoint subsets v S A, B V, let [A, B] denote the set of all edges with one end in A and the other end in B. For any set C V, the induced subgraph C is the maximal subgraph of G with vertex set C.

6 Saenpholphat and Zhang [25] introduced the concept of connected resolving set and in this context they introduced the concept of distance similar vertices. Two vertices u and v of a connected graph G are defined to be distance similar if d(u, x) = d(v, x) for all x V (G) {u, v}. Hence two vertices u and v in a connected graph G are distance similar if and only if either uv / E(G) and N(u) = N(v) or uv E(G) and N[u] = N[v]. Also, distance similarity in a connected graph G is an equivalence relation on V (G). Therefore this relation gives a unique partition of V (G). We observe that if U is a distance similar equivalence class of G, then {d(x, v) : v U} = 1 for all x V U. These observations lead to the following concepts. Let G = (V, E) be a connected graph. A proper subset S of V is called a distance similar set if {d(u, v) : v S} = 1 for all u V S. A distance similar set S is called a maximal distance similar set if any set S 1 with S S 1 V, is not a distance similar set of G. The maximum cardinality of a maximal distance similar set of G is called the distance similar number of G and is denoted by ds(g). The minimum cardinality of a maximal distance similar set in G is called the lower distance similar number of G and is denoted by ds (G). Any distance similar set S of G with S = ds(g) is called a ds-set of G. A nonempty subset S of V is called a pairwise distance similar set (pds-set) if either S = 1 or any two vertices in S are distance similar. The maximum cardinality of a maximal pairiv

7 wise distance similar set in G is called the pairwise distance similar number of G and is denoted by Φ(G). The minimum cardinality of a maximal pairwise distance similar set in G is called the lower pairwise distance similar number of G and is denoted by Φ (G). Let V 1, V 2,..., V k be the distance similar equivalence classes of G. Then Φ(G) = max 1 i k V i and Φ (G) = min 1 i k V i. In Chapter 2, we obtain a condition for S V to be a distance similar set of a graph G and a condition for a maximal distance similar set to be a distance similar equivalence class of G. We characterize bipartite graphs and unicyclic graphs with ds(g) = 1. We give a relation connecting dim(g) and cardinality of maximal distance similar sets. We characterize graphs with distance similar number equal to (G), n 2, n 3 and d(g). We also determine the distance similar number for several product graphs. We show that the set of all maximal distance similar sets which are contained in any neighborhood N(u) forms a partition of N(u). We also prove that the distance similar number of any graph can be computed in polynomial time. In Chapter 3, we initiate a study of pairwise distance similar set and pairwise distance similar number of a graph. Let Φ(G) and Φ (G) denote the pairwise distance similar number and lower pairwise distance similar number of a graph G. We present several basic results on these parameters. We obtain a characterization of graph with Φ(G) = (G) and Φ (G) = (G). We present sharp v

8 lower and upper bounds of Φ(G) for product graphs. Further we characterize graphs with Φ(G) = n 2 and Φ(G) = n 3. One of the basic problems in graph theory is to select a minimum set S of vertices in such a way that each vertex in the graph is uniquely determined by its distances to the chosen vertices. This concept was introduced by Slater [29] who called such a set as a locating set. The same concept was independently discovered by Harary and Melter [16] who used the term resolving set. By an ordered set of vertices we mean a set W = {w 1, w 2,, w k } on which the ordering (w 1, w 2,, w k ) has been imposed. For an ordered subset W = {w 1, w 2,, w k } of V (G), we refer to the k-vector (ordered k-tuple) r(v W ) = (d(v, w 1 ), d(v, w 2 ),, d(v, w k )) as the (metric) representation of v with respect to W. The set W is called a resolving set for G if r(u W ) = r(v W ) implies u = v for all u, v V (G). Hence if W is a resolving set of cardinality k for a graph G of order n, then the set {r(v W ) : v V (G)} consists of n distinct k-vectors. A resolving set of minimum cardinality is called a basis for G. The metric dimension of G is defined to be the cardinality of a basis of G and is denoted by dim(g). The definition of metric dimension of a graph depends on the concept of resolving set W, in which an order is imposed on the elements of W. We define the concept distance pattern distinguishing sets in which W is considered simply as a set and for any vertex v the set of all distances from v to W is associated. vi

9 Let M be a nonempty subset of V (G) and let u V (G). The distance pattern of u with respect to M is given by f M (u) = {d(u, v) : v M}. If f M is an injective function on V, then the set M is called a distance pattern distinguishing set (DP D-set) of G. If G admits a DP D-set, then G is called a DP D-graph. The minimum cardinality of a DP D-set in a DP D-graph G is the DP D-number of G and is denoted by ϱ(g). In Chapter 4, we study distance pattern distinguishing sets and distance pattern distinguishing number of a graph. We obtain several necessary conditions for distance pattern distinguishing sets, and characterize some families of graphs which admit a distance pattern distinguishing set. We also obtain distance pattern distinguishing number of several families of graphs. Further, we give a relation connecting distance pattern distinguishing number and metric dimension of a graph. We define the concept local distance pattern distinguishing set (LDP D-set) and LDP D-number of a graph G. We obtain LDP D-number of several families of graphs. Maximum matchings in bipartite graphs have several interesting applications. Let G = (V, E) be a bipartite graph with bipartition V 1 and V 2, where V 1 V 2. Hall [15] proved that there exists a matching M in G such that V 1 is matched to a subset of V 2 under M if and only if N(S) S for all S V 1. The condition N(S) S for all S V is called Hall s condition. Motivated by this condition Chartrand and Lesniak [7] defined a subset U of V to be nondeficient if N(S) S for every vii

10 nonempty subset S of U. We introduce the concept of nondeficient number of G. The nondeficient number nd(g) of a graph G is defined to be the maximum cardinality of a nondeficient set of G. Any nondeficient set U of G with U = nd(g) is called a nd-set of G. In Chapter 5, we initiate a study of nondeficient number of graphs. We also present sharp lower and upper bounds for nondeficient number. We obtain several bounds for nd(g) in terms of well known graph theoretic parameters such as β 0 and β 1. Further we give a relation connecting nondeficient number and order of a maximum critical independent set. viii

