Some Results on Tenacity of Graphs

Transcription

1 WSEAS TRANSACTIONS o MATHEMATICS Fegwei Li Some Results o Teacity of Graphs Fegwei Li Shaoig Uiversity Departmet of mathematics Shaoig, Zhejiag P R Chia fegweili@eyoucom Abstract: The teacity of a icomplete coected graph G is defied as T (G) = mi{ S +m(g S) (G S) : S V (G), (G S) > 1}, where (G S) ad m(g S), respectively, deote the umber of compoets ad the order of a largest compoet i G S This is a reasoable parameter to measure the vulerability of etworks, as it takes ito accout both the amout of work doe to damage the etwork ad how badly the etwork is damaged I this paper, we firstly give some results o the teacity of gear graphs After that, the teacity of the leicographic product of some special graphs are calculated We also give the eact values for the teacity of powers of paths Fially, the relatioships betwee the teacity ad some vulerability parameters, amely the itegrity, toughess ad scatterig umber are established Key Words: Teacity, Vulerability, Cartesia product, Gear graph, Powers of graphs, T -set 1 Itroductio The vulerability of a commuicatio etwork, composed of processig odes ad commuicatio liks, is of prime importace to etwork desigers As the etwork begis losig liks or odes, evetually there is a loss i its effectiveess Thus, commuicatio etworks must be costructed to be as stable as possible, ot oly with respect to the iitial disruptio, but also with respect to the possible recostructio of the etwork The commuicatio etwork ofte has as cosiderable a impact o a etwork s performace as the processors themselves Performace measures for commuicatio etworks are essetial to guide the desigers i choosig a appropriate topology I order to measure the performace, we are iterested i the followig performace metrics (there may be others): (1) the umber of elemets that are ot fuctioig, () the umber of remaiig coected sub-etworks, (3) the size of a largest remaiig group withi which mutual commuicatio ca still occur The commuicatio etwork ca be represeted as a udirected ad uweighted graph, where a processor (statio) is represeted as a ode ad a commuicatio lik betwee processors (statios) as a Supported by the NSFC(No108716) ad the Foudatio of Zhejiag Educatioal Committee(NoY015964) edge betwee correspodig odes If we use a graph to model a etwork, there are may graph theoretical parameters used to describe the vulerability of commuicatio etworks Most otably, the verte-coectivity ad edgecoectivity have bee frequetly used The difficulty with these parameters is that they do ot take ito accout what remais after the graph is discoected Cosequetly, a umber of other parameters have bee itroduced that attempt to cope with this difficulty, icludig toughess ad edge-toughess i [10,0,5], itegrity ad edge-itegrity i [4,5,6], teacity ad edge-teacity i [1,8,9,11,1,16,18,1], rupture degree i [14,15,17] ad scatterig umber i [,3,5] Ulike the coectivity measures, each of these parameters shows ot oly the difficulty to break dow the etwork but also the damage that has bee caused For coveiece, we recall some parameters of [] Let G be a fiite simple graph with verte set V (G) ad edge set E(G) For S V (G), let (G S) ad m(g S), respectively, deote the umber of compoets ad the order of a largest compoet i G S A set S V (G) is a cut set of G, if either G S is discoected or G S has oly oe verte We shall use for the smallest iteger ot smaller tha, ad for the largest iteger ot larger tha A edge is said to be subdivided whe it is replaced by a path of legth two coectig its eds, ad the iteral verte i this path is a ew verte E-ISSN: Issue 9, Volume 11, September 01

