on the connectivity and the conditional diameter of graphs and digraphs

On the Connectivity and the Conditional Diameter of Graphs and Digraphs

C. Balbuena and A. Carmona

Departament de MatemAtica Aplicada Ill, Universitat Politknica de Catalunya, 08034 Barcelona, Spain

J. FBbrega and M. A. Fiol

Departament de Maternatica Aplicada i Telemhtica, Universitat PolitQcnica de Catalunya, 08034 Barcelona, Spain

Recently, it was proved that if the diameter D of a graph G is small enough in comparison with its girth, then G is maximally connected and that a similar result also holds for digraphs. More precisely, if the diameter D of a digraph G satisfies D 5 21 - 1, then G has maximum connectivity ( K = 6 ) . and if D 5 21, then it attains maximum edge-connectivity ( A = 6 ) , where I is a parameter which can be thought of as a generalization of the girth of a graph. In this paper, we study some similar conditions for a digraph to attain high connectivities, which are given in terms of what we call the conditional diameter or P-diameter of G. This parameter measures how far apart can be a pair of subdigraphs satisfying a given property P, and, hence, it generalizes the standard concept of diameter. As a corollary, some new sufficient conditions to attain maximum connectivity or edge-connectivity are derived. It is also shown that these conditions can be slightly relaxed when the digraphs are bipartite. The case of (undirected) graphs is managed as a corollary of the above results for digraphs. In particular, since I 2 1, some known results of Plesnik and Znhm are either reobtained or improved. For instance, it is shown that any graph whose line graph has diameter D = 2 (respectively, D I 3) has maximum connectivity (respectively, edge-connectivity.) Moreover, for graphs with even girth and minimum degree large enough, we obtain a lower bound on their connectivities. CI 7996 John Wiley & Sons, Inc.

1. INTRODUCTION

The study of some parameters related to the connectivity o f (di)graphs has recently proved to be o f some interest in the design of reliable and fault-tolerant interconnection or communication networks. This is because such networks are usually modeled by graphs or digraphs-de- pending on the bidirectional or unidirectional nature o f their links-whose vertices and edges represent, respectively, the processing elements (nodes of the network) and the communication links between them. Thus, a common requirement i s the fault-tolerance of the network, i.e., the ability o f the system to work even if some nodes and/or links fail. See. for instance, the survey o f

Bermond et al. [ 51. This fact has encouraged many authors to study sufficient conditions for a graph or digraph to have high connectivity. In particular, i t i s interesting to know under which circumstances the (di)graph i s maximally connected or edge-connected, i.e.. its connectivity or edge-connectivity equals i t s minimum degree. Most o f these conditions are stated in terms o f other relevant parameters in network design, such as the number of vertices, minimum and maximum degrees, diameter (which measures the maximum communication delay between nodes), and girth. For instance, a well-known result o f Chartrand [ 8 ] states that if G i s a graph on n vertices and with minimum degree 6 L Ln/2J then G i s maximally edge-connected. Since Chartrand's paper,

NETWORKS, Vol. 28 (1996) 97-105 (c 1996 John Wiley & Sons. Inc. CCC 0028-3045/96/020097-09

97

98 BALBUENA ET AL.

many results of a similar nature have been derived by different authors. See. e.g.. the works of Ayoub and Frisch [ I ] , Lesniak [? I ] , Goldsmithand White [17],Goldsmith and Entringer [ 161, Volkmann [ 26. 271. and the authors [ 4 J . Roughly speaking. the conditions studied in these papers are of the following type:

( a ) Given the number of vertices, the minimum degree is large enough.

Other approaches are, however. of a different nature and involve other parameters. Thus, in the papers of Es- fahanian [ lo] . Fiol [13]. lmase et al. [19]. and Soneoka et al. [ 24.25 1. we can find the following type ofconditions:

( b ) Given the diameter and the maximum degree, the number of vertices is large enough.

Moreover, taking the girth into consideration-or, in the case of digraphs. a new parameter of a similar signif- icance-Soneoka et al. “4. 251, FAbrega et al. [ I I . 12, 141 and Balbuena et al. [ 31 considered the following condition:

( c ) Given the girth. the diameter is small enough.

Some mixed-type conditions have also been considered in the papers of Fiol [I31 and the authors [3 ] . For instance,

(b-c) Given the diameter, the maximum degree, and the girth, the number of vertices is large enough.