11 ACKNOWLEDGEMENT First of all, I thank Almighty God for His abundant blessings. I am indebted and grateful to my supervisor Prof. S. Arumugam, Director, n-cardmath, Kalasalingam University, Krishnankoil for his invaluable guidance, inspiration and fruitful discussions during the course of this research work. My special thanks are due to our collaborators Prof. Dr. B.D. Acharya and Dr. K.A. Germina. S. B. Rao, I wish to express my thanks to the Department Research Committee Members for their helpful suggestions. I thank the Department of Science and Technology (DST), Government of India, New Delhi for providing financial assistance through Research Fellowship during the period August August 2011 at Kalasalingam University. I am extremely thankful to the Chairman, A.K. Group of Institutions Kalvivallal Thiru T. Kalasalingam, Chancellor Ilayavallal Thiru K. Sridharan and Vice-Chancellor Dr. S. Saravanasankar, Kalasalingam University for providing the necessary facilities during the period of my research. I also thank the staff members and research scholars of n-cardmath for their kind cooperation. Words are inadequate to express my gratitude to my parents and other members of my family for their affectionate blessings and support throughout my academic career. I heartily thank my friends for their constant encouragement to finish this work successfully. affection. It is my pleasure to thank Prof. S. Arumugam s family for their Finally, I extend my thanks to Mr. K. Alaguraj, for typesetting the thesis in an excellent manner. R.ANANTHA KUMAR ix

12 TABLE OF CONTENTS TITLE PAGE NO. ABSTRACT iii LIST OF FIGURES xii LIST OF SYMBOLS xiii CHAPTER 1. PRELIMINARIES INTRODUCTION BASIC GRAPH THEORY DISTANCE RELATED CONCEPTS ORGANIZATION OF THE THESIS 13 CHAPTER 2. DISTANCE SIMILAR SETS 15 IN GRAPHS 2.1 INTRODUCTION BASIC RESULTS AN ALGORITHM FOR COMPUTING ds(g) GRAPHS WITH ds(g) = DISTANCE SIMILAR SETS AND RESOLVING SETS BOUNDS FOR DISTANCE SIMILAR NUMBER DISTANCE SIMILAR SETS AND GRAPH OPERATIONS CONCLUSION AND SCOPE 38 x

13 TITLE PAGE NO. CHAPTER 3. PAIRWISE DISTANCE SIMILAR 39 SETS IN GRAPHS 3.1 INTRODUCTION BASIC RESULTS CONCLUSION AND SCOPE 48 CHAPTER 4. DISTANCE PATTERN 49 DISTINGUISHING SETS IN GRAPHS 4.1 INTRODUCTION BASIC RESULTS DP D-NUMBER OF GRAPHS METRIC DIMENSION AND DP D-NUMBER EMBEDDING A GRAPH INTO A DP D-GRAPH LOCAL DP D-SETS LDP D-GRAPHS AND UNIVERSAL VERTICES CONCLUSION AND SCOPE 90 CHAPTER 5. NONDEFICIENT SETS IN GRAPHS INTRODUCTION BASIC RESULTS BOUNDS NONDEFICIENT SETS AND GRAPH OPERATIONS CONCLUSION AND SCOPE 112 REFERENCES 113 LIST OF PUBLICATIONS 117 VITAE 118 xi

14 LIST OF FIGURES FIGURE TITLE PAGE NO. NO. 2.1 Graph with ds = 4 and ds = Example for distance similarity is not hereditary Example for illustration of Algorithm Graph with Φ = 4 and Φ = Graphs with unequal and equal Φ and Φ Graph with ds = 4 and Φ = Graph with Φ = n DP D-Graph Graph which is LDP D but not DP D Tree with n ( I c N(I c ) ) = 4 and nd(t ) = Graph G xii

15 LIST OF SYMBOLS ω α deg(v) or d(v) d(g) or diam(g) dim(g) ϱ(g) ds(g) β 1 g β 0 or α l(p ) ds (G) Φ (G) δ µ(g) nd(g) η(g) Φ(G) r rank(g) T k (G) clique number critical independence number degree of v diameter dimension of G distance pattern distinguishing number of G distance similar number of G edge independence number girth independence number length of the path P lower distance similar number of G lower pairwise distance similar number of G maximum degree minimum degree Mycielskian of a graph G nondeficient number of G nullity of G pairwise distance similar number of G radius rank of G trestled graph of G of index k xiii

16 CHAPTER 1 PRELIMINARIES 1.1 INTRODUCTION In this chapter we collect some basic definitions and theorems on graphs which are needed for the subsequent chapters. For graph theoretic terminology, we refer to Chartrand and Lesniak [7]. In Section 1.2 we present some of the basic definitions in graph theory. In Section 1.3 we present the fundamentals of distance similar vertices and nondeficient sets in graphs. In Section 1.4 we give an overview of the organization of the remaining chapters of the thesis. 1.2 BASIC GRAPH THEORY Definition A graph G is a finite nonempty set of objects called vertices together with a set of unordered pairs of distinct vertices of G called edges. The vertex set and the edge set of G are denoted by V (G) and E(G) respectively. The edge e = {u, v} is said to join the vertices u and v. We write e = uv and say that

17 u and v are adjacent vertices; u and e are incident, as are v and e. If e 1 and e 2 are distinct edges of G incident with a common vertex, then e 1 and e 2 are adjacent edges. The number of vertices in G is called the order of G and the number of edges in G is called the size of G. The order and size of G are denoted by n and m respectively. A graph is trivial if its vertex set is a singleton. Definition Let G = (V, E) be a graph and let v V. A vertex u is called a neighbor of v in G if uv is an edge of G. The set N(v) of all neighbors of v is called the open neighborhood of v. Thus N(v) = {u V : uv E}. The closed neighborhood of v in G is defined as N[v] = N(v) {v}. If S V, then N(S) = N(v) and N[S] = N(S) S. Definition The degree of a vertex v in a graph G is defined to be the number of edges incident with v and is denoted by deg(v) or d(v). In other words deg(v) = N(v). The minimum and maximum degrees of vertices of G are denoted by δ and respectively. A vertex of degree zero in G is called an isolated vertex and a vertex of degree one is called a pendant vertex or a leaf. An edge e in a graph G is called a pendant edge if it is incident with a pendant vertex. Any vertex which is adjacent to a pendant vertex is called a support vertex. A vertex of degree n 1 is called an universal vertex. 2 v S