2 WSEAS TRANSACTIONS o MATHEMATICS Fegwei Li A subset S of V is called a idepedet set of G if o two vertices of S are adjacet i G A idepedet set S is a maimum if G has o idepedet set S with S > S The idepedece umber of G, β(g), is the umber of vertices i a maimum idepedet set of G A subset S of V is called a coverig of G if every edge of G has at least oe ed i S A coverig S is a miimum coverig if G has o coverig S with S > S The coverig umber, α(g), is the umber of vertices i a miimum coverig of G We use Body ad Murty [] for termiology ad otatios ot defied here For comparig, the followig graph parameters are listed The coectivity is a parameter defied based o Quatity (1) The coectivity of a icomplete graph G is defied by κ(g) = mi{ S : S V (G), (G S) > 1}, ad that of the complete graph K is defied as 1 Both toughess ad scatterig umber take ito accout Quatities (1) ad () The toughess ad scatterig umber of a icomplete coected graph G are defied by S t(g) = mi{ : S V (G), (G S) > 1} (G S) ad s(g) = ma{(g S) S : S V (G), (G S) > 1}, respectively The itegrity is defied based o Quatities (1) ad (3) The itegrity of a graph G is defied by I(G) = mi{ S + m(g S) : S V (G)} Both the teacity ad rupture degree take ito accout all the three quatities The teacity ad rupture degree of a icomplete coected graph G are defied by ad S + m(g S) T (G) = mi{ : (G S) S V (G), (G S) > 1} r(g) = ma{(g X) X m(g X) : X V (G), (G X) > 1}, respectively Ad the teacity ad rupture degree of the complete graph K is defied as ad 1, respectively The correspodig edge aalogues of these cocepts are defied similarly, see [6, 0, 1] From the above defiitios, we ca see that the coectivity of a graph reflects the difficulty i breakig dow a etwork ito several pieces This ivariat is ofte too weak, sice it does ot take ito accout what remais after the correspodig graph is discoected Ulike the coectivity, each of the other vulerability measures, ie, toughess, scatterig umber, itegrity, teacity ad rupture degree, reflects ot oly the difficulty i breakig dow the etwork but also the damage that has bee caused Further, we ca easily see that the teacity ad rupture degree are the two most advaced oes amog these parameters whe measurig the vulerability of etworks Amog the above parameters, the teacity is a reasoable parameter ca be used for measurig the vulerability of etworks [1,7,8,10,11,17,18] A verte cut set S of a graph G is called a T-set of G if it satisfies that T (G) = S +m(g S) (G S) I [11], Moazzami et al compared the itegrity, coectivity, bidig umber, toughess, ad teacity for several classes of graphs I [9], Choudum et al studied the teacity of complete graph products ad grids I [16], Li, Ye ad Li discussed the teacity ad rupture degree for permutatio graphs of complete bipartite graphs Li, Ye ad Sheg gave eact values of rupture degree for some useful graphs i [15] Cheg et al [8] determied the maimum teacity of trees ad uicyclic graphs with give order ad show the correspodig etremal graphs These results are helpful i costructig stable etworks with lower costs Cozzes et al[11] studied the teacity of Harary Graphs I [18], Ma proved that computig the teacity of a graph is NP-hard i geeral So, it is a iterestig problem to determie teacity for some special graphs I this paper, we cosider the problem of computig the teacity of graphs I Sectio, we give some results o the teacity of gear graphs After that, the teacity of the leicographic product of some special graphs, such as P [P ], C [P ] ad K 1, 1 [P ] are calculated i sectios 3 We also give the eact values for the teacity of powers of paths i sectio 4 Fially, the relatioships betwee the teacity ad some vulerability parameters, amely the itegrity, toughess ad scatterig umber are established i sectios 5 E-ISSN: Issue 9, Volume 11, September 01

