Selçuk J. Appl. Math. Selçuk Journal of Vol. 10. No. 1. pp. 107-120, 2009 Applied Mathematics
Computing the Tenacity of Some Graphs Vecdi Aytaç
Computer Engineering Department, Ege University — 35100, Bornova — Izmir, Turkey e-mail: vecdi.aytac@ ege.edu.tr
Received: January 5, 2009
Abstract. In communication networks, “vulnerability” indicates the resistance of a network to disruptions in communication after a breakdown of some proces-sors or communication links. We may use graphs to model networks, as graph theoretical parameters can be used to describe the stability and reliability of communication networks. In an analysis of the vulnerability of such a graph (or communication network) to disruption, two quantities (there may be others) that are important are:
(1) the number of the components in the unaffected graph, (2) the size of the largest connected component.
In particular, it is crucial that the first of these quantities be small, while the second is large, in order for one to say that the graph has tenacity. The concept of tenacity was introduced as a measure of graph vulnerability in this sense. The tenacity of a graph is defined as
() = min ½
|| + ( − )
( − ) : ⊂ () and ( − ) 1 ¾
where the ( − ) is the number of components of − and ( − )is the number of vertices in a largest component of G. In this paper we give some bounds for tenacity and determination of the tenacity of total graphs of specific families of graphs and combinations of these graphs.
Keywords: Network, Vulnerability, Graph Theory, Stability, Connectivity and Tenacity.
2000 Mathematics Subject Classification: 05C99, 05C40, 68R10, 90C27, 90C35, 90B12.
1. Introduction
A communication network is composed of processors and communication links. Network designers attach importance the reliability and stability of a network. If the network begins losing communication links or processors, then there is a loss in its effectiveness. This event is called as the vulnerability of communication network.
The vulnerability of communication networks measures the resistance of a net-work to a disruption in operation after the failure of certain processors and com-munication links. Cable cuts, processor interruptions, software errors, hardware failures, or transmission failure at various points can interrupt service for a long period of time. But network designs require greater degrees of stability and re-liability or less vulnerability in communication networks. Thus, communication networks must be constructed to be as stable as possible, not only with respect to the initial disruption, but also with respect to the possible reconfiguration of the network.
The communication network often has as considerable an impact on a network’s performance as the processors themselves. Performance measures for communi-cation networks are essential to guide the designers in choosing an appropriate topology. In order to measure the performance we are interested in the following performance metrics (there may be others):
(1) the number of the components of the remaining graph, (2) the size of the largest connected component.
The communication network can be represented as an undirected and unweighted where a processor (station) is represented as a node and a communication link between processors (stations) as an edge between corresponding nodes. A graph G is denoted by = ( () ()), where () is vertices of set of G and () is edges of set of G. The number of vertices and the number of edges of the graph G are denoted by | ()| = , |()| = respectively. Tree, star, cycle, mesh, complete and hypercube graph are popular communication networks.
A complete graph K is a fully connected (or all to all) network. In a complete
graph, each vertex is directly connected to all other vertices. So each vertex has n-1 connections.
A mesh graph is a cartesian product of path graphs and . Communication
is allowed only between neighbouring vertices. In 2D mesh, all interior vertices are connected to four other vertices.
A cycle is a linear array with the end nodes linked. So each vertex has degree 2.
An -dimensional hypercube consists of 2 vertices and 2−1 edges. The
each vertex degree and diameter of are .
An n-dimensional star graph consists of n! vertices and (( −1)!) 2 edges.
has a smaller degree and diameter than the hypercube.
A tree is a connected graph that contains no subgraph isomorphic to a cycle. If we use a graph to model a network, there are many graph theoretical para-meters used to describe the stability and reliability of communication networks
including connectivity, integrity, toughness, binding number and tenacity [1-4, 6-10, 12]. Definitions of some graph parameters are given below.
Definition 1.1: (): The connectivity of a graph G is defined by () = min
≤ (){||}
for which − is disconnected or trivial.
Definition 1.2: (): The toughness of a graph G is defined by () = min ≤ () ½ || ( − ) ¾
where S is a vertex cut of G and ( − ) is the number of the components of G-S.
Definition 1.3: (): The integrity of a graph is given by () = min
⊆ (){|| + ( − )}
where ( − ) is the maximum number of vertices in a component of − . Definition 1.4: (): The tenacity of a graph is defined as
() = min ⊆ () ½ || + ( − ) ( − ) : ⊂ and ( − ) ¾
where the ( − ) is the number of components of − and ( − ) is the number of vertices in a largest component of G.