18 Definition A walk W in a graph G is an alternating sequence u 0, e 1, u 1,..., u k 1, e k, u k of vertices and edges of G, beginning and ending with vertices, such that e i = u i 1 u i, for 1 i k. The vertices u 0 and u k are called the origin and terminus of W respectively and W is called a u 0 -u k walk. The walk W is also denoted by (u 0, u 1, u 2,..., u k 1, u k ). If u 0 = u k, the walk is closed and open otherwise. The number of edges in a walk is called the length of the walk. A path P of length k (denoted by P k ) is a walk (u 0, u 1, u 2,..., u k 1, u k ) in which all the vertices u 0, u 1, u 2,..., u k 1, u k are distinct. Definition A cycle C k of length k 3 in a graph G is a closed walk (u 0, u 1, u 2,..., u k 1, u 0 ) in which all the vertices u 0, u 1, u 2,..., u k 1 are distinct. A cycle C k is called even or odd according as k is even or odd. A graph G having no cycle is called an acyclic graph. A graph having exactly one cycle is called an unicyclic graph. The length of a shortest cycle (if any) in a graph G is called its girth and denoted by g(g). Definition A graph G is said to be connected if every pair of distinct vertices of G are joined by a path. A graph G that is not connected is called a disconnected graph. A maximal connected subgraph of G is called a component of G. Thus a disconnected graph is a graph having more than one component. 3

19 Definition A graph G is complete if every pair of distinct vertices of G are adjacent in G. A complete graph on n vertices is denoted by K n. A clique in G is a complete subgraph of G. The maximum order of a clique in G is called the clique number of G and is denoted by ω(g) or simply ω. A clique H in G with V (H) = ω is called a maximum clique in G. Definition The complement G of a graph G is the graph with vertex set V (G) such that two vertices are adjacent in G if and only if they are not adjacent in G. Definition A nontrivial connected graph having no cut vertex is called a block. A block of a graph G is a subgraph of G which itself is a block and is maximal with respect to this property. For any real number x, x denotes the largest integer less than or equal to x and x denotes the smallest integer greater than or equal to x. Definition A graph H is called a subgraph of G if V (H) V (G) and E(H) E(G). A subgraph H of a graph G is a proper subgraph of G if either V (H) V (G) or E(H) E(G). A spanning subgraph of G is a subgraph H of G with V (H) = V (G). 4

20 For a set S of vertices of G, the induced subgraph denoted by S or by G[S], is the maximal subgraph of G with vertex set S. Thus two vertices of S are adjacent in S if and only if they are adjacent in G. For any two disjoint subsets A, B V, let [A, B] denote the set of all edges with one end in A and the other end in B. Definition Two graphs G 1 and G 2 are said to be isomorphic (written as G 1 = G2 ) if there exists a bijection φ : V (G 1 ) V (G 2 ) such that uv E(G 1 ) if and only if φ(u)φ(v) E(G 2 ). Such a function φ is called an isomorphism from G 1 to G 2. Definition A bipartite graph G = (X, Y, E) is a graph whose vertex set V (G) can be partitioned into two nonempty subsets X and Y such that each edge of G has one end in X and the other end in Y. The pair (X, Y ) is called a bipartition of G. Definition A complete bipartite graph K r,s is a bipartite graph G with bipartition X, Y such that X = r, Y = s and every vertex in X is adjacent to every vertex in Y. The graph K 1,r is called a star. When r > 2, the vertex of degree r in K 1,r is called its center. The graph G obtained from K 1,r and K 1,s by joining their centers by an edge is called a bistar and is denoted by B(r, s). 5

21 Definition A k-partite graph G is a graph whose vertex set V can be partitioned into k nonempty subsets V 1, V 2,..., V k such that no edge has both ends in any one subset V i. If further G contains every edge uv where u V i, v V j, i j, then G is called a complete k-partite graph and is denoted by K n1,n 2,...,n k where V 1 = n 1, V 2 = n 2,..., V k = n k. Definition A subset S V is said to be independent if no two vertices in S are adjacent. The independence number β 0 (G) is the maximum cardinality of an independent set in G. Definition A connected acyclic graph is called a tree. A disconnected graph in which each component is a tree is called a forest. Definition Two graphs G 1 and G 2 are said to be disjoint if they have no vertex in common. They are said to be edge disjoint if they have no edge in common. Definition vertex sets. Let G 1 and G 2 be two graphs with disjoint 1. The union G = G 1 G 2 has V (G) = V (G 1 ) V (G 2 ) and E(G) = E(G 1 ) E(G 2 ). A graph G consisting of k, k 2 disjoint copies of a graph H is denoted by G = kh. 6

22 2. The join G = G 1 +G 2 has V (G) = V (G 1 ) V (G 2 ) and E(G) = E(G 1 ) E(G 2 ) {uv : u V (G 1 ), v V (G 2 )}. 3. The Cartesian product G = G 1 G 2 has V (G) = V (G 1 ) V (G 2 ), and two vertices (u 1, u 2 ) and (v 1, v 2 ) of G are adjacent in G if and only if either u 1 = v 1 and u 2 v 2 E(G 2 ) or u 2 = v 2 and u 1 v 1 E(G 1 ). 4. The corona of G 1 and G 2 is defined to be the graph G = G 1 G 2 formed from one copy of G 1 and V (G 1 ) copies of G 2 where the i th vertex of G 1 is adjacent to every vertex in the i th copy of G 2. In particular, the corona H K 1 is the graph constructed from a copy of H, where for each vertex v V (H), a new vertex v and a pendant edge vv are added. 5. The Lexicographic product G H of two graphs G and H has V (G H) = V (G) V (H) and two vertices (u, x), (v, y) of G H being adjacent whenever uv E(G), or u = v and xy E(H). 6. For three or more disjoint graphs G 1, G 2,..., G n the sequential join G 1 + G G n is the graph (G 1 + G 2 ) (G 2 + G 3 ) (G n 1 + G n ). Definition A split graph is a graph whose vertex set can be partitioned into two sets V 1 and V 2 such that V 1 forms a complete graph and V 2 is an independent set. Definition The hypercube Q n is the graph whose vertices are the n-dimensional binary vectors, where two vertices are 7