3 WSEAS TRANSACTIONS o MATHEMATICS Fegwei Li Computig the Teacity of Gear Graphs Geared systems are used i dyamic modellig These are graph theoretic models that are obtaied by usig gear graphs Similarly the cartesia product of gear graphs, the complemet of a gear graph, ad the lie graph of a gear graph ca be used to desig a gear etwork We kow that the teacity is a reasoable parameter to measure the vulerability amog these parameters Cosequetly these cosideratios motivated us to ivestigate the teacity of gear graphs Now we give the followig defiitios Defiitio 1 [] The wheel graph with spokes, W, is the graph that cosists of a -cycle ad oe additioal verte, say u, that is adjacet to all the vertices of the cycle I Figure 1, we display W 6 u Figure 1: Wheel graph W 6 Defiitio [7] The gear graph G is a graph obtaied from the wheel graph W by subdividig each edge of the outer -cycle of the W just oce It is easily see that the gear graph G has + 1 vertices ad 3 edges I Figure we display G 6 ad we call the verte u ceter verte of G Now we give the teacity of a gear graph Theorem 3 Let G be a gear graph The T (G ) = 1 Proof O oe had, G is a icomplete coected graph with V (G ) = + 1 umber of vertices, let S be a cut set of G with S =, the the remaiig graph G S has at most + 1 compoets, ad so, m(g S) u Figure : Gear graph G 6 Sice + 1 1, + 1 So, must be at most So we get that T (G ) mi{ }, + 1 where Now we cosider the fuctio f() = = + ( + 1) + 1 ( + 1) f () = ( ( + 1)) ( + 1) 3 Sice < + 1, we have f () < 0, ad so f() is a decreasig fuctio Thus the fuctio f() = + ( + 1) ( + 1) takes its miimum value at =, ad f mi () = 1 Sice this value ca be achieved for each, we have T (G ) 1 (1) O the other had, we let S deotes the coverig set of G The S = α(g ) = ad (G S ) = β(g ) = + 1 So m(g S ) = 1 From the defiitio of teacity, we have T (G ) S + m(g S ) (G S ) = + 1 = 1 () + 1 Cosequetly, by usig (1) ad (), we have T (G ) = 1 I the et, we will study the teacity of complemet of gear graph G Firstly, we itroduce the cocept of the complemet of a graph E-ISSN: Issue 9, Volume 11, September 01

4 WSEAS TRANSACTIONS o MATHEMATICS Fegwei Li Defiitio 4 [] The complemet of a graph G is a graph G o the same vertices such that two vertices of G are adjacet if ad oly if they are ot adjacet i G Theorem 5 Let G be a gear graph The T (G ) = Proof We kow that a gear graph G ca be costructed from a wheel graph W by subdividig each edge of the outer cycle of the W just oce Let S be a set of vertices of the outer -cycle i W, ad let S be a set of vertices which are added to the outer -cycle i G Let u be a ceter verte Sice S is a idepedet set of G, these vertices form a complete graph with order i G Similarly, sice S {u} is a idepedet set of G, these vertices form a complete graph with order + 1 i G Moreover the graph G cotais some edges joiig K +1 to K It is obvious that the verte u i G is ot adjacet to ay verte i K So we have two cases: Case 1 If we remove the vertices of S i G, the we have oly oe compoet which is graph K +1 The ad so m(g S ) = V (K +1 ) = + 1 S + m(g S ) = + 1 (3) (G S ) Case If we remove the vertices of S i G, the we have two compoets which are graphs K ad K 1 The ad so m(g S ) = V (K ) = S + m(g S ) = (4) (G S ) By usig (3) ad (4) we have T (G ) = mi{, + 1} = Now we cosider the teacity of the cartesia product of two graphs Defiitio 6 () The Cartesia product of two graphs G 1 ad G, deoted by G 1 G, is defied as follows: V (G 1 G ) = V (G 1 ) V (G ), two vertices (u 1, u ) ad (v 1, v ) are adjacet if ad oly if u 1 = v 1 ad u is adjacet to v i G or u 1 is adjacet to v 1 i G 1 ad u = v Observe that if G 1 ad G are coected, the G 1 G is coected Lemma 7 [5] Let G be a gear graph The t(k G ) = 1 Lemma 8 [5] Let m 3 ad 3 be positive itegers The t(g m G ) = m + m + m + m Lemma 9 [1] If G is a icomplete coected graph, β(g) is the idepedece umber of G ad t(g) is the toughess of G, the we have T (G) t(g) + 1 β(g) Theorem 10 Let G be a gear graph The T (K G ) = Proof β(k G ) = + 1 O oe had, by lemmas 7 ad 9, we kow that 1 T (K G ) t(k G ) + β(k G ) = (5) O the other had, we ca take the coverig set S of K G istead of a verte cut of K G The S = α(k G ) = + 1 ad (K G S) = β(k G ) = + 1 So m(k G S) = 1 From the defiitio of teacity, we have T (K G ) S + m(k G S) (K G S) = (6) Cosequetly, by usig (5) ad (6), we have T (K G ) = + +1 Theorem 11 Let m 3 ad 3 be positive itegers The T (G m G ) = 1 Proof It is obvious that β(g m G ) = m + m + + 1, α(g m G ) = m + m + O oe had, by lemmas 8 ad 9, we kow that T (G m G ) t(g m G ) + 1 β(g m G ) E-ISSN: Issue 9, Volume 11, September 01

