Some diameter notions in lexicographic product of graphs

Many graphs such as hypercubes, star graphs, pancake graphs, grids, tori etc are known to be good interconnection network topologies. In any network topology, the vertices represent the processors and the edges represent links between the processors. Two most important criteria - efﬁciency and reliability of network models - can be studied with the help of graph theoretical techniques. The lexicographic product is a well studied graph product. The distance notions such as various diameters of a graph help to analyze the efﬁciency of any interconnection network. In this paper, we study some distance notions such as wide diameter, diameter variability and diameter vulnerability of lexicographic products that could be used in the design of interconnection networks.


Introduction
The processors of a parallel and distributed system and the connections between the processors can be represented as an interconnection network. The topological structure of an interconnection network can be modelled by a connected graph where the vertices and edges represent sites of the network and the physical communication links respectively. Many graph theoretic parameters that are useful to study the efficiency and reliability of an interconnection network are discussed in [6].
A simple graph G = (V, E) with |V | = n and |E| = m is denoted as G = (n, m). The degree of a vertex u in G, d G (u) or simply d(u), is the number of edges incident with u in G. The minimum degree and the maximum degree of a graph G are denoted by δ(G) and ∆(G) respectively. The distance between u and v in G, denoted by d G (u, v), is the length of a shortest path joining u and v in G. The diameter of a graph G, diam(G), is the maximum distance between any two vertices in G. The diameter often measures efficiency of a network with maximum time -delay or signal degradation. The diametral vertices of G are two vertices u, v ∈ V (G) such that d(u, v) = diam(G). A subset S ⊆ V (G) of vertices is an independent set if no two vertices of S are joined by an edge in G. The independent domination number of a graph G, γ i (G), is the minimum cardinality of a maximal independent set in G. The vertex connectivity, κ(G) of a graph G is the minimum number of vertices whose removal from G makes the graph either disconnected or K 1 . The edge connectivity, κ ′ (G) of a graph G is the minimum number of edges whose removal makes the graph disconnected. The network fault tolerance capacity can be measured by studying the connectivity of the corresponding graph. A good network must be hard to disrupt even if some vertices or edges are being attacked and the transmissions between the processors must remain connected. For all notions not given here, see [13].
The lexicographic product H 1 • H 2 of any two graphs H 1 and H 2 is the graph with the vertex- . The necessary and sufficiency condition for the lexicographic product of two graphs H 1 • H 2 to be connected is that H 1 is connected. If H 1 = K n , then diam(H 1 • H 2 ) = diam(H 1 ) and diam(K n • H) = 2, [7]. In [14], Yang et al. studied the connectivity of the lexicographic product of graphs and they have proved that if H 1 = (n 1 , m 1 ) is a connected simple graph and H 2 = (n 2 , m 2 ) is any simple graph then: Let H 1 * H 2 be any of the graph products. For any vertex u ∈ H 1 , the subgraph of For every integer w, 1 ≤ w ≤ κ(G), any collection of 'w' internally vertex disjoint paths between two vertices u and v of G is termed as the w-container and it is denoted by C w (u, v). In C w (u, v), the parameter w is the width of the container. The length of the container is the length of the longest path in C w (u, v). The w-wide diameter D w (G) of a graph G is the minimum number l such that there is a C w (u, v) of length at most l between any pair of distinct vertices u and v in G.
The wide diameter of a graph is D κ(G) (G). This concept was introduced by Hsu [6] to unify the concepts of diameter and connectivity. The wide diameter of some networks are studied in [9] and [5].
www.ejgta.org Some diameter notions in lexicographic product of graphs | Chithra M R et al.
Vulnerability measures maximum routing delay that can happen because of vertex or edge faults. Diameter can be used to measure the maximum delay in routing. In this context, the vertex fault diameter and the edge fault diameter are defined and studied by several authors. The vertex [8]. Chung and Garey [3] proposed the problem of determining the diameter vulnerability of a graph. In [15] Ye et al. improves the result of Peyrat [10] and gave a bound as 4 √ 2t−6 < f ′ (G) ≤ max{59, 5 √ 2t+7} for t ≥ 4. The concept of fault diameter was introduced by Krishnamoorthy and B. Krishnamurthy [8]. The problem of diameter vulnerability is proved to be NP-complete by Schoone et al. [11].
The diameter of a graph may change by the addition or the deletion of edges. The following notations denote the diameter variability of a graph G. Let k ≥ 1 be any positive integer. D −k (G) is the minimum number of edges to be added to G to decrease the diameter by (at least) k and D 0 (G) is the maximum number of edges that can be deleted from G so that the diameter is not altered. In [1], [2], the diameter variability of the product graphs are discussed. In [12], Wang et al. studied the diameter variability of cycles and tori. Graham and Harary studied the diameter variability of hypercubes in [4].
In this paper, we study the wide diameter, the diameter vulnerability and the diameter variability of the lexicographic product of graphs. We consider both H 1 and H 2 to be connected  Proof. The proof is divided into three cases.
There exists a container of length at most l between any two vertices u i and u k in G, since there exists a container of length l in G, Since the length of the container in G is l, there exists a pair of vertices u x and u y in G such that the path joining u x and u y is of length exactly equal to l. Then C w|V (H)| ((u x , v j ), (u y , v j )) in G ′ is of length exactly equal to l.
Consider the vertices u i and u a in H 1 . By the assumption there exists a container of length at most l in between u i and u a in G.
is of length same as that of P 1 . Again, by the structure of the lexicographic product, there exists w |V (H)| internally vertex disjoint paths between (u i , v j ) and (u a , v b ) which is of length at most l. Since the length of the container in G is l, there exists a pair of vertices u x and u y in G such that the path joining u x and u y is of length exactly equal to l. So , the result follows.