23 adjacent if and only if they differ in exactly one coordinate. Alternatively, Q n is K 2 if n = 1, while for n 2, Q n = Q n 1 K 2. Definition [17] Given an arbitrary graph G, the trestled graph of index k, denoted by T k (G), is the graph obtained from G by adding k-copies of K 2 for each edge uv of G and joining u and v to the respective end vertices of each K 2. Definition [23] For a graph G = (V, E), the Mycielskian of G is the graph µ(g) with vertex set V V {u}, where V = {v i : v i V } and is disjoint from V and E = E {v i v j : v iv j E} {v i u : v i V }. Definition [13] Let G 0 be a graph with V (G 0 ) = {v 1, v 2,..., v k } and let G 1, G 2,..., G k be k disjoint graphs. The composition graph G = G 0 [G 1, G 2,..., G k ] is formed as follows: We replace each vertex v i in G 0 with the graph G i and make each vertex of G i adjacent to each vertex of G j whenever v i is adjacent to v j in G 0. In particular the graph P k [G 1, G 2,..., G k ] is called the sequential join of the graphs G 1, G 2,..., G k. Definition [9] The core of a graph G is obtained by successively deleting end vertices until none remain. 8

24 1.3 DISTANCE RELATED CONCEPTS One concept that pervades all of graph theory is that of distance, and distance is used in isomorphism testing, graph operations, extremal problem on connectivity and diameter. One of the fundamental problems in the study of chemical structure is to determine ways to represent a set of chemical compounds such that distinct compounds have distinct representations. This problem is solved by using the concept of resolving sets in [10]. Definition The distance d G (u, v) or d(u, v) between two vertices u and v of a connected graph G is defined to be the length of a shortest path joining u and v in G. The eccentricity of a vertex v of a connected graph G is defined as e(v) = max{d(u, v) : u V (G)}. The radius of G is defined as rad(g) = min{e(v) : v V (G)} and the diameter of G is defined as diam(g) = d(g) = max{e(v) : v V (G)}. Consequently, diam(g) is the maximum distance between any two vertices of G. In [29] and later in [30], Slater introduced the concept of a locating set for a connected graph G. He called the cardinality of a minimum locating set as the location number of G. Independently, Harary and Melter [16], discovered these concepts as well but used the term resolving set and metric dimension. Applications of resolving sets arise in various areas including coin weighing problem [28], drug discovery [10], robot navigation [19], network discovery and verification [2], connected joins in graphs [27] and strategies for the mastermind game [11]. 9

25 Definition Let G be a connected graph. By an ordered set of vertices we mean a subset W = {w 1, w 2,..., w k } V (G) on which the ordering (w 1, w 2,..., w k ) has been imposed. For an ordered subset W V (G), we refer to the k-vector (ordered k-tuple) r(v W ) = (d(v, w 1 ), d(v, w 2 ),..., d(v, w k )) as the metric representation of v with respect to W. The set W is called a resolving set for G if r(u W ) = r(v W ) implies that u = v for all u, v V (G). Hence if W is a resolving set of cardinality k for a graph G of order n, then the set {r(v W ) : v V (G)} consists of n distinct k-vectors. A resolving set of minimum cardinality for a graph G is called a basis for G. Definition The metric dimension of G is defined to be the cardinality of a minimum resolving set of G and is denoted by dim(g). A resolving set W of G is a minimal resolving set if no proper subset of W is a resolving set of G. The upper metric dimension of G is defined to be the maximum cardinality of a minimal resolving set of G and is denoted by dim + (G). For any connected graph G of order n, we have 1 dim(g) dim + (G) n 1. The minimum metric dimension problem is to find a basis of G. Garey and Johnson [10] noted that the minimum metric dimension problem is NP-complete for general graphs by a reduction from 3-dimensional matching. An explicit reduction from 3-SAT was given by Khuller et al. [19]. 10

26 The metric dimension of some standard graphs are listed below. [10] dim(g) = 1 if and only if G = P n. [10] dim(g) = n 1 if and only if G = K n, where n 2. [19] For the cycle C n, n 3, dim(c n ) = 2. [8] For the graph K r,s, r, s 1, dim(k r,s ) = r + s 2. Theorem [19] Let T = (V, E) be a tree which is not a path. For any v V, let l v denote the number of components S of T {v} such that the induced subgraph S {v} is a path with v as origin. Then dim(t ) = (l v 1). l v >1 Theorem [19] The metric dimension of a d-dimensional grid (d 2) is d. For a survey of results in metric dimension, we refer to Chartrand and Zhang [8] and Hernando et al. [18]. Definition [25] Two vertices u and v of a connected graph G are defined to be distance similar if d(u, x) = d(v, x) for all x V (G) {u, v}. Proposition [25] Two vertices u and v in a connected graph are distance similar if and only if either uv / E(G) and N(u) = N(v) or uv E(G) and N[u] = N[v]. Proposition [25] Distance similarity in a connected graph G is an equivalence relation on V (G). 11

27 Proposition [25] If U is a distance similar equivalence class of a connected graph G, then U is either independent in G or in G. Proposition [25] Let G be a nontrivial connected graph of order n. If G has k distance similar equivalence classes, then dim(g) n k. We observe that if U is a distance similar equivalence class of G, then {d(x, v) : v U} = 1 for all x V U. Definition [7] A collection {S 1, S 2,..., S n } of finite nonempty sets has a system of distinct representatives (SDR) if there exist n distinct elements x 1, x 2,..., x n such that x i S i, 1 i n. Theorem [7] A collection {S 1, S 2,..., S n } of finite nonempty sets has an SDR if and only if for each integer k with 1 k n, the union of any k of these sets contains at least k elements. Definition A set of pairwise independent edges of G is called a matching in G. The number of edges in a maximum matching of G is the edge independence number β 1 (G) of G. If M is a matching in a graph G with the property that every vertex of G is incident with an edge of M, then M is a perfect matching in G. Definition A vertex that is incident with no vertex of M is called an M-vertex. Let M be a matching in a graph G. An M-alternating path of G is a path whose edges are alternatively in 12