Definition 1.5: [5] For a connected graph G, we define the distance ( ) between two vertices u and v as the minimum of the lengths of the u-v paths of G.
Definition 1.6: [5] A subset S of V is called an independent of G if no two vertices of S are adjacent in G. An independent set S is a maximum if G has no independent set S ’ with | ’| ||. The independence number of G, (), is the number of vertices in a maximum independent set of G.
Definition 1.7: [5] A subset S of V is called a covering of G if every edge of G has at least one end in S. A covering S is a minimum covering if G has no covering S ’ with | ’| ||. The covering number, (), is the number of vertices in a minimum covering of G.
Theorem 1.1: [5, 13] For any graph G of order n, () + () =
In the next section, known results on the tenacity of specific families of graphs, on bounds for the tenacity, on relationships between tenacity and other parame-ters are given. In Section 3 we consider with the relations between the tenacity and the others graph theoretical parameters. After that we give our theorem relating to tenacity and the independence number. Finally, we formulize the tenacity values of total graph of some basic graphs. We find also tenacity of graphs that are obtained from corona operation of total graphs of some graphs. Throughout this work, the first integer larger than or equal to x is denoted by de, the first integer small than or equal to x is denoted by bc, and the absolute value of x is denoted by ||.
2. Results on Tenacity
In this section, we will review some of the known results.
Theorem 2.1: [9, 10] If G is a spanning subgraph of H, then () ≤ (). Theorem 2.2: [9, 10] For any graph G, () ≥ () + 1
() , where () is the independence number of G.
Theorem 2.3: [9, 10] If G is not complete, then () ≤ − () + 1 () Theorem 2.4: [9, 10] For any graph G, () ≥ () + 1
().
Theorem 2.5: [9, 10] For any nontrivial, non complete graph G with n vertices and any vertex v,
( − ) ≥ () −12. Theorem 2.6: [7, 9 ,10]
(a) For every integer ≥ 2, () =
(
1 if n is odd + 2
if n is even (b) For any positive integer n, () =
2+ 2 2 .
(c) For any even integers n and m, (× ) =
+ 2 . (d) For any even integer n, (× 2) =
+ 1 . (e) For any positive integer n, (× ) = − 1 +
1 .
3. Bound for Tenacity of a Graph
In this section, we consider with the relations between the tenacity and the toughness, connectivity and integrity.
Theorem 3.1: Let G be a connected graph such that () = and () = . Then
≤( − 1) ( + 1) 2 − − 1
Proof: Denote the order of G by n. Then ( − ) + || ≤ . It is easy to see that
( − ) ≥ − || ( − )
From the definition of () we know that || + ( − ) ≥ . Therefore ( − ) ≤ − || − ( − ) + 1
= + 1 − (|| + ( − )) ≤ + 1 − On the other hand, since || ≥ 1, we have ( − ) − || ≤ − .
(1) || + ( − ) ≥ || + − || ( − ) = + ( ( − ) − 1) || ( − ) ≥ + ( ( − ) − 1) ( ( − ) − + ) ( − ) = 2 − + 2( − ) − ( − ) ( − + 1) ( − ) = ( − ) + 2 − ( − )− ( − + 1) ≥ ( − ) + 2 − ( − )− ( + 1 − ) From the definition of () we know that () ≤ || + ( − )
( − ) . On the other hand, it is not difficult to see that || + ( − ) ≤ + 1 − ( − ). Therefore () ≤ + 1 − ( − )
( − ) which implies that ( − ) ≤ + 1 + 1. So, we know that ( − ) ≥ 2 and ( − ) ≤ + 1 + 1. We write these results in above inequality (1). Therefore we have the following inequality;
|| + ( − ) ≥ 2 + (2 − ) + 1 + 1 − ( + 1 − ) + ( + 1 − ) − 2 ≥ (2 − 1) + 1 + 1 ( − 1) ( + 1) 2 − ≥ + 1 ≤ ( − 1) ( + 1) 2 − 1 − 1 The proof is completed.
Theorem 3.2: Let G be a connected graph such that () = , () = and () = . Then
≤( + 1) − 1
Proof: As showing in the proof of the above Theorem 3.1, we have ( − ) ≤ + 1
+ 1 from the definition of T(G). || ( − ) ≥ || + 1 + 1 || ≥ ≥ + 1 + 1 ( + 1) − 1 ≥ ≤ ( + 1) − 1 The proof is completed.
Corollary 3.1: For any graph G, t(G) + 1
() ≤ () ≤
( + 1) − 1. Proof: This result is easily seen from Theorem 3. 2 and Theorem 2. 1.