This paper concentrates upon conditions of type ( c ) , but instead of using the standard diameter. it is shown that many of the previous results can be extended by considering the so-called conditional diameter. This parameter measures the maximum distance among sub( di)graphs satisfying a given property. So, its consideration could be of some interest if. in some applications. we need to minimize the communication delays between the network clusters modeled by such sub( di)graphs. The new results are derived for general and bipartite (di)graphs in Sections 2 and 3. respectively. The remainder of this section is devoted to recall some basic concepts and to fix the terminology.

From now on, G stands for a simple digraph, i.e., without loops or multiple edges, with set of vertices C’( G ) and set of (directed) edges E( G). If G is bipartite, we will write I,‘(G) = C’(, U [.TI, where L‘” and C‘, denote the partite (or stable) vertex sets. I f s E V ( G), let (x) and r + ( x) denote. respectively, the sets of vertices adjacent to and from x. Their cardinalities are the in-degree of x, 6- ( s) = 6;( x) = I r - (s) I , and the oirt-degree ofx , 6 (x) = 6L(x-) = l r + ( . x ) l . The minimum in-degree and out-

degree will be denoted by 6 + = 6’( G ) and 6 - = 6 - ( G) . respectively. and the niiriirniirn ckgrw of G‘ by 6 = 6( G ) = min { 6 - , 6’ } . For any pair of vertices s, 1’ E 17(G) . a path from x to 1’ is called an s + 1’ path. The distuncc ,/iorn .x l o js is denoted by d( x, y) = d(;( x, J,) and D = D( G ) - mux,,., I ((,) { d ( x , J)) stands for the diutiwtcv of G. A digraph G is said to be (stronglj,) connected when for any pair of vertices .I-, J’ E 1’( G ) there always exists an A- +

jp path. Throughout the paper, G stands for a connected digraph. so that 6( G ) 2 1. As usual, the conncctivilj- (or v t~r~ei - -~ .or i r ic~c~l i l ’ i l !~) and e ~ ~ : c - c ~ o r i i i t ~ ~ ~ r i ~ ~ i l ! ~ of G are denoted by K = K ( G ) and X = A ( G ) , respectively. I t is well known that K i X I 6; see, for instance. Geller and Harary [ 151. Hence. G is said to be mu.\-irnul/~* conncwcd when K = X = 6 and rnu.vimullj* cc~~~~c.-c.onncc.tcd when X = 6.

Given two subdigraphs G I , Gz C G, the distuncc. from GI to Gr is defined as

-

The distance between two subsets of vertices I’? C 1’( G ) , denoted by d( I ’ , , I ; ) . is defined analogously and. clearly. d ( G l , G I ) = d( I ’ ( G l ) . C’(GZ)). lfoneofthe subdigraphs or bertex sets. say GI or I’,. consists of a single vertex .\-. we simply write d ( . i , G?) or d ( x , V 2 ) respectively .

Given a property P of a pair ( G I . Gz) of subdigraphs of G, we introduce the concept of condiliotiul drurnclcr or P-dititnctc~r- defined by

D,,, = max { d( G I , G,) : ( G I , G r ) satisfy P 1 . ( I ) G,.(qCCJ

For instance. if ‘7’ is the property of G,, i = I , 2. being trivial ( i.e.. isolated vertices ). the conditional diameter D , coincides with the standard diameter D.

Similar notation and results apply. and are well known. for (undirected) graphs. For all the definitions not given here. we refer the reader to the book of Chartrand and Lesniak [ 91. However, for our purposes. we will deal with a (simple) graph G by considering its u.s.sor~ialcd.~jirnmtitr-ic~ digrupli G*. i.e., the digraph obtained from G by replacing each edge xj- E E ( G ) by the two (directed ) edges (s, J.)

and ( J-, x ) forming a “digon.” The basic reason is that K( G* ) = K ( G), and since a minimum edge-disconnecting set cannot contain digons. also A ( G * ) = A ( G).

To study the connectivity ofgraphs and digraphs. some of the authors [ 1 I ] (see also [ 141) introduced a new parameter related to the number ofshort paths, the definition of which is as follows:

CONNECTIVITY /DIAMETER OF GRAPHS AND DIGRAPHS 99

1.

2.

l f ' d ( x , y ) < I,, rhe shortest .Y + .I* puth is itniqite and (here ure at rnost x dierenr x + ypaths of length d ( x , y) + 1; l f d ( s, y ) = I,, there is on/!* onc shortest x + palh.