28 M and not in M. An M-augmenting path is an M-alternating path both of whose end vertices are M vertices. Definition [3] A subgraph H of G is called an elementary subgraph if every component of H is either a cycle or an edge. Definition [5] The rank and the nullity of a graph G, denoted by rank(g) and η(g) respectively, are defined to be the rank and nullity of the adjacency matrix of G. 1.4 ORGANIZATION OF THE THESIS Saenpholphat and Zhang [25] introduced the concept of connected resolving set and in this context they introduced the concept of distance similar vertices and distance similar equivalence class of a connected graph G. In Chapter 2, we introduce the concept of distance similar set and distance similar number ds(g) of a graph. We prove that ds(g) can be computed in polynomial time. We characterize bipartite graphs and unicyclic graphs with ds(g) = 1. We obtain a relation connecting dim(g) and ds(g). We also characterize graphs with distance similar number equal to (G), n 2, n 3 and d(g). We also determine the distance similar number of various graph products. Hernando et al. [18] called a pair of vertices satisfying the distance similarity condition as twins and introduced the concept of the twin graph of a graph G. Motivated by this, in Chapter 3, we initiate a study of pairwise distance similar set and pairwise distance 13

29 similar number Φ(G) of a graph. We obtain a characterization of graphs with Φ(G) = (G), Φ (G) = (G), Φ(G) = n 2 and Φ(G) = n 3. We obtain bounds for the pairwise distance similar number of product graphs. Chapter 4 is devoted to the study of distance pattern distinguishing sets and distance pattern distinguishing number of a graph. We characterize a few families of graphs which admit a distance pattern distinguishing set. We also determine distance pattern distinguishing number of several families of graphs. We present some embedding techniques to embed a given graph into a graph which admits a distance pattern distinguishing set. We also investigate the relation between distance pattern distinguishing number and metric dimension of a graph. Further, we initiate a study of local distance pattern distinguishing set (LDP D-set) and LDP D-number of a graph. Also we obtain LDP D-number of several families of graphs. Chartrand and Lesniak [7, Page 235] defined a subset U of V to be nondeficient if N(S) S for every nonempty subset S of U. In Chapter 5, using the concept we introduce the nondeficient number of a graph. We present sharp lower and upper bounds for the nondeficient number of a graph in terms of well known graph theoretic parameters. Further we obtain a relation connecting nondeficient number and the order of a maximum critical independent set of a graph. 14

30 CHAPTER 2 DISTANCE SIMILAR SETS IN GRAPHS 2.1 INTRODUCTION Saenpholphat and Zhang [25] introduced the concept of connected resolving set and in this context they introduced the concept of distance similar vertices and obtained several basic results. Two vertices u and v of a connected graph G are defined to be distance similar if d(u, x) = d(v, x) for all x V (G) {u, v}. Thus u and v are distance similar if and only if either uv / E(G) and N(u) = N(v) or uv E(G) and N[u] = N[v]. Distance similarity in a connected graph G is an equivalence relation on V (G). If U is a distance similar equivalence class of a connected graph G, then U is either independent in G or in G. Proposition [25] Let G be a nontrivial connected graph of order n. If G has k distance similar equivalence classes, then dim(g) n k. The content of this chapter has been accepted for publication in Utilitas Mathematica.

31 We observe that if U is a distance similar equivalence class of G, then {d(x, v) : v U} = 1 for all x V U. Motivated by this observation we introduce the concept distance similar set and distance similar number ds(g) of G and initiate a study of this parameter. We characterize bipartite graphs and unicyclic graphs with ds(g) = 1. We present several fundamental results on these concepts and also an algorithm which computes ds(g) in O(n 4 )-time. We also obtain a characterization of graphs with distance similar number equal to (G), n 2, n 3 and d(g). We determine the distance similar number of several graph products. 2.2 BASIC RESULTS Definition Let G = (V, E) be a connected graph. A proper subset S of V is called a distance similar set of G if {d(u, v) : v S} = 1 for all u V S. A distance similar set S is a maximal distance similar set if any set S 1 with S S 1 V, is not a distance similar set of G. The maximum cardinality of a distance similar set of G is the distance similar number of G and is denoted by ds(g). The minimum cardinality of a maximal distance similar set of G is called the lower distance similar number of G and is denoted by ds (G). Any distance similar set S of G with S = ds(g) is called a ds-set of G. 16

32 We start with an example to illustrate the concept of distance similar set and distance similar number. Example (1) For the graph G 1 given in Figure 2.1, S 1 = {a, b, c, d} and S 2 = {x} are maximal distance similar sets and ds(g 1 ) = 4 and ds (G 1 ) = 1. a b c d G 1 x Figure 2.1: Graph with ds = 4 and ds = 1. (2) Let T be a tree. For any support vertex v, let l(v) denote the number of leaves adjacent to v. Then ds(t ) = max{l(v) : v is a support vertex of T }. In particular for the path P n, we have ds(p n ) = ds (P n ) = 1 for all n 4. (3) For the complete bipartite graph K m,n with m, n 2 and with bipartition X, Y, both X and Y are maximal distance similar sets. Hence ds(k m,n ) = max{m, n} and ds (K m,n ) = min{m, n}. 2 if n = 3 or 4 (4) For the cycle C n, we have ds(c n ) = 1 if n 5. 17