Theorem 3.3: Let G1and G2be two graphs with n vertices. If (1) ≥ (2),
then (1) ≤ (2).
Proof: We have () ≤ − () + 1 () from Proposition 2. 3. In this case if the value of () enlarges, then tenacity becomes small. For two graphs having the same number of vertices, if (1) ≥ (2) then it is easy to see that
(1) ≤ (2). Therefore, we have (1) ≤ (2).
Theorem 3.4: For any graph G with n vertices, if ( ) ≤ 2 then () = () + 1
() .
Proof: If ( ) ≤ 2, in the same time the graph G does not contain 22as an
Case 1. ( − ) = 1.
Clearly, || = () and ( − ) = (). Hence,
() = min ½ || + ( − ) ( − ) ¾ = min ½ () + 1 () ¾ =() + 1 () Case 2. ( − ) = 2.
Clearly, || = () − 1 and ( − ) = (). Hence,
() = min ½ || + ( − ) ( − ) ¾ = min ½ () − 1 + 2 () ¾ =() + 1 () This result is the same as the result of Case1.
Case 3. ( − ) 2.
Clearly, || = − ( − ) and ( − ) = 1. Hence,
() = min ½ || + ( − ) ( − ) ¾ = min ½ − ( − ) + ( − ) 1 ¾ =
It is easy to see that () + 1
() . Hence, we obtain that () =
() + 1 () . 4. Tenacity and Operations on Total Graphs
In this section, firstly, we will give a definition of total graph of a graph and corona operation on graphs. After that we will give some results about the tenacity of (), (), (1), ()◦ (2) and ()◦ (2).
Definition 4.1: [11] The vertices and edges of a graph are called its elements. Two elements of a graph are neighbors if they are either incident or adjacent. The total graph () of the graph = ( () ()), has vertex set () ∪ (), and two vertices of () are adjacent whenever they are neighbors in G. It is easy to see that () always contains both G and Line graph () as a induced subgraphs. Total graph is the largest graph that is formed by the adjacent relations of elements of a graph. It is important from this respect [11].
Theorem 4.1: The tenacity value of () is defined as,
( ())
2 ( ())
2 + 1
Proof: If we remove r vertices from graph (), then the number of the
re-maining connected components is at mostj 2 k
+ 1. In this case the order of the largest remaining component is ( − ) ≥ 2 − 1 − ¥
2 ¦ + 1 ( ()) ≥ min ( + 2−1− 2+1 2+ 1 )
The function [ + (2 − 1 − ) ((2) + 1)] [(2) + 1] takes its minimum value at = 2 − 1. Consequently,
( ())
2 ( ())
2 + 1 The proof is completed.
Corollary 4.1: For the graph () ( ()) = min
( 2 − ( ()) ( ()) + 1§ 2 ¨ ) . Proof: The number of vertices, the independence number and the covering number of () are ( ()) = 2 − 1, ( ()) = » ( ()) 3 ¼ and ( () = 2 − 1 − ( ()) respectively. There are three cases.
Case1. ( () − ) = 1
Clearly, || = ( ()) and ( () − ) = ( ()). Hence,
( ()) = min ½ || + ( () − ) ( () − ) ¾ = min ½ () + 1 ( ()) ¾ = 2 − ( ())) ( ()) Case 2. ( () − ) = 2 Clearly, || = − 1 and ( () − ) = l 2 m . Hence, ( ()) = min ½ || + ( () − ) ( () − ) ¾ = min ( + 1 § 2 ¨ ) = + 1§ 2 ¨ Case 3. 3 ≤ ( () − ) ≤ 2 − 1 − (3 + 2) = 2 − 6
( ()) = min ½ || + ( () − ) ( () − ) ¾ ≤ min ½ 2 + ( ()) − 9 2 ¾ ≤2 + ( (2 )) − 9
We claim that this result is greater than the above results. Firstly, we show that this result is greater than the Case 1.
2 − ( ())) ( ()) ? 2 + ( ()) − 9 2 2 + ( ()) − 9 2 = 2 − ( ()) + ( − 9) 2 It is easy to see that 2 − ( ()) + ( − 9)
2
2 − ( ()))
( ()) for ≥ 9.
Now we show that this result is greater than the Case 2. + 1 § 2 ¨ ? 2 + ( ()) − 9 2 = 3 − (( ()) + 9) 2 3 − (( ()) + 9) 2 − 2( + 1) ? 0 (3 − ( () − 13) − 4 2 = (2 + ( () − 13) − 4 2
For ≥ 9 and ( () , 2 + ( () − 13 0. Thus, (2 + ( () −
13) ≥ 9 and (2 + ( () − 13) − 4 0. Hence, the proof of the claim is
completed.