5 WSEAS TRANSACTIONS o MATHEMATICS Fegwei Li = m + m = 1 (7) m + m O the other had, we let S deotes the coverig set of G m G The S = α(g m G ) = m + m + ad (G m G S) = β(g m G ) = m + m So m(g m G S) = 1 From the defiitio of teacity, we have T (G m G ) S + m(g m G S) (G G S) = m + m = 1 (8) m + m Cosequetly, by usig (7) ad (8), we have T (G m G ) = 1 Defiitio 1 [] The lie graph L(G) of a graph G is a graph such that each verte of L(G) represets a edge of G, ad ay two vertices of L(G) are adjacet if ad oly if their edges are icidet, meaig they share a commo ed verte i G Theorem 13 Let G be a gear graph The T (L(G )) = + 1 Proof It is obvious that β(l(g )) = ad α(l(g )) = O oe had, let S be a cut set of L(G ) with S =, the the remaiig graph L(G ) S has at most compoets, ie ad so Sice 3 that (L(G ) S), m(l(g ) S) 3 1, must be at most Thus we get T (L(G )) mi{ where Now we cosider the fuctio f() = + 3 f () = + 3 }, = ( + 6) 4( 6) 3 Sice < 6, we have f () < 0, ad so f() is a decreasig fuctio So, the fuctio f() = ( + 6) takes its miimum value at =, ad f mi () = +1 Sice this value ca be achieved for each, we have T (L(G )) + 1 (9) O the other had, we let S deotes the coverig set of L(G ) The S = α(l(g )) = ad (L(G ) S ) = β(l(g )) = So m(l(g ) S ) = 1 From the defiitio of teacity, we have T (L(G )) S + m(l(g ) S ) (L(G ) S ) = + 1 (10) Cosequetly, by usig (9) ad (10), we have T (L(G )) = +1 3 Computig the Teacity of Leicographic Product of Graphs I this sectio, the teacity of the leicographic product of some special graphs, P [P ], C [P ] ad K 1, 1 [P ] are calculated Defiitio 14 [5] The leicographic product G 1 [G ] of two graphs G 1 ad G is a graph such that: V (G 1 [G ]) = V (G 1 ) V (G ), two distict vertices (u 1, v 1 ) ad (u, v ) of G 1 [G ] are adjacet if ad oly if either u 1 is adjacet to u i G 1 or u 1 = u ad v 1 is adjacet to v i G The leicographic product is also kow as the compositio The leicographic product is ot commutative ad is coected wheever G 1 is coected Theorem 15 Let P be the path with order ( 4), the the teacity of P [P ] is T (P [P ]) = { (+) if is eve if is odd Proof P [P ] is a coected graph with V (P [P ]) = umber of vertices We distiguish two cases: Case 1 Whe is eve Let S be a cut set of P [P ] with S =, it is obvious that Subcase 11 Whe 3, we ote that P [P ] S has at most compoets This implies m(p [P ] S) E-ISSN: Issue 9, Volume 11, September 01

6 WSEAS TRANSACTIONS o MATHEMATICS Fegwei Li Sice, must be at most Hece, we have that T (P [P ] mi{ where Now we cosider the fuctio f() = + f () = + }, = (4 + ) 4( 4) 3 Sice < 4, we have f () < 0, ad so f() is a decreasig fuctio Thus the fuctio f() = (4 + ) takes its miimum value at =, ad f mi () = (+) Sice this value ca be achieved for each eve, we have T (P [P ]) ( + ) Subcase 1 Whe =, we ote that P [P ] S has eact compoets This implies m(p [P ] S) Hece, we have that S + m(p [P ] S) (P [P ] S) It is easily see that whe 4, + So, i this case we have T (P [P ]) ( + ) + ( + ) (11) O the other had, as show i Figure 3, we let S = {((u i, v 1 ), (u i, v )) i =, 4,, }, the S =, (P [P ] S ) = ad m(p [P ] S ) = 4 From the defiitio of teacity, we have T (P [P ]) S + m(p [P ] S ) (P [P ] S ) ( + ) = (1) Cosequetly, by usig (11) ad (1), we kow that i this case ( + ) T (P [P ]) = (13) Case Whe is odd Let S be a cut set of P [P ] with S =, it is obvious that Note that P [P ] S has at most + compoets This implies m(p [P ] S) + Sice +, must be at most 1 We have that + + T (P [P ] mi{ }, where 1 Now we cosider the fuctio f() = + + f () = + + = ( + 4) ( + ) 8( ) ( + ) 3 Sice 1 <, we have f () < 0, ad so f() is a decreasig fuctio Thus the fuctio f() = ( + 4) ( + ) takes its miimum value at = 1, ad f mi () = Sice this value ca be achieved for each odd, we have T (P [P ]) (14) O the other had, as show i Figure 3, we let S = {((u i, v 1 ), (u i, v )) i =, 4,, 1}, the S = 1, (P [P ] S ) = +1 ad m(p [P ] S ) = From the defiitio of teacity, we have T (P [P ]) S + m(p [P ] S ) (P [P ] S ) = (15) Hece, by combig (14) ad (15), we kow that i this case T (P [P ]) = (16) Cosequetly, by usig (13) ad (16), we have T (P [P ]) = { (+) if is eve if is odd E-ISSN: Issue 9, Volume 11, September 01

