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The Diameter of an Interval Graph Is Twice of Its Radius Tarasankar Pramanik, Sukumar Mondal and Madhumangal Pal

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The Diameter of an Interval Graph Is Twice of Its Radius Tarasankar Pramanik, Sukumar Mondal and Madhumangal Pal World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:5, No:8, 2011 The diameter of an interval graph is twice of its radius Tarasankar Pramanik, Sukumar Mondal and Madhumangal Pal 5 7 10 G =(V,E) 1238 9 Abstract—In an interval graph the distance between 4 6 two vertices u, v is de£ned as the smallest number of edges in a path joining u and v. The eccentricity of a vertex v is the maximum Fig. 2. An interval representation of Figure 1 among distances from all other vertices of V . The diameter (δ) and radius (ρ) of the graph G is respectively the maximum and minimum among all the eccentricities of G. The center of the graph G is the set C(G) of vertices with eccentricity ρ. In this context our aim is to An interval graph and its interval representation are shown ρ = δ establish the relation 2 for an interval graph and to determine in Figure 1 and Figure 2 respectively. the center of it. Interval graphs arise in the process of modeling real life Keywords—Interval graph, interval tree, radius, center. situations, specially involving time dependencies or other restrictions that are linear in nature. This graph and various subclass thereof arise in diverse areas such as archeology, I. INTRODUCTION molecular biology, sociology, genetics, traf£c planning, VLSI N undirected graph G =(V,E) is an interval graph if design, circuit routing, psychology, scheduling, transportation A the vertex set V can be put into one-to-one correspon- and others. Recently, interval graphs have found applications dence with a set of intervals I on the real line R such that two in protein sequencing [7], macro substitution [2], circuit rou- vertices are adjacent in G if and only if their corresponding tine [8], £le organization [1], job scheduling [1], routing of intervals have non-empty intersection. The set I is called two points nets [6] and many others. An extensive discussion an interval representation of G and G is referred to as the of interval graphs also appears in [5]. Thus interval graphs intersection graph of I [5]. Let I = {i1,i2,...,in}, where have been studied intensely from both the theoretical and ic =[ac,bc] for 1 ≤ c ≤ n, be the interval representation of algorithmic point of view. the graph G, ac is the left endpoint and bc is the right end The notion of a center in a graph is motivated by a large point of the interval ic. Without any loss of generality assumed class of problems collectively referred to as the facility- the following: location problems where one is interested in identifying a (a) an interval contains both its endpoints and that no subset of the vertices of the graph at which certain facilities two intervals share a common endpoint [5], are to be located in such a way that for every vertex in the (b) intervals and vertices of an interval graph are one graph, the distance to the nearest facility is minimum. G =(V,E) d(u, v) and the same thing, For a connected graph , the distance u v (c) the graph G is connected, and the list of sorted between vertices and is the smallest number of edges in u v endpoints is given and a path joining and . v ∈ V e(v) (d) the intervals in I are indexed by increasing right The eccentricity of a vertex , is denoted by and is de£ned by endpoints, that is, b1 <b2 < ···<bn. e(v)=max{d(u, v):u ∈ V }. The diameter δ(G) (or simply δ), radius ρ(G) (or simply ρ) and the center C(G) of a graph G are de£ned as follows: 1g 3g 5g 7g 9g @ @ δ(G)=max{e(v):v ∈ V }, @ @ ρ(G)=min{e(v):v ∈ V }, International Science Index, Mathematical and Computational Sciences Vol:5, No:8, 2011 waset.org/Publication/3440 @ @ g g g g g and C(G)={v ∈ V : e(v)=ρ(G)}. 