Intuitively, this parameter measures how far away from a vertex set F (of cardinality smaller than 6 - x ) one can move. Note that, since G has no loops, I, 2 I for any x. Obviously, the same definition applies for a graph G (considering undirected paths). In this case. it turns out that the parameter I = lo( G * ) is tightly related to the girth g of G. Indeed, one can readily check that I = [ ( g - 1 ) /21 .

2. CONNECTIVITY AND CONDITIONAL DIAMETER

As we explained in the Introduction, some of the derived conditions to attain a high connectivity require one to have a small diameter in comparison with the girth or parameter I,. Thus, in [ I 11 and [ 141. it was proved that if G is a digraph with minimum degree 6 > I , diameter D , parameter I,, and connectivities K and X then

~ 2 6 - x if D S 2 1 , - 1 ; ( 2 )

X 2 6 - x if D I 21,. ( 3 )

This result has some interesting corollaries for both graphs and digraphs: see Jolivet [ 201. Plesnik [ 221, Soneoka et al. [24,25]. and the above-mentioned references [ I I , 141. Some improvements of these results can be obtained if other relevant parameters of G are known. For instance, in [ 3, 131, the order of G is also taken into account. In this section, we show that, even without any further information on G , the above conditions in ( 2 ) and ( 3 ) can be generalized by using some conditional diameters instead ofthe standard one D. The main result of this section is the following theorem:

Theorem 2.1. Lct G bc u digruph bcith minirnirm dc>grcc 6 = min { 6 +, 6 } > I . parurntwr I,( G ) = I,. and con- nectivitics K und A. Then,

( a ) I ~ ' K < 6 - K , there e.uist tw'o indirced .subdigraphs G I . G z C G , ~ i i t h d t ( G , ) ~ 6 ' - ~ a n d b - ( G ~ ) 2 6 - - ~ ,

(6) I f ' X < 6 - x, rherc e.uist two indicced siibdigruphs GI, GI C G, with 6'(Gl) 2 6 + - X a n d 6 - ( G 2 ) 2 6 - - A.

Proyf: We can assume that G # K,* since, by definition, K ( K : ) = b ( K , * ) = n - 1. To prove ( a ) , let F b e a minimum order disconnecting set of vertices, i.e.. I FI = K

such thUl d( GI, G . 2 ) 2 21,;

stid1 that d(G1, Gz) 2 21, + I .

and G - F is not connected. Let the set V ( G ) \ F be partitioned into two disjoint nonempty sets V - , V + such that G - F has no edges from V" to V ' . Let p = max{d(x, F) : x € V } and p' = max{d(F, x ) : x E I.'+ } . Then. the vertices of V - and V + can be, respectively. partitioned into subsets V , , 1 5 i I p and V ; , I 5; I p', according to their distance to and from F, i.e., 1 ~ ' , = { s E V - : d ( x , F ) = i } a n d V I , = { x E V + : d ( F , s) = j } (Vo = V b = F). Without lossofgenerality, suppose that p I p' (otherwise, use the converse digraph G).

Under the above conditions, it is proved in [ 1 1 . Th. 3.11 that if p 5 I , - 1 then K = I FI 2 6 - x, which contradicts our initial assumption. Hence, we can assume that p 2 I,. Let us consider the (nonempty) set V ( I,) = { s E c'- : d ( x , F) 2 I,}. We claim that every vertex x of the subdigraph GI = ( V ( I,)), induced by V ( I,), has out-degree 6L,(x) 2 6' - K . The result is trivial if d ( x , F) > l r , since, then. I " ( x ) C V ( G l ) . Hence, we can assume that d ( x , F) = I,. If 6&(x) = I r + ( x ) fl V(l,)l < 6t.(x) - K , vertex x has at least K + 1 out-neighbors in V,r-l, say sI , x 2 . . . . , x , , I . Now, for each x, , 1 I i I K

+ 1, let/,' be a vertex in F a t minimum distance from x, . Since I FI = K , we must have/; =,J for some i Z J , and, hence, there exist two disjoint x + I,: (shortest) paths of length I,. which contradicts the definition of this parameter. Therefore, 6t. , (x) 2 62.(x) - K 2 6' - K as claimed and, hence. a + ( GI ) 2 6' - K . Similarly, since p' 2 p 2 Is, the subdigraph G2, induced by the set of vertices V'( I,) = { x E 1'' : d ( F , x) 2 I , } , satisfies 6-(Gz) 2 6 - - K .