7 WSEAS TRANSACTIONS o MATHEMATICS Fegwei Li u 1 u u 3 u 1 u v 1 v (u 1, v 1 ) (u, v 1 ) (u 3, v 1 ) (u 1, v 1 ) (u, v 1 ) (u 1, v ) (u, v ) (u 3, v ) (u 1, v ) (u, v ) Figure 3: Graphs P, P ad P [P ] Theorem 16 Let C be the cycle with order ( 4), the the teacity of C [P ] is T (C [P ]) = { (+) (+3) 1 if is eve if is odd Proof C [P ] is a coected graph with V (C [P ]) = umber of vertices We distiguish two cases to complete the proof Case 1 Whe is eve Let S be a cut set of C [P ] with S =, it is obvious that 4 Note that C [P ] S has at most compoets This implies Sice that m(c [P ] S), must be at most Hece, we have T (C [P ] mi{ where Now we cosider the fuctio f() = + f () = + }, = (4 + ) 4( 4) 3 Sice < 4, we have f () < 0, ad so f() is a decreasig fuctio Thus the fuctio f() = (4 + ) takes its miimum value at =, ad f mi () = (+) Sice this value ca be achieved for each eve, we have T (C [P ]) ( + ) (17) O the other had, as show i Figure 4, we let S = {((u i, v 1 ), (u i, v )) i = 1, 3,, 1}, the S =, (P [P ] S ) = ad m(p [P ] S ) = From the defiitio of teacity, we have T (P [P ]) S + m(p [P ] S ) (P [P ] S ) ( + ) = (18) Cosequetly, by usig (17) ad (18), we kow that i case 1 ( + ) T (P [P ]) = (19) Case Whe is odd Let S be a cut set of C [P ] with S =, it is obvious that 4 Subcase 11 Whe 5, we ote that C [P ] S has at most compoets This implies Sice that m(c [P ] S), must be at most + 1 We have T (C [P ] mi{ where + 1 Now we cosider the fuctio f() = f () = + + }, = ( 4 + 4) ( ) 16(1 ) ( ) 3 < 0, so, f() is a decreasig fuctio The fuctio f() = ( 4 + 4) ( ) takes its miimum value at = + 1, ad f mi () = (+3) 1 Sice this value ca be achieved for each odd, we have T (C [P ]) ( + 3) 1 Subcase 1 Whe = 4, we ote that C [P ] S has eact compoets This implies m(c [P ] S) 1 E-ISSN: Issue 9, Volume 11, September 01

8 WSEAS TRANSACTIONS o MATHEMATICS Fegwei Li Hece, we have that S + m(c [P ] S) (C [P ] S) It is easily see that whe 5, 3 + So, i case, we have T (C [P ]) ( + 3) ( + 3) 1 (0) O the other had, as show i figure 4, we let S = {((u i, v 1 ), (u i, v )) i = 1, 3,, }, 1 ad the S = 1, (P [P ] S ) = m(p [P ] S ) = 4 From the defiitio of teacity, we have T (P [P ]) S + m(p [P ] S ) (P [P ] S ) ( + 3) = 1 (1) Cosequetly, by usig (0) ad (1), we have the followig result i case T (P [P ]) = ( + 3) 1 () Hece, by combig (19) ad (), we have T (C [P ]) = { (+) (+3) 1 if is eve if is odd u 1 u u 3 u 1 u v 1 v (u 1, v 1 ) (u, v 1 ) (u 3, v 1 ) (u 1, v 1 ) (u, v 1 ) (u 1, v ) (u, v ) (u 3, v ) (u 1, v ) (u, v ) Figure 4: Graphs C, P ad C [P ] Theorem 17 Let K 1, 1 be the star with order ( 4), the the teacity of K 1, 1 [P ] is T (K 1, 1 [P ]) = 4 1 Proof K 1, 1 [P ] is a coected graph with V (K 1, 1 [P ]) = umber of vertices O oe had, it is easy to see that the coectivity of K 1, [P ] is κ(k 1, 1 [P ]) = As show i Figure 5, the verte set S = {(u, v 1 ), (u, v )} is the uique cut set which satisfies the coectivity of K 1, 1 [P ] If we remove the S from K 1, 1 [P ], the the remaiig graph has (K 1, 1 [P ] S) = 1 compoets ad the order of the largest compoet of K 1, 1 [P ] is m(k 1, [P ] S) = Hece, by the defiitio of teacity, we kow that T (K 1, 1 [P ]) S + m(k 1, 1[P ] S) (K 1, 1 [P ] S) 4 = 1 (3) O the other had, let S be a T -set of K 1, [P ], if S has more tha two vertices, it is easy to see that ad (K 1, 1 [P ] S ) 1 m(k 1, [P ] S ) 1 Hece, by the defiitio of teacity, we kow that T (K 1, 1 [P ]) = S + m(k 1, 1 [P ] S ) (K 1, 1 [P ] S ) 4 1 (4) Cosequetly, by usig(3) ad (4), we have T (K 1, 1 [P ]) = Teacity of Powers of Paths For a iteger k 1, the k-th power of a graph G, deoted by G k, is a supergraph with V (G k ) = V (G) ad E(G k ) = {(u, v) : u, v V (G), u v ad d G (u, v) k} The secod power of a graph is also called its square We otice that G 1 is just G itself So, we let k i the followig As a useful etwork, power of cycles ad paths have arouse iterests for may etwork desigers CA Barefoot, et al gave the eact values of itegrity E-ISSN: Issue 9, Volume 11, September 01