33 Observation (1) If S is any distance similar set of G and u N(S) S, then u is adjacent to every vertex in S. Thus S N(u) and hence 1 ds (G) ds(g) (G). Also ds(g) = n 1 if and only if (G) = n 1. (2) A proper subset S of V (G) is a distance similar set of G if and only if the edge induced subgraph [S, N(S) S] is a complete bipartite graph. Observation Any distance similar equivalence class is obviously a distance similar set. However, the converse is not true. For the graph G 1 given in Figure 2.1, S = {a, b, c, d} is a maximal distance similar set. Since S is neither an independent set nor a clique in G 1, it follows from Proposition that S is not a distance similar equivalence class. Lemma Let S be a maximal distance similar set in G. Then S is a distance similar equivalence class if and only if S is an independent set or a clique in G. Proof. If S is a distance similar equivalence class, then the result follows from Proposition Conversely, let S be a maximal distance similar set in G such that S is an independent set or a clique in G. Let v S and let U be the distance similar equivalence class such that v U. Let w S {v}. Since S is a distance similar set, d(x, v) = d(x, w) for all x V S. Now, let x S {v, w}. Since S N(u) for some u V, it follows that d(x, v) = d(x, w) = 2 18

34 if S is independent and d(x, v) = d(x, w) = 1 if S is a clique. Thus v and w are distance similar vertices and hence w U. Hence S U. Thus U is a distance similar set and since S is a maximal distance similar set, S = U. The following lemma gives a necessary and sufficient condition for a set S to be a distance similar set. Lemma Let G be any nontrivial connected graph and let S be a proper subset of V with S 2. Then S is a distance similar set of G if and only if N(x) S = N(y) S for all x, y S. Proof. Suppose N(x) S = N(y) S for all x, y S. Let v V S and d(v, S) = k. Let P : (v = v 0, v 1,..., v k ) be a shortest path joining v and S, so that v k S. Since N(x) S = N(y) S for all x, y S it follows that v k 1 is adjacent to all the vertices of S. Hence {d(v, w) : w S} = 1 and S is a distance similar set of G. Conversely, let S be a distance similar set of G and let x, y S. If N(x) S N(y) S, let w N(x) S and w / N(y) S. Then {d(w, v) : v S} 2, which is a contradiction. Corollary Let S 1 and S 2 be two distance similar sets in G such that S 1 2, S 2 2 and S 1, S 2 N(u). If S 1 S 2, then S 1 S 2 is a distance similar set of G. Corollary The set of all maximal distance similar sets which are contained in N(u) forms a partition of N(u). 19

35 Proposition Let S be any distance similar set of G with S 3 and let v S. Then S {v} is a distance similar set if and only if d S (v) = 0 or S 1. Proof. If d S (v) = 0 or S 1, then {d(v, v i ) : v i S {v}} = {2} or {1}. Also, since S is a distance similar set, {d(w, v i ) : v i S {v}} = 1 for all w V {S {v}}. Hence S {v} is a distance similar set of G. Conversely, if 1 d S (v) < S 1, then there exist vertices v i, v j S such that vv i E(G) and vv j / E(G). Hence S {v} is not a distance similar set of G. Corollary Let S be any distance similar set of G. Then the following statements are equivalent: (i) S {v} is a distance similar set for all v S. (ii) S is an independent set or a clique in G. (iii) S is a distance similar equivalence class. Observation Distance similarity is not a hereditary property. For example, for the graph G 2 given in Figure 2.2, S 1 = {v 2, v 3, v 4 } is a distance similar set but the subsets {v 2, v 3 } and {v 3, v 4 } of S 1 are not distance similar sets. Also maximality of distance similar sets is not equivalent to 1-maximality. For example, for the graph G 2 given in Figure 2.2, S 2 = {v 7 } is a distance similar set, which is 1-maximal but not maximal, since S 2 {v 8, v 9 } is a maximal distance similar set. 20

36 v 2 v 7 v 1 v 8 v 3 v 5 v 6 v 11 v 4 G 2 v 9 v 10 Figure 2.2: Example for distance similarity is not hereditary. Proposition For any separable graph G, ds (G) = 1. Proof. Let v be a cut vertex of G. We claim that {v} is a maximal distance similar set of G. Suppose there exists a distance similar set S such that v S and S 2. Let u V (G) be such that S N(u). Then S V (G 1 ) where G 1 is the block of G containing v and u. Now let G 2 be a block of G such that v V (G 2 ) and G 2 G 1. Then for any y V (G 2 ) N(v), we have d(y, v) = 1 and d(y, w) 2 for all w S {v}, which is a contradiction. Hence {v} is a maximal distance similar set of G and ds (G) = AN ALGORITHM FOR COMPUTING ds(g) In this section we prove that the distance similar number ds(g) can be computed in O(n 4 ) time. Given two vertices u, v V with v N(u), we first give an algorithm to find the maximal distance similar set S such that v S and S N(u). 21

37 Algorithm Input : A vertex u V (G) and v N(u). Output: Maximal distance similar set S such that S N(u) and v S. S 0 = N(u), S 1 = {x N(u) : N(x) N(u) = N(v) N(u)} i = 1 While S i S i 1 do S i = S i 1 S i, Si v = N(v) S i If Si v = then S i+1 = S i (N(S ( i ) S i) ) else S i+1 = (N(x) S i ) N(S i Sv i ) end Output S i x S v i Theorem Let G = (V, E) be a connected graph, let u V and v N(u). Let S k be the output of Algorithm Then S k is a maximal distance similar set with v S k N(u). Proof. Clearly N(u) = S k S k 1 S k 2 S 1, where the sets S k, S k 1, S k 2,..., S 1 are mutually disjoint. Clearly v S k and S k N(u). Now let y V S k. If y / N(u), then it follows from the definition of S 1 that {d(y, z) : z S 1 } = 1. Since S k S 1 we have {d(y, z) : z S k } = 1. If y N(u), then y S i for some i, 1 i k 1. We claim that y is adjacent to all the vertices of S i+1 or to no vertex of S i+1. If Si v =, then S i+1 = S i (N(S i ) S i), and in this case y is adjacent to no vertex of S i+1. If Si v, then 22