Thus tenacity of () is the minimum value of the obtained results from the
above cases. The proof is completed. Corollary 4.2: For the graph (),
3
( ())≤ ( ()) ≤
2 − 6 + ( ())
2 .
Proof: This corollary gives the upper and below bounds for the tenacity of () to us. There are two cases.
Case 1. ( () − ) ≥ 1
Clearly, || ≥ 2 and ( () − ) ≤ ( ()). Hence,
( ()) = min ½ || + ( () − ) ( () − ) ¾ ≥ min ½ 3 ( ()) ¾ = 3 ( ())→ ( ()) ≥ 3 ( ())
Case 2. This case is the same as Case 3 in the proof of the Corollary 4. 1. Hence, it is easy to see the right hand of the inequality.
The proof is completed.
Theorem 4.2: The tenacity value of () is defined as,
( ()) 4 − 1
( ())
Proof: If we remove r vertices from graph (), then the number of the
remain-ing connected components is at mostj 2 k
. In this case the order of the largest re-maining component is (−) ≥2 − ¥ 2 ¦ . So, ( ()) ≥ min ( +2− 2 2 )
The function [ + (2 − ) (2)] (2) takes its mimimum value at = 4. Consequently,
( ()) 4 − 1
( ())
The proof is completed.
Corollary 4.3: For the graph (),
( ()) = min ⎧ ⎪ ⎨ ⎪ ⎩ 2 − ( ()) + 1 ( ()) ⎧ ⎪ ⎨ ⎪ ⎩ 2( + 2) if is even + 3 ¥ 2 ¦ otherwise ⎫ ⎪ ⎬ ⎪ ⎭ ⎫ ⎪ ⎬ ⎪ ⎭ Proof: The proof of Corollary 4.3 is done similarly as in the proof of Corollary 4.1.
Corollary 4.4: For the graph (),
5
( ())≤ ( ()) ≤
2 − 7 + ( ())
2 .
Proof: The proof of Corollary 4.4 is done similarly as in the proof of Corollary 4.2.
Theorem 4.3: Let (1) be the total graph of 1. ( (1)) =
+ 2 . Proof: The number of vertices, the independence number and the covering number of (1) are ( (1)) = 2 + 1, ( (1)) = and ( (1)) =
+ 1 respectively. Obtained new n vertices from the definition of total graph are adjacent the other − 1 vertices. So, (1) contains a complete graph
with n vertices. If ( (1) − ) = 1, then || = + 1and ( (1) − ) =
Fig.2. (1) Graph
If v vertex and n vertices which are v1, v2,. . . , v in Fig. 2 are removed from
(1), then we have n components which have only one vertex. In this case
tenacity of (1) is minimal. Thus the proof is completed.
Definition 4.2: [5, 11] The corona 1 ◦2was defined by Frucht and Harary
as the graph G obtained by taking one copy of G1 of order n1 and n1 copies of
G2, and then joining the i’th node of G1to every node in the i’th copy of G2.
Theorem 4.4: Let () and (2) be the total graphs of P and P2
respec-tively. For ()˚ (2), ( ()◦ (2)) = ( ()) + 4 ( ()) where ( ()) = 2 − 1 − » 2 − 1 3 ¼ . Proof: There are three cases.
Case 1. || = ( ()).
Clearly, when we remove the vertices of the covering set of the () from () ◦ (2), obtained components are K4or (2) = 3. So, ( ()˚ (2)−) =
4 and ( ()◦ (2) − ) = (( ()). Hence, ( ()◦ (2)) = min ½ ( ()) + 4 ( ()) ¾ = ( ()) + 4 ( ()) Case 2. || ≥ ( ()) + 1.
Clearly, when we remove more than the vertices of the covering set of the () from ()◦ (2), obtained components are (2) = 3. So, ( () ◦ ( 2) − ) = 3 and ( ()◦ (2) − ) = (( ()). Hence, ( ()◦ (2)) ≥ min ½ ( ()) + 4 ( ()) ¾ ≥( ( ( ()) + 4 )) Thus the Case 1 is the minimum in respect of obtained this value.
Case 3. || ( ()).
In this Case, the number of removed vertices is reduced but the number of vertices in the largest component is increased and the number of the components is decreased. As to Case 1, numerator is increased but denominator is decreased, so the obtained value is larger than that of Case 1.