9 WSEAS TRANSACTIONS o MATHEMATICS Fegwei Li u 1 u u 3 u 1 u (u, v 1 ) v 1 v (u 1, v 1 )(u, v 1 )(u 3, v 1 ) (u 1, v 1 ) (u 1, v ) (u, v )(u 3, v ) (u 1, v ) (u, v ) Figure 5: Graphs K 1, 1, P ad K 1, 1 [P ] of powers of cycles i [3], ad determied the coectivity, bidig umber ad toughess of powers of cycles[4] Verte-eighbor-itegrity of powers of cycles were studied i [13] by Cozzes ad Wu I [19] Moazzami gave the eact values for the teacity of powers of cycles Zhag ad Yag[4] studied the bidig umber of the Powers of Paths ad cycles I this sectio, we cosider the problem of computig the teacity of powers of paths It is easy to see that P k = K if k + 1 So, i the followig lemmas, we suppose that k Lemma 18 If S is a miimal T -set for the graph P k, k, the S cosists of the uio of sets of k cosecutive vertices such that there eists at least oe verte ot i S betwee ay two sets of cosecutive vertices i S Proof We assume that the vertices of P k are labeled by 0, 1,,, 1 Let S be a miimal T -set of P k ad j be the smallest iteger such that T = {j, j + 1,, j + t 1} is a maimum set of cosecutive vertices such that T S Relabel the vertices of P k as v 1 = j, v = j + 1,, v t = j + t 1,, v = j 1 Sice S V (P k ) ad T V (P k ), v does ot belog to S Sice S must leave at least two compoets of G S, we have t 1, ad so v t+1 v Therefore, {v t+1, v } S = Now suppose t < k Choose v i such that 1 i t, ad delete v i from S yieldig a ew set S = S {v i } with S = S 1 By the defiitio of P k (1 k ) we kow that the edges v i v ad v i v t+1 are i P k S Cosider a verte v p adjacet to v i i P k S If p t + 1, the p < t + k So, v p is also adjacet to v t+1 i P k S If p <, the p k + 1 ad v p is also adjacet to v i P k S Sice t < k, the v ad v t+1 are adjacet i P k S Therefore, we ca coclude that deletig the verte v i from S does ot chage the umber of compoets, ad so ad Thus, we have S + m(p k S ) (P k S ) (P k S ) = (P k S) m(p k S ) m(p k S) + 1 S 1 + m(p k S) + 1 (P k S) = S + m(p k S) (P k S) = T (P k ) This is cotrary to our choice of S Thus we must have t k Now suppose t > k Delete v t from the set S yieldig a ew set S 1 = S {v t } Sice t > k, the edge v t v is ot i P k S 1 Cosider a verte v p adjacet to v t i P k S 1 The, p t + 1 ad p t + k, ad so v p is also adjacet to v t+1 i P k S 1 Therefore, deletig v t from S yields ad So, (P k S 1 ) = (P k S) m(p k S 1 ) m(p k S) + 1 S 1 + m(p k S 1 ) (P k S 1 ) S 1 + m(p k S) + 1 (P k S) = S + m(p k S) (P k = T (P k ), S) which is agai cotrary to our choice of S Thus, t = k, ad so S cosists of the uio of sets of eactly k cosecutive vertices Lemma 19 There is a T -set S for the graph P k, such that all compoets of P k S have order m(p k S) or m(p k S) 1 Proof Amog all T -sets of miimum order, cosider those sets with maimum umber of miimum order compoets, ad we let s deote the order of a miimum compoet Amog these sets, let S be oe with the fewest compoets of order s i P k Suppose s m(p k S) Note that all of the compoets must be sets of cosecutive vertices Assume that C p is a smallest compoet E-ISSN: Issue 9, Volume 11, September 01

10 WSEAS TRANSACTIONS o MATHEMATICS Fegwei Li The V (C p ) = s, ad without loss of geerality, let C p = {v 1, v,, v s } Suppose C e is a largest compoet, ad so V (C e ) = m(p k S) = m ad let C e = {v j, v j+1,, v j+m 1 } Let C 1, C,, C a be the compoets with vertices betwee v s of C k ad v j of C e, such that C i = p i for 1 i a, ad let C i = {v i1, v i,, v ipi } Now we costruct the verte set S as S = S {v s+1, v 1p1 +1, v p +1,, v a p a+1 } Therefore, S = S, ad So we have Therefore, {v 11, v,, v a1, v j } m(p k S ) m(p k S) (P k S ) = (P k S) S + m(p k S ) (P k S ) S + m(p k S) (P k S) T (P k ) = S + m(p k S ) (P k S ) But, P k S has oe less compoets of order s tha P k S, a cotradictio Thus, all compoets of