Finally, since any path from V - to I.'' goes through F ,

( b ) The proof of this case goes along the same lines as ( a ) . Now, let E be a minimum edge-disconnecting set of G, i.e.. 1 El = X and G - E is not connected. We again consider a partition of V( G ) consisting of two disjoint nonempty sets L - . I,'*, such that the only edges from I / - to I." are those in E. Let F = { /E V - : (f,f") E E } , F '= { / ' € V ' : ( , f , / ' ) E E } , V, = { x € V - : d ( x , F ) = i ) , O < i s p , a n d V ; = { x € V + : d ( F ' , x ) = j } , O 5; 5 p', and assume that p I p'. Within this framework, i t w a s p r o v e d i n [ l l ] t h a t i f p s I , - 1 t h e n X r 6 - a , so that we can assume that p 2 I,. Then, using the same notation as in case (a ) , and reasoning similarly, it is proved that the induced subdigraphs GI = (V(I , ) ) and G2 = ( V ' ( l r ) ) satisfy 6 + ( G l ) 2 6 + - A, 6- (G2) 2 6- - A, and d( GI, G,) 2 21, + I , as claimed.

d(G1, G2) 2 d ( V ( G l ) , F) + d(F, V ( G 2 ) ) 2 21,.

At this point. to give some corollaries of the above theorem, it is useful to consider some conditional diameters. To begin with, let D,, 0 5 Y I 6, denote the 'P- diameter, where P is the property of both subdigraphs G I , G2 C G being induced and with minimum degrees 6+( GI ), 6 - ( G,) 2 u. With this notation, note that Do = D and. clearly,

100 BALBUENA ET AL.

From now on, it is assumed that 6 > I , since, otherwise. the digraph is obviously maximally connected.

Corollary 2.2. Let G he a digraph with minimum degree 6, paraineter I,, and connectivitie.r K and X. Thoi ,

( a ) K L 6 - A tj'Dn+] I 21, - 1; ( 6 ) X 2 6 - A $D,+I I 21,.

Proyf.' To prove ( a ) , suppose that G has K s 6 - A

- I . Then. by Theorem 2 . l (a ) , there exist two induced subdigraphs G I , Gz such that & ' (GI ) 2 6' - K L 6' - 6 + K + 1 2 A + 1, d-(G,) 2 K + I , and d ( G l , G,) 2 21,. contradicting the hypothesis on D,,, . Case ( b ) is proved similarly by using Theorem 2.1 ( b ) .

Note that, because of (4) . Corollary 2.2 implies the known results ( 2 ) and (3 ) . Now, let us consider the case A = 0. Since each subdigraph with 6' 2 1 or 6 - 2 1 contains a directed cycle. we will make use of the conditional diameter D(, = DP, where the condition P is now to be a (directed ) cycle.

Corollary 2.3. Let G he a digraph \titi1 rninitniini dc<yrcc 6, puraineter lo = I, and ~~oii~i~~~,t ivit ie.s K and A. Then.

( a ) K = 6 $D,. I 21 - 1, ( b ) X = 6 i / 'Dc. I 21.

When the girth g of the digraph is known (i.e., the minimum among the lengths of its directed cycles), Cor- ollary 2.2 leads to the following result, which, roughly speaking, says that the connectivity or edge-connectivity is at least 6 - A, even if there exist some vertices with eccentricity greater than or equal to 21, or 21, + 1, respectively. To be more precise, let us denote by n*( 6, g) the minimum number of vertices of a digraph with minimum degree h 6 and girth at least g. A trivial general bound is n*( 6, g ) 2 g + 6 - 1. Moreover, given a subset of vertices S C 1'( G). let D.\. be the 'P-diameter, where P is the property of both subdigraphs G, , i = I , 2, being a vertex in 1'( G)\S. Thus.

Corollary 2.4. Let G be u digraph nilh rninimiim degree 6, parumeter I,, girth g. and connc.ctiviries K and A. Then.

( a ) K 2 6 - A $DS I 21, - I jbr sotne S C C'(G). IS1 = n * ( ~ + 1,g)- 1;

( 6 ) X L 6 - A ij'Ds I 21,for .some S C V ( G ) . I S ( = n * ( ? r + l , g ) - 1 .