38 S i+1 = ( x S v i ) (N(x) S i ) N(S i Sv i ). If y Sv i, then y is adjacent to all the vertices of S i+1, and if y S i Si v, then y is adjacent to no vertex of S i+1. Since y N(u), it follows that either d(y, z) = 1 for all z S i+1 or d(y, z) = 2 for all z S i+1. Also S k S i+1 and hence {d(y, z) : z S k } = 1. Thus S k is a distance similar set of G. Now let S be any distance similar set in G such that v S N(u). We claim that S S k. Suppose there exists a vertex y such that y S and y / S k. Hence y S i for some i, 1 i k 1. If Si v =, then S i = N(S i 1 ) S i 1. Since y S i, there exists z S i 1 such that zy E(G). Also zv / E(G), and both y, (( v S, which is a contradiction. ) If Si v ), then S i = S i 1 (N(x) S i 1 ) N(S i 1 Svi 1 ). If y N(S i 1 x S v i 1 S v i 1 ), then there exists z S i 1 Sv i 1 such that zy E(G). Also zv ( / E(G) and both) z, v S, which is a contradiction. If y / (N(x) S i 1 ), then there exists y 1 Si 1 v such that yy 1 / x S v i 1 E(G). Also y 1 v E(G), which is a contradiction. Therefore S S k. Thus S k is a maximal distance similar set with v S k N(u). Illustration We illustrate Algorithm with an example. For the graph G 3 given in Figure 2.3, we find the maximal distance similar set S such that S N(u) and v S. From Algorithm 2.3.1, we have S 0 = {v, v 1, v 2, v 3, v 4 }, S 1 = {v, v 1, v 2 }, S 1 = {v 3, v 4 } and S1 v =. Since S1 v =, S 2 = {v, v 1 }, S 2 = {v 2 } and S2 v =. Since S2 v =, S 3 = S 2 (N(S 2) S 2 ) = S 2 = S 2. Therefore S 3 = {v, v 1 } 23

39 is the output of Algorithm and is a maximal distance similar set containing v. v v 1 u v 2 v 5 v 6 v 3 v 4 G 3 Figure 2.3: Example for illustration of Algorithm Theorem O(n 4 ) time. For any graph G, ds(g) can be computed in Proof. Let u V (G) and let v N(u). Let f(u) = max{ S : S is a maximal distance similar set and S N(u)}. Given the adjacency list of the graph G, the sets S 0, S 1, S i and Si v in Algorithm can be determined in O(n) time each. Also the set S i+1 can be determined in O(n 2 ) time both when Si v = and Si v. Since the while loop is executed at most n times, it follows that the algorithm takes O(n 3 ) time to find the maximal distance similar set S N(u) containing v. Hence f(u) can be computed in O(n 3 ) time. Now, since ds(g) = max{f(u) : u V (G)}, it follows that ds(g) can be computed in O(n 4 ) time. 24

40 2.4 GRAPHS WITH ds(g) = 1 In this section we present several families of graphs with ds(g) = 1. Theorem Let G = (V, E) be a bipartite graph of order at least three with bipartition V 1, V 2. Then ds(g) = 1 if and only if for any set S V 1 or S V 2 with S 2, the induced subgraph S N(S) is not a complete bipartite graph. Proof. Suppose ds(g) = 1. If there exists S V 1 such that S 2 and S N(S) is a complete bipartite graph, then S is a distance similar set of G and hence ds(g) 2, which is a contradiction. Also, since G is a bipartite graph, for any subset S with S V 1 or S V 2, the vertex induced graph S N(S) and the edge induced subgraph [S, N(S) S] are isomorphic and hence the converse follows from (2) of Observation Corollary For the n-dimensional hypercube Q n, n 3, ds(q n ) = 1. Corollary Let T be a tree. Then ds(t ) = 1 if and only if every support vertex in T has exactly one adjacent leaf. Theorem Let G be any graph with δ(g) 2 and g(g) 5, where g(g) is the girth of G. Then ds(g) = 1. Proof. Suppose G has a ds-set S with S 2. Then S N(u) for some u V (G) S. Further since g(g) 5, S is independent. 25

41 Now, since δ(g) 2, there exists a vertex v N(S) S with v u and both u and v are adjacent to all the vertices in S. Thus G contains a cycle C 4, a contradiction. Hence ds(g) = 1. Theorem Let G be a unicyclic graph with cycle C. Then ds(g) = 1 if and only if G satisfies the following: (i) Every support vertex in G has exactly one adjacent leaf. (ii) If C = C 3, then at most one vertex of C 3 has degree two and if C = C 4, then at most two vertices of C 4 have degree two and they are adjacent. Proof. Suppose ds(g) = 1. Since the set of all leaves adjacent to any support vertex of G forms a distance similar set, (i) follows. If G does not satisfy (ii), then the two vertices in C 3 with degree two or the two nonadjacent vertices in C 4 with degree two form a distance similar set of G, a contradiction. Therefore G satisfies the conditions (i) and (ii). Conversely, let G be a unicyclic graph satisfying the conditions (i) and (ii). Suppose ds(g) 2 and let S be any ds-set of G. Since S 2 and any two vertices of S lie on a cycle, it follows that S V (C). Further S is a ds-set of the cycle C and since ds(c n ) = 1 if n 5, it follows that C = C 3 or C = C 4. Now condition (ii) implies that [S, N(S) S] is not a complete bipartite graph, which is a contradiction. Thus ds(g) = 1. 26

42 2.5 DISTANCE SIMILAR SETS AND RESOLVING SETS In Proposition 2.1.1, Saenpolphat and Zhang presented a lower bound for metric dimension of a graph in terms of order and number of equivalence classes of a graph. In this section we obtain a sharp and better lower bound for the metric dimension of a graph G by using maximal distance similar sets of G. Theorem Let S be any maximal distance similar set in G with S 2 and let S N(u). Then dim(g) dim( S {u} ) 1. Proof. Let W be a basis of G. Let v 1, v 2 S. If W S =, then r(v 1 W ) = r(v 2 W ). Hence it follows that W S and any pair of vertices in S is resolved by a vertex in W S. Hence W S or (W S) {u} is a resolving set of S {u}, so that dim( S {u} ) W S + 1. Now, let T be any basis for S {u}. Then W 1 = (W (W S)) (T S) is a resolving set of G and hence T S W S. Thus dim( S {u} ) W S and hence dim( S {u} ) = W S or W S + 1. Now dim(g) = W W S dim( S {u} ) 1. Corollary Let S 1, S 2,..., S k be disjoint maximal distance similar sets of G with S i 2. Then dim(g) k (dim( S i + K 1 ) 1). i=1 27