P k S have order m(p k S) or m(p k S) 1 So, m(p k S) = k( 1) By the above two lemmas we give the eact values of teacity of the powers of paths Theorem 0 Let P k be a powers of a path P ad = r(k + 1) + s for 0 s < k + 1 The T (P k ), if k + 1 k(r 1)+ = k(r 1) r r, if > k + 1 ad s=0 kr+ kr r+1 r+1, if > k + 1 ad s 0 Proof If k + 1, the P k = K, so, T (P k ) = If > k + 1, let S be a miimum T -set of P k By Lemmas 18 ad 19 we kow that ad S = k( 1) m(p k S) = We distiguish two cases: k( 1) Case 1 If s = 0, the = r(k + 1), by m(p k k( 1) S) = 1 We kow that r Thus, by the defiitio of teacity we have k( 1) T (P k k( 1) + ) = mi{ r} Now we cosider the fuctio k( 1) k( 1) + f() = f () = k k k k + k 3 = 3 Sice 3 > 0, we have f () 0 if ad oly if g() = k k 0 Sice the root of the equatio g() = k k = 0 is = +k k Whe +k k, it is easily see that +k k > r, so, if r, we have g() 0, so, f () 0, ad so f() is a decreasig fuctio ad the miimum value occurs at the boudary Thus = r The, k(r 1) T (P k k(r 1) + r ) = r Case If s 0, the = r(k + 1) + s, by m(p k k( 1) S) = 1 We kow that r + 1 Thus, by the defiitio of teacity we have k( 1) T (P k k( 1) + ) = mi{ r+1} Now we cosider the fuctio k( 1) k( 1) + f() = f () = k k k k + k 3 = 3 Sice 3 > 0, we have f () 0 if ad oly if g() = k k 0 Sice the root of the equatio g() = k k = 0 is = +k k Whe +k k, it is easily see that +k k > r + 1, so, if r + 1, we have g() 0, so, f () 0, ad so f() is a decreasig fuctio ad the miimum value occurs at the boudary Thus = r + 1 The, kr T (P k kr + r+1 ) = r + 1 E-ISSN: Issue 9, Volume 11, September 01

11 WSEAS TRANSACTIONS o MATHEMATICS Fegwei Li 5 Relatioships Betwee Teacity ad Some Other Vulerability Parameters I this sectio, the relatioships betwee the teacity ad some vulerability parameters, amely the itegrity, toughess ad scatterig umber are established Theorem 1 [] For ay graph G of order, β(g) + α(g) = Theorem If G is a icomplete coected graph, I(G) is the itegrity of G ad β(g) is the idepedece umber of G, the we have T (G) I(G) β(g) Proof Suppose that S is a T -set of G The, by the defiitio, we have T (G) = S + m(g S) (G S) It is obvious that (G S) β(g), S +m(g S) I(G) So we have T (G) = S + m(g S) (G S) I(G) β(g) The result i Theorem is best possible, this ca be show by the gear graph G = G Lemma 3 [5] If G is a icomplete coected graph, I(G) = κ(g) + 1 if ad oly if κ(g) = α(g) Theorem 4 Let G be a icomplete coected graph, if κ(g) = α(g), the we have T (G) = κ(g) + 1 β(g) Proof Let we select the maimum coverig set S be a cut-set of G The, S = α(g), ad by Theorem 14, we have (G S) = α(g) = β(g), m(g S) = 1, by the defiitio of teacity, we have = κ(g) + 1 β(g) O the other had, by Theorem ad Lemma 3, we have T (G) I(G) β(g) = κ(g) + 1 β(g) Thus, whe κ(g) = α(g), we have T (G) = κ(g)+1 β(g) Theorem 5 If G is a icomplete coected graph, t(g) is the toughess of G ad α(g) is the coverig umber of G, the we have T (G) t(g)(1 + 1 α(g) ) Proof Suppose that S is a T -set of G The, by the defiitio, we have T (G) = S + m(g S) (G S) It is obvious that (G S) β(g), m(g S) 1, ad S α(g) So we have T (G) = S +m(g S) (G S) = S (G S) ( S +m(g S) S ) t(g)(1 + m(g S) S ) t(g)(1 + 1 α(g) ) The result i Theorem 5 is best possible, this ca be show by the graph G = K 1, 1 Theorem 6 If G is a icomplete coected graph, s(g) is the scatterig umber of G ad β(g) is the idepedece umber of G, the we have s(g) + 1 T (G) + 1 κ(g) Proof Let S be a cut-set of G The, by the defiitio of scatterig umber, we have s(g) (G S) S S + m(g S) + 1 (G S) The, by the defiitio of teacity, we have T (G) S + m(g S) (G S) = α(g) + 1 β(g) T (G) S + m(g S) (G S) + 1 (G S) (G S) E-ISSN: Issue 9, Volume 11, September 01