Proof.' ( a ) Suppose that G has K 5 6 - A - 1 . Then, by Corollary 2.2(a), DT+I t 21,. Hence, there exist two

induced subdigraphs G I . G2, with &'(GI) 2 K + I and 6 - ( G,) 2 A + 1. at distance d( G I , GZ) 2 21,. Therefore. I b'(G,)I 2 n * ( ~ + 1, g) , i = I , 2, since clearly G, has girth at least g . Let x, and ji, 1 5 j I n*( A + 1. g). be different vertices of GI and Gr, respectively. Hence, d(,u,, j;) 2 21, for any j, and so at least one vertex of each pair (x,. y,) must belong to S, which yields the contradiction I SI L n*( A + I , 8). Case ( b ) is proved analogously by using Corollary 2.2( b) . w

Cacceta and Haggkvist [ 71 conjectured that the minimum number of vertices n( 6. g ) of a digraph with minimum degree exactly 6 and girth equal to g is 6(g - 1 ) + I . This conjecture has been proved for some small val- ues of 6 or g. Moreover, Hamidoune [ 181 proved it in the case of vertex transitive digraphs. Note that if the conjecture is true, then n*( 6, g ) = n(6 , g). In the case A

= 0, we obtain the following conditions to have maximum connectivities:

Corollary 2.5. Lvt G he u digruph with tninimirin &grw 6, puratnc~tcr I, girth <g. and c,oiinc~~tiriities K and A. Then.

( u ) K = 6 1f'D.q 5 21 - 1 ,fi)r soine S C V ( G ) , I Sl =

g - I ; ( b ) X = 6 if' Ds I 31./i)r soinc S C k'(G), I SI =

g - I .

Theorem 2. I provides some similar corollaries for ( undirected ) graphs, with the appropriate changes. First, recall that. givcn a graph G' with girth g, its corresponding symmetric digraph G* has parameter I = I0(G*) = ~ ( g - 1 ) / 2 J. Then, ( 2 ) and ( 3 ) yield the following results. already given by Soneoka et al. [ 24, 251:

( g - 3. g even;

g - 1, godd ( 6 )

g - 2, geven. X = 6 if D I

Taking A = 0 in Corollary 2.2. we then get the following stronger results:

Corollary 2.6. Lct G be u graph )titli iniiiir?iiiin degree 6, girth g, rind ionnc.c.tivitic>.y K and A. Then.

g - I , g d d

I: - 2, g e v m . ( b ) X = 6 I/ 'D, I


The conditional diameter D , measures now the maximum distance among all nontrivial induced subgraphs, since in the associated symmetric digraph G* we have 6' = 6 - .

This corollary can also be given in terms of the diameter of the line graph LG. Recall that in the line graph LG of a graph G each vertex represents an edge of G and two vertices are adjacent iff their corresponding edges are. Let us consider the edges x l y l , x2y2 E €( G ) . Then, the distance between the corresponding vertices of LG satisfies d L G ( x I y l , x 2 y z ) = d G . ( x I y I , x2y2) + 1, so that, in terms of the conditional diameter D , , we have D ( L G ) = D , + I . Hence,

Corollary 2.1. Lel G be a gruph with minimum degree 6. girth g, and connectivities K and A. Then,

g - 1, godd

g - 2, geven; ( a ) K = 6 i f D ( L G ) I

g odd ( b ) X = 6 i f D ( L G ) I

These results generalize those of Soneoka et al. [ ( 5 ) and ( 6 ) ] , given in terms of D ( G ) instead o f D( LG). In particular, since for any (simple) digraph g 2 3, we get that any graph whose line graph has diameler D = 2 ( rc- specrively, D I 3 ) has maximitm connectivity ( respcxStively, edge-conneclivity). The result concerning the edge-connectivity generalizes that given by Plesnik [ 221, stating that any graph with diameter two is maximally edge-connected.

Keeping in mind that, given a graph G, its corresponding symmetric digraph G* has girth g( G* ) = 2, Corollary 2.5 leads to the following result. where D, stands for the conditional diameter DS when S = { u 1 :

Corollary 2.8. Let G be a graph Milk minimum degree 6. girlh g. and connectivilirs K and A. Then.

( a ) K = 6 i fDv

I g - 2 , g o d d 5 for some u E V ( G ) ;

( g - 3, g even

( h ) X = 6 if'Dv

g - I , godd

g - 2, g even for some u E V ( G ) .

Again, since g 2 3, Corollary 2.8(b) tells us that if there exists a vertex v such that d ( x , y ) 5 2 for any x , y E V ( G)\ { u } then X = 6. This result was first given by Plesnik and Zn6m in [ 231.