Theorem 2.1. If G is a connected non-complete graph and H is a connected graph, then
Then there exists a container of width κ(G) between any two vertices of G which is of length at most k. Then, by Lemma 2.1, there exists a container of width κ(G) ×|V (H)| between any two vertices of G ′ which is of length at most k.
There exists a container of length at most k joining (u i , v 1 ) and (u j , v 1 ). More over there exists a container of width at least κ(G) between (u i , v 1 ) and (u j , v 1 ) where all the internal vertices are of the form (u a , v 1 ), a ∈ {1, 2, ..., x, y, ..., n 1 }. If Thus there exist a container of width κ(G) which is of length at most k joining u i and u j in G.

Diameter vulnerability of the lexicographic product of graphs
Let u x , u y be a pair of diametral vertices of G, by a path u x − u x+1 − u x+2 − ... − u y−1 − u y . Let G ′′ be the subgraph obtained from G ′ after the deletion of κ ′ (G ′ )−1 edges from G ′ . Let us consider the following cases.
Clearly, this length is at most diam(G).
Case 1b: Let κ ′ (G ′ ) − 1 edges be deleted from H-layer of G ′ at u i . Then, the deleted edges are of the form is a path of length two in G ′′ . Thus the diam(G ′ ) is unaltered by this type of deletion.
Case 1c: Let κ ′ (G ′ ) − 1 edges deleted from G ′ be any arbitrary collection of edges. Consider a pair of diametral vertices (u Figure 1). Consider a pair of diametral vertices (u x , v w ) and (u y , v z ) in G ′ . Since, we have already considered Cases 1a and 1b, there exist a path of length diam(G ′ ) between (u x , v w ) and (u y , v z ) in where the vertex (u x , v w ) in ux H-layer will be adjacent to at least one vertex (say)(u x+1 , v p ) in u x+1 H-layer, the vertex (u x+1 , v p ) in u x+1 H-layer will be adjacent to at least one vertex (say) (u x+2 , v q ) in u x+2 H-layer and so on (see Figure 2). ((u a , v w ), (u a , v n 2 )) ≤ diam(H) and d G ′′ ((u a , v n 2 ), (u b , v w )) ≤ f ′ (G). Similarly, the distance between any two vertices in G ′ is at most f ′ (G) + diam(H).

Remark:
Consider H • P 3 where H is the graph obtained by taking two copies of K n , n > 3 which is joined by an edge. For this graph f ′ (H • P 3 ) = 5, since f ′ (H) = 3 and diam(P 3 ) = 2. Thus the above bound is strict for an infinite family of graphs.
Proof. Let S be a collection of κ(G ′ ) − 1 vertices in G ′ . When S is deleted from G ′ the new subgraph obtained is denoted as G ′′ . Let u x , u y be a pair of diametral vertices of G, by a path u x − u x+1 − u x+2 − ... − u y−1 − u y . Let us consider the following cases.  (u y , v a ). Thus, the diam(G ′ ) remains the same after removing vertices in S. {1, 2, 3, ..., n 1 }. Let n 2 − 1 vertices from S be deleted. Clearly the distance between any two vertices in G ′ is not affected by the removal of these vertices.