Corollary 4.5: Let () and (2) be the total graphs of P and P2
respec-tively. For ()◦ (2),
( ()◦ (2)) ( ())
Proof. There are two cases. Case1. ( ()) =
( ()) + 1
(( ())
. We have from Corollary 4.1, ( ()) =
( ()) + 1
(( ())
and from Theorem 4.4 ( ()◦ (2)) =
( ()) + 4
( ())
. Let us show that + 4 + + 1 , where = ( ()) and = ( ()). + 1 − + 4 + ? 0 ( + 1) ( + ) − ( + 4) ( + ) ? 0 2+ − 3 ( + ) For → 2+ − 3 ( + ) 2+ − 3 ( + ) ? 0 ( − 2) ( + ) ? 0 Because of 2, ( − 2)
( + ) 0. So, the claim is true. Case 2. ( ()) = + 1 § 2 ¨ . From Corollary 4. 1, ( ()) = + 1 § 2
¨ and from Theorem 4. 4 ( () ◦ (
2)) =
( ()) + 4
( ())
. Let us show that + 4 2 − 1 2( + 1) , where ( ()) = and 2 − 1 = 2( + 1) − + 4 ? 0 2( + 1) − + 4 = 2 + 2 + (− − 4) = + 2 + ( − 4)
Since + 2 + ( − 4) 0, the claim is true. Consequently, the proof is completed.
Theorem 4.5: Let T(C) and T(P2) be the total graphs of C and P2
respec-tively. For ()◦ (2), ( ()◦ (2)) = ( ()) + 4 ( ()) where ( ()) = 2 − ¹ ( ()) 3 º
Proof: The proof of Theorem 4. 5 is done similarly as in the proof of Theorem 4. 4.
Corollary 4.6: Let T(C) and T(P2) be the total graphs of Cand P2
respec-tively. For ()◦ (2),
( ()◦ (2)) ( ())
Proof: It is done similarly as in the proof of Corollary 4. 5. 5. Conclusion
If a system such as a communication network is modeled by a graph G, there are many graph theoretical parameters used to describe the stability and reliability of communication networks including connectivity, integrity, toughness, binding number and tenacity. Two ways of measuring the stability of a network is through the ease with which one can disrupt the network, and the cost of a disruption. Connectivity has the least cost as far as disrupting the network, but it does not take into account what remains after disruption. One can say that the disruption is less harmful if the disconnected network contains more components and much less harmful if the affected components are small. One can associate the cost with the number of the vertices destroyed to get small components and the reward with the number of the components remaining after destruction. The tenacity measure is compromise between the cost and the reward by minimizing the cost: reward ratio. Thus, a network with a large tenacity performs better under external attack. The results of this paper suggest that tenacity is a more suitable measure of stability in that it has the ability to distinguish between graphs that intuitively should have different measures of stability.
References
1. Bagga K.S. - Beineke L.W. — Goddard W.D. - Lipman M.J. and Pippert R.E., A Survey of Integrity, Discrete Applied Math. 37/38, 13-28 ,1992.
2. Bagga K.S. - Beineke L.W. and Pippert R.E, Edge Integrity: A Survey, Discrete Math., 124, 3-12(1994).
3. Barefoot C.A, —Entringer R., and Swart H.C., Vulnerability in Graphs-A Compar-ative Survey. J.Comb.Math.Comb.Comput. 1, 13-22, 1987.
4. Barefoot C.A, —Entringer R., and Swart H.C., Integrity of Trees and Powers of Cycles, Congressus Numeratum 58, 103-114, 1987.
5. Chartrand G. - Lesniak L,.Graphs and Digraphs, California Wadsworth & Brooks, 1986.
6. Chvatal V., Tough Graphs and Hamiltonian Circuits, Discrete Math.5, 215- 218, 1973.
7. Choudum S. A. and Priya N., Tenacity of Complete Graph Products and Grids, Networks 34, 192-196, 1999.
8. Cozzens M., Stability Measures and Data Fusion Networks, Graph Theory Notes of New York XXVI, pp.8-14, 1994.
9. Cozzens M. - Moazzami D.and Stueckle S., the Tenacity of A Graph, Graph The-ory, Combinatorics, Algorithms and Applications. Vol 2. 1995. Proceedings of the seventh quadrennial international conference on the theory and applications of graphs, Kalamazoo, MI, USA, June15, 1992. New York, NY.Wiley, pp.1111-1122.10.
10. Cozzens M. - Moazzami D. and Stueckle S., the Tenacity of Harary Graphs, J.Comb. Math. Comb. Comput. 16, 33-56, 1994.
11. Harary F., Graph Theory , Addison-Wesley Pub. California, 1971.
12. Dündar P., - Aytac A., Integrity of Total Graphs via Some Parameters, Mathe-matical Notes Vol.76, N5 (November), p.665-672, 2004.
