On Certain Connected Resolving Parameters of Hypercube Networks *

Given a graph , a set is a resolving set if for each pair of distinct vertices there is a vertex such that . A resolving set containing a minimum number of vertices is called a minimum resolving set or a basis for . The cardinality of a minimum resolving set is called the resolving number or dimension of and is denoted by . A resolving set is said to be a star resolving set if it induces a star, and a path resolving set if it induces a path. The minimum cardinality of these sets, denoted respectively by  = , G V E G   , W V     , w d v  G dim   , u v V G 


Introduction
A query at a vertex discovers or verifies all edges and non-edges whose endpoints have different distance from In the network verification problem [1], the graph is known in advance and the goal is to compute a minimum number of queries that verify all edges and non-edges.This problem has previously been studied as the problem of placing landmarks in graphs or determining the metric dimension of a graph [2].Thus, a graph-theoretic interpretation of this problem is to provide representations for the vertices of a graph in such a way that distinct vertices have distinct representations.This is the subject of the papers [3][4][5].

v . v
For an ordered set of vertices and a vertex in a connected graph , the code or representation of with respect to W is the -vector where is the distance between the vertices  , d x y x and .The set is a resolving set for if distinct vertices of have distinct codes with respect to .Equivalently, for each pair of distinct vertices there is a vertex such that The minimum cardinality of a re-solving set for G is called the resolving number or dimension and is denoted by .

An Overview of the Paper
The concept of resolvability in graphs has previously appeared in literature.Slater [4,5] introduced this concept, under the name locating sets, motivated by its application to the placement of a minimum number of sonar detecting devices in a network so that the position of every vertex in the network can be uniquely determined in terms of its distance from the set of devices.He referred to a minimum resolving set as a reference set and called the cardinality of a minimum resolving set as the location number.Independently, Harary and Melter [3] discovered this concept, but used the term metric dimension, rather than location number.Later, Khuller et al. [2] also discovered these concepts independently and used the term metric dimension.These concepts were rediscovered by Chartrand et al. [6] and also by Johnson [7] while attempting to develop a capability of large datasets of chemical graphs.It was noted in [8] that determining the metric dimension of a graph is NP-complete.It has been proved that the metric dimension problem is NP-hard [2] for general graphs.Manuel et al. [9] have shown that the problem remains NP-complete for bipartite graphs.There are many applications of resolving sets to problems of network discovery and verification [1], pattern recognition, image processing and robot navigation [2], geometrical routing protocols [10], connected joins in graphs [11] and coin weighing problems [12].This problem has been studied for trees, multi-dimensional grids [2], Petersen graphs [13], torus networks [14], Benes networks [9], honeycomb networks [15], enhanced hypercubes [16] and Illiac networks [17].
Many resolving parameters are formed by combining resolving property with another common graph-theoretic property such as being connected, independent, or acyclic.The generic nature of conditional resolvability in graphs provides various ways of defining new resolving parameters by considering different conditions.In general, a connected graph can have many resolving sets.It is interesting to study those resolving set whose vertices are located close to one another.A resolving set of is connected if the subgraph induced by W is a nontrivial connected subgraph of .The minimum cardinality of a connected resolving set is called connected resolving number and it is denoted by [18].In this paper we introduce a new resolving parameter called star resolving number.A resolving set is said to be a star resolving set if the subgraph induced by is a star and a path resolving set [19] if induces a path.In this paper we show the existence of star and path resolving sets in hypercube networks.

Topological Properties of Hypercube Networks
The hypercube is a very popular, versatile and vertextransitive interconnection network.When the dimension of hypercube increases, the cardinality of its vertex set increases exponentially.The effectiveness of parallel computers is often determined by its communication network.The interconnection network is an important component of a parallel processing system.A good interconnection network should have less topological network cost and meanwhile keep the network diameter as shorter as possible [20].
and  0 1  [21].It has been proved in [22] The bound is tight for , and it is not tight for .A laborious calculation verifies that is resolved by the 4-vertex set {00000, 00011, 00101, 01001}.Caceres

Star Resolving Number
We begin this section by defining a star and a star resolving set.
any graph G .In a star resolving set the maximum distance between any two locations (vertices) is 2.
We now proceed to identify a star resolving set in a hypercube network It is clear that there are four copies of We denote them as .

Note that vertices in
The next result which we state as Lemma 1 is crucial to our work.We omit the proof as this result has been proved in [16] for enhanced hypercubes.
be the image of x .Let be any vertex in w and be the images of are equidistant from every vertex of and in particular from every vertex of This implies Proof.We prove the theorem by induction on .r Base Case: Let and where 0 1 and 2 It follows from the definition of hypercube edges that 0 is adjacent to both 1 and It is easy to check that 1 W is a resolving set for Figure 3 shows the distinct codes of vertices in with respect to Since induces it is a star resolving set for .
Now assume that the result is true for the hypercube Let where be a star resolving set for Here 0 w is the hub and it is adjacent to all Moreover Divide into four copies 1 .
There exist vertices either in The same argument applies to the following cases.
. y We need to prove that for some in be the images of x and respectively.
y Case 2.1: The proof is similar to Case 2.

Case 4:
Let x and y be the images of x and rey spectively.There are three possibilities  conclusion will follow by Case 1 and Case 2.
Case 5: Let and be the images there exist a such that .

Path Resolving Number
In this section we determine a path resolving number for hypercube networks.

Conclusion
In this paper we have introduced a new resolving parameter called a star resolving number.We have determined the star resolving number and path resolving number for hypercube networks.The problem is open for architectures like Benes and Butterfly networks.

5 QFigure 1 .
Figure 1.(a) Binary representation; (b) Decimal representation.et al. [22] have determined dim   r Q for small values of by computer search; the values are shown in the following table:

Figure 3 .
Figure 3. (a) Resolvingset W 1 in Q 3 ; (b) Codes of vertices of Q 3 with respect to W 1 .
that they are equidistant from every vertex of 0 in particular from every vertex of P Since there exist a path in Q