www.ejgta.org
Some diameter notions in lexicographic product of graphs | Chithra M R et al.
Case 2c: Let S be any arbitrary collection of vertices. Consider a pair of diametral vertices (u x , v p ) and (u y , v q ) in G. Let the κ(G ′ )−1 vertices from G ′ be deleted. Then, , since we have already considered the case of the deletion of vertices of the form (u i , v p ) where i ∈ {1, 2, 3, ..., n 1 }, there exist at least one vertex (say) (u i , v j ) for each j ∈ {1, 2, ... , n 2 } and are adjacent to the vertices (u r , v p ) where p ∈ {1, 2, 3, ..., n 2 }. Thus the diam(G ′ ) remains the same.
Then κ(G ′ ) ≥ 2n 2 . We shall prove the theorem by considering the following sub cases.
Case 3b: Let S be any arbitrary collection of vertices.
vertices which form a vertex cut of G and p ∈ {1, 2, 3, ..., n 2 − 1}, be deleted. Now, from the G -layer at v n 2 in G ′ , only κ(G) − 1 vertices can be deleted, otherwise G ′′ becomes disconnected. Figure 3). Thus, f (G ′ ) ≤ f (G). Consider a pair of diametral vertices (u x , v w ) and (u y , v z ) in G ′ . Let the κ(G ′ ) − 1 vertices be deleted. Since, we have already considered Cases 3a, there exist a path of length diam(G ′ ) between where the vertex (u x , v w ) in ux H-layer will be adjacent to at least one vertex (say)(u x+1 , v p ) in u x+1 H-layer, the vertex (u x+1 , v p ) in u x+1 H-layer will be adjacent to at least one vertex (say) (u x+2 , v q ) in u x+2 H-layer and so on (see Figure 4). Thus, the diam(G ′ ) remains the same after removing vertices in S. From the above cases, the result follows. Proof. Consider a pair of diametral vertices (u x , v w ) and (u y , v z ) in G ′ where u x and u y in G are joined by a path u x − u x+1 − u x+2 ...u y−1 , u y . Let the edges (u i , v p ) − (u i , v q ) where p, q ∈ {1, 2, ..., n 2 } and i ∈ {1, 2, ..., n 1 } in G be deleted to get G ′′ . Then,

Diameter variability of the lexicographic product of graphs
. Thus, the distance between any two vertices in G ′′ is not affected by the removal of these edges. Proof. Let u x , u y be a pair of diametral vertices of G by a path u is even or odd respectively. Thus, diam(G ′ ) = diam(G ′′ ). Hence, D 0 (G ′ ) ≥ n 1 m 2 + n 2 2 m 1 − (n 1 m 2 + m 1 n 2 + 2m 1 m 2 ) = n 2 2 m 1 − (m 1 n 2 + 2m 1 m 2 ).
Proof. Let d G (u x , u y ) = diam(G) and let e l edges are added to G to decrease the diameter of G by k + 2. Consider a pair of diametral vertices (u x , v q ) and (u y , v r ) in G ′ . Let the e l edges u Proof. Let d G (u x , u y ) = diam(G) and let e l edges are added to G to decrease the diameter of G by k, where added edges are not incident on the diametral vertices of G. Consider a pair of diametral vertices (u x , v q ) and (u y , v r ) in G ′ . Let the e l edges whose end vertices are of the form Thus, the distance between any two vertices is at most diam(G ′ ) − k.  Then, the corollary follows from the above result.

Concluding Remarks and Further Scope
Two main interconnection network models -grids and tori-motivated us to study the graph product structures from the view point of interconnection models. We have seen several papers in which the distance notions have been studied and the graph product considered mainly in those papers was the Cartesian product. In [14], connectivity of Lexicographic product is studied and this motivated us to think the Lexicographic product as a network model. In this paper, we have studied wide diameter, diameter variability and fault diameter of the lexicographic product of graphs since it is important in the design of interconnection networks and we established some bounds for these parameters. We have noted that H 1 • H 2 has better wide diameter, diameter variability, fault diameter as compared to that of H 1 . Hence H 1 • H 2 can be a better network model as compared to that of H 1 . One can extend this work by characterizing the graphs for which the equality of the bounds is attained. We have discussed the diameter notions based on connectivity. One may think of these notions based on some other graph parameter which may be helpful to study the reliability and efficiency of the model.