2 4 6 8 10 The center of a graph may be a single vertex or more than one vertex. This shows in the following £gure. The graph in Figure 3(a) has only one center with center node 2 while the G =(V,E) Fig. 1. An interval graph graph in Figure 3(b) has two centers and the center nodes are 3 and 4. T. Pramanik and M. Pal are with the Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Mid- A. Survey of related works napore 721102, INDIA e-mail: [email protected]. S. Mondal is with Raja N. L. Khan Women’s College, Midnapore – 721102, It is both well-known and easy to observe that the center INDIA e-mail: [email protected]/[email protected]. of an arbitrary graph G =(V,E) can be computed by the International Scholarly and Scientific Research & Innovation 5(8) 2011 1412 scholar.waset.org/1307-6892/3440 World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:5, No:8, 2011 4i hold: (i) av <au <bv, 1 2 3 i i i (ii) av <bu <bv, (iii) au <av <bu, 3(a) (iv) au <bv <bu. The following lemma is true for a given interval graph, G = i3 (V,E). c u v w ∈ V i1 2i c i5 i6 Lemma 1 ([15]): If the vertices , , be such that u<v<win the IG-ordering and u is adjacent to w then v 4 i is also adjacent to w. For each vertex v ∈ V let H(v) represent the highest 3(b) numbered adjacent vertices of v. If no adjacent vertex of v exists with higher IG number than v then H(v) is assumed to Fig. 3. (a) The center with one vertex; (b) The center with two vertices. be v. In other words, H(v)=max{u :(v,u) ∈ E,u ≥ v}. For a given interval graph G, let a tree T (G)=(V,E) following brute force approach: perform breadth-£rst search of be de£ned such that E = {(u, H(u)) : u ∈ V,u = n},n be G starting in turn, at every vertex of G. Clearly, this procedure the root of T (G). This tree is called the interval tree IT. The takes O(|V |×|E|) time. various properties of interval tree are available in [11], [12], For some particular classes of graphs, such as for trees [14]. The most important property is as follows: [4], outerplaner graphs [3] etc., linear time algorithms can be Lemma 2 ([13]): For a connected interval graph there ex- devised to compute the center. For the interval graph with n ists a unique interval tree T (G). m O(n + m) vertices and edges. Olariu [9] has presented an h level time sequential algorithm where the input is an adjacency 10 0 list that takes O(n + m) space. In [9], all maximal cliques h h 8 9 1 are used as input. But the generation of all maximal cliques h h h h is slightly complicated. In [10], Olariu et al. have presented 4 5 6 7 @ 2 an O(log n) time and O(n) processor parallel algorithm to h h@ h £nd the center of an interval graph if the endpoints list is 1 2 3 3 given. But Pal et. al. [14] have designed optimal algorithm Fig. 4. The interval tree of the interval graph of Figure 2 to compute center and diameter of an interval graph, which was an improvement over Olariu’s algorithm, since with same For each vertex v of interval tree, de£ne level(v) to be the input Olariu’s algorithm takes O(n + m) time. Also Pal et distance of v from the vertex n in the tree. al. have presented some properties on interval graphs and some Let Nl be the set of vertices which are at a distance l from properties of diameter and center, but no relation between the vertex n. Thus Nl = {u : d(u, n)=l} where d(u, n) is radius and diameter has been established. the distance between u and n in the interval tree and N0 is the Based on the proposed sequential algorithms two O(n/P + singleton set {n}.Ifu ∈ Nl then d(u, n)=l and the vertex u log n) time parallel algorithms have been designed for the is at level l of the interval tree. Thus, the vertices at level l of same problems using P processors and O(n) space on the the interval tree are the vertices of Nl. It follows from Lemma EREWPRAM if the sorted endpoints list is given. It is 1, that the vertices of Nl are consecutive integers. Hence the improvement over the algorithm of [10], because, if the sorted path starting from the vertex 1 and ending at the vertex n in endpoints list is given as an input, then the complexity of the T (G) is called the main path. The main path is represented algorithm of [10] remains unchange. by dotted lines in Figure 4. In this paper we have established the relation between the De£ne the height h of the tree T (G) by radius and the diameter of the interval graph. h =max{level(v):v ∈ V }. The distance between any two vertices of G can be deter- II. INTERVAL TREE AND ITS PROPERTIES mined from the following result. International Science Index, Mathematical and Computational Sciences Vol:5, No:8, 2011 waset.org/Publication/3440 Lemma 3 ([14]): Given u, v ∈ V,v = n, let w be the vertex G =(V,E) V = {1, 2,...,n}, |V | = n, |E| = m Let , be at level(v)+1 on the path from u to n and w = H(w).
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