43 Corollary Let S 1, S 2,..., S k be disjoint maximal distance similar sets of G with S i 2 and each S i is an independent set or a clique. Then dim(g) k ( S i 1). i=1 Observation The lower bound for dim(g) given in Corollary is sharp. Consider the graph G obtained from the star K 1,k and k copies G 1, G 2,..., G k of the path P 3, where G i = (x i, y i, z i ), by joining all the pendent vertices of the star to all the vertices of each G i. Then V (G i ), 1 i k, are maximal distance similar sets of G and dim( V (G i ) {u i } ) = 2 where u i is a pendent vertex of the star K 1,k. Further W = {x 1, x 2,..., x k } is a basis of G and hence dim(g) = k = k (dim( V (G i ) {u i } ) 1). Further for i=1 any distance similar equivalence class S i in G, we have S i = 1 and hence the bound for dim(g) given in Proposition reduces to the trivial inequality dim(g) 0. This shows that the bound given in Theorem is better than the one given in Proposition BOUNDS FOR DISTANCE SIMILAR NUMBER Since ds(g) (G) n 1 and ds(g) = n 1 if and only if (G) = n 1, it follows that ds(g) n 2 for any graph G with (G) n 1. The following theorem gives a characterization of all graphs with ds(g) = (G). Theorem Let G be any connected graph of order n. Then ds(g) = (G) if and only if (G) n 2 and G = G1 + K n (G), where G 1 is any graph of order (G). 28

44 Proof. Suppose ds(g) = (G). Let S be a ds-set of G. Then S = N(u) where u V (G) S and d(u) = (G). Now let v V S and let d(v, S) = t. Let P = (v = v 0, v 1, v 2,..., v t ) where v t S be a shortest path. Now v t 1 is adjacent to v t in S and since S is a ds-set, v t 1 is adjacent to all vertices in S. Thus d(v t 1 ) = (G) and t = 1. Thus every vertex in V S is adjacent to every vertex in S and V S is an independent set in G. Hence G is isomorphic to S + K n (G). Further S and V S are distance similar sets in G and hence (G) V S, so that (G) n 2. Conversely, suppose G is isomorphic to G 1 + K n (G), V (G 1 ) = (G) and (G) n 2. Then V (G1 ) is a distance similar set of G with V (G 1 ) = (G). Hence it follows from (1) of Observation that ds(g) = (G). We now proceed to characterize graphs G with ds(g) = n 2, n 3 and d(g). Theorem Let G be any graph of order n with n 5. Then ds(g) = n 2 if and only if G is isomorphic to 2K 1 + H where H is any graph of order n 2 with (H) < n 3. Proof. Suppose ds(g) = n 2. Then (G) = n 2 by (1) of Observation Thus ds(g) = (G) and the result follows from Theorem Theorem Let G be a graph of order n with n 6. Then ds(g) = n 3 if and only if G is isomorphic to one of the following graphs. 29

45 (i) The graph obtained from any graph H of order n 3 and the path P 3 by joining exactly one pendent vertex of P 3 to all the vertices of H. (ii) The graph obtained from any graph H of order n 3 and K 2 K 1 by joining exactly one vertex of K 2 and the vertex of K 1 to all the vertices of H. (iii) The graph H + 3K 1 or H + (K 2 K 1 ), where H is any graph of order n 3 with (H) n 5 and H does not contain K 2,n 5 as a subgraph when (G) = n 2. Proof. Let G be any connected graph of order n 6 with ds(g) = n 3. It follows from Observation that n 3 (G) n 2. Now let S be a ds-set of G. Let u V S be such that S N(u) and let V S = {u, v, w}. Case 1. (G) = n 3. Then it follows from Theorem that G = H + 3K 1 where H is any graph of order n 3. Hence G is isomorphic to the graph given in (iii). Case 2. (G) = n 2. In this case the vertex u is adjacent to at most one of the vertices v, w and hence {u, v, w} = K 3 or P 3 or K 2 K 1. Suppose {u, v, w} = K 3. Since G is connected, the vertices v and w are adjacent to all the vertices of S. Also, if S contains a subgraph K 2,n 5 and V 1 is a partite set of K 2,n 5 with V 1 = n 5, then 30

46 V 1 (V S) is a distance similar set of G of cardinality n 2, which is a contradiction. Hence G is isomorphic to the graph given in (iii). If {u, v, w} = P 3 = (u, v, w), then v is nonadjacent to the vertices of S since (G) = n 2. Also if w is adjacent to all the vertices of S, then S {v} is a distance similar set of cardinality n 2, a contradiction. Hence G is isomorphic to the graph given in (i). Now we assume that {u, v, w} = K 2 K 1. Without loss of generality let K 2 K 1 = (u, v) {w}. Since G is connected, w is adjacent to all the vertices of S. Now v is either adjacent to u only or adjacent to all the vertices of S. If v is adjacent to u only, then G is isomorphic to the graph given in (ii). Now suppose v is adjacent to all the vertices of S. If S contains a subgraph K 2,n 5 and V 1 is a partite set of K 2,n 5 with V 1 = n 5, then V 1 (V S) is a distance similar set of G of cardinality n 2, which is a contradiction. Hence G is isomorphic to the graph given in (iii). Conversely, let G be a graph as given in the theorem. It follows from (1) of Observation and Theorem that ds(g) n 3. Further V (H) is a ds-set of G of cardinality n 3. Thus ds(g) = n 3. Theorem Let G be any graph of order n 4 and diameter d. Then the following holds: (i) If d 3, then ds(g) n d and ds(g) = n d if and only if G can be obtained from a path P of length d by replacing any vertex v i of P by any graph H of order ds(g) and joining every vertex of H to the neighbors of v i in P. 31