12 WSEAS TRANSACTIONS o MATHEMATICS Fegwei Li So, we have (G S) + 1 T (G) + 1 O the other had, we kow that S κ(g) Thus (G S) S + 1 T (G) + 1 κ(g) By the defiitio of scatterig umber ad the choice of S, we kow that s(g) = ma{(g S) S } + 1 T (G) + 1 κ(g) The result i Theorem 6 is best possible, this ca be show by the graph G = K 1, 1 6 Coclusio If a system such as a commuicatio etwork is modeled by a graph G, there are may graph theoretical parameters used to describe the vulerability of commuicatio etworks icludig coectivity, itegrity, toughess, bidig umber, teacity ad rupture degree Two ways of measurig the vulerability of a etwork is through the ease with which oe ca disrupt the etwork, ad the cost of a disruptio Coectivity has the least cost as far as disruptig the etwork, but it does ot take ito accout what remais after disruptio Oe ca associate the cost with the umber of the vertices destroyed to get small compoets ad the reward with the umber of the compoets remaiig after destructio The teacity measure is compromise betwee the cost ad the reward by miimizig the cost: reward ratio Thus, a etwork with a large teacity performs better uder eteral attack I this paper, we have obtaied the eact values or bouds for the teacity of some special graphs Ackowledgemets: The author is thakful to aoymous referees for their costructive suggestios ad critical commets, which led to this improved versio Refereces: [1] Vecdi Aytaç, Computig the teacity of some graphs, Selçuk J Appl Math, Vol10, 009,(1):pp [] J A Body ad U S R Murty, Graph Theory with Applicatios, Macmilla, Lodo ad Elsevier, New york, 1976 [3] C A Barefoot, R Etriger ad Heda C Swart, Itegrity of trees ad powers of cycles, Cogress Numer, Vol58, 1987, pp [4] C A Barefoot, R Etriger ad H Swart, Vulerability i graphs - A comparative survey, J Combi Math Combi Comput, Vol1, 1987, pp 1- [5] K S Bagga, L W Beieke, W D Goddard, M J Lipma ad R E Pippert, A survey of itegrity Discrete Appl Math, Vol37/38, 199, pp 13-8 [6] K S Bagga, L W Beieke, M J Lipma ad R E Pippert, Edge-itegrity- a survey, Discrete math, Vol14, 1994, pp 3-1 [7] A Bradstädt, V B Le ad J P Spirad, Graph classes: a survey, SIAM, Philadelphia, PA, 1999 [8] T C E Cheg, Yikui Li, Chuadog Xu ad Sheggui Zhag, Etreme teacity of graphs with give order ad size, arxiv reprit arxiv: v1 [mathco](011) [9] S A Choudum ad N Priya, Teacity of Complete Graph Products ad Grids, Networks, Vol34, 1999, pp [10] V Chvátal, Tough graph ad hamiltoia circuits, Discrete Math, Vol5, 1973, pp 15-8 [11] M Cozzes D Moazzami ad S Stueckle, the Teacity of Harary Graphs, JComb Math Comb Comput, Vol16, 1994, pp [1] M Cozzes, D Moazzami ad S Stueckle, The teacity of a graph, Proc 7th Iteratioal Coferece o the Theory ad Applicatios of Graphs, Wiley, New York, 1995, pp [13] M B Cozzes ad Shu-Shih Y Wu, Verte- Neighbor-Itegrity of Powers of Cycles, Ars Combiatoria, Vol48, 1998, pp [14] A Kirlagic, The Rupture Degree ad Gear Graphs, Bull Malays Math Sci Soc, Vol3, 009, (1):pp [15] F W Li, Q F Ye ad B H Sheg, Computig rupture degrees of some graphs, WSEAS Trasactios o Mathematics, Vol11, 01, (1): pp 3-33 [16] F W Li, Q F Ye ad X L Li, Teacity ad rupture degree of permutatio graphs of complete bipartite graphs, Bull Malays Math Sci Soc, Vol 34, 011, (3): pp [17] Y K Li, S G Zhag ad X L Li, Rupture degree of graphs, It J Comput Math, Vol8, 005, (7):pp [18] D E Ma, The teacity of trees, PhDThesis, Northeaster Uiversity, 1993 E-ISSN: Issue 9, Volume 11, September 01

13 WSEAS TRANSACTIONS o MATHEMATICS Fegwei Li [19] D Moazzami, Vulerability i graphs - A comparative survey, J Combi Math Combi Comput, Vol 30, 1999, pp 3-31 [0] Y H Peg, C C Che ad K M Koh, O the edge-toughess of a graph (I), Southeast Asia Math Bull, Vol1, 1988, pp [1] B L Piazza, F S Robertst ad S K Stueckle, Edge-teacious etworks, Networks, Vol5, 1995, pp 7-17 [] S G Zhag, X L Li ad X L Ha, Computig the scatterig umber of graphs, It J Comput Math, Vol 79, 00,():pp [3] S G Zhag, Z G wag, Scatterig umber i graphs, Networks, Vol37, 001, ():pp [4] X K Zhag ad C M Yag, The bidig umber of the Powers of Path ad circuit, Joural of guagdog istitute for atioalities, Vol4, 1993, pp [5] E Asla ad A Kirlagic, Computig The S- catterig Number ad The Toughess for Gear Graphs, BullSocMath Baja Luka, Vol 18, 011, pp 5-15 E-ISSN: Issue 9, Volume 11, September 01