The above result can be improved in the case of even g , i.e., we can allow more than one vertex with eccentricity greater than or equal to g - 3 or g - 2. Moreover, we prove that the sufficient conditions in ( 5 ) and (6) can also be relaxed for minimum degree 6 5 4.

Theorem 2.9. Let G be a graph with even girth g. mini- mitm ckgree 6. diamerer D, and conneclivities K and A.

(a .1 ) I / ' K = 1 and 6 2 2 , then D 2 g; ( a . 2 ) I ~ K = 2 [ ~ = 31 and 6 2 3[6 2 41, then D 2

g - 1; ( a . 3 ) I f K < 6 and 6 L 5 , then eirher D 2 g - 1 , or D

= g - 2 und ihere cxisi two induced subgraphs G , , G2 C G , M ~ i t h 6 ( G I ) , 6 ( G ' 2 ) ~ 6 - ~ + 2 , s i t c h t h a t d ( G l , G 2 ) = g - 2 .

(b .1) I f ' X = I a n d 6 = 2 , t h e n D > g + l ; ( h . 2 ) !/'A = 2 [ X = 31 and 6 2 3 [ 6 2 41, then D L g; ( b . 3 ) I f ' X < 6 and 6 2 5 , lhcw either D 2 g. or D =

g - 1 and thew erist t M w induced subgraphs G I , G2 C G, wilh 6(GI ) , 6(Gz) 2 6 - X + 2 , such thai d ( G 1 , G 2 ) = g - 1 .

Proul.' ( a ) Using the same notation as in Theorem 2.1, we have p' 2 p 2 I = [ ( g - 2 ) / 2 ] , so that D 2 g - 2. Moreover, if either p 2 I + I or p' 2 I + 1, then D 2 g - 1. So, assume that p = p' = 1. In this case, we have the two following results, on which our proof is based:

( i ) We have I r ( x ) r l V/I 2 6 ( x ) - K + 1 for any x E V/ , and 1 r(y) n V ; I 2 6 ( y ) - K + 1 for any y E V i . We prove this by contradiction. Consider that x E V/ and assume that 1 r(x) n V,( I 6(x) - K . Then, there exist K

+ 1 vertices x, E F(x), I I i I K + 1, of which x , , . . . , x, belong to V/- I . Let,/,' E Fdenote the vertex at minimum distance from x, . As 1 FJ = K , there exist some i # j such that/,' = A = f. Then. there is a closed walk of length d ( f , x,) + d( f , x,) + 1 s 21 + 1, which is impossible since g = 21 + 2. The reasoning on y E V ; is similar.

( i i ) If there exist vertices x E V/ and J E F, such that d ( x , J ' ) = I and d ( x , F\ { f } ) L I + 1, then D L g - I . Indeed. take yl E V ; and y2 E r(yl ) n V ; (f0 by (i) .) If d ( f ; y , ) 2 1 + 1 for some i = 1, 2, then d ( x , y , ) 2 21 + 1 = g - 1 and, hence, D 2 g - I . Otherwise, we get a closed walk of length d ( f ; y I ) + d ( f , y 2 ) + 1 = 21 + 1, contradicting that g = 21 + 2.

We are ready now to prove the theorem. If K = I , then ( i ) gives I r ( x ) rl V/I 2 6(x), which is impossible. There- fore, our first assumption cannot hold and so p' 2 p 2 I + I . This proves case (a.1). If 6 2 3 and K = 2, then ( i ) gives I r ( x ) n V,l = 6 ( x ) - 1 and, hence, vertex xsatisfies the hypothesis in ( i i ) . If 6 2 4 and K = 3, then ( i ) gives I r (x) n Vll 2 6(x) - 2 and now any vertex xI E r ( x ) n V/ satisfies such a hypothesis. This completes the proof of case (a.2). Finally, if 6 2 5 and K -= 6, let us consider

102 BALBUENA ET AL.

the nonempty set I;. and suppose that D = g - 2. Then. ( i ) implies that 1 r(x) f l V/- I 1 I K - I . Moreover, if I r(x) r l L ; . l 1 = K - I , each vertex y E r (x ) f l V, satisfies hypothesis ( i i ) , and, hence. D 2 g - I . a contradiction. Thus, I I'(s) n V/ I I I K - 2 and, hence, 6,,(x) = 1 r(x) f l I'/l 2 6 - K + 2. where G I = (C;). Consequently. 6( G I ) 2 d - K + 2. (The reasoning on G7 = ( V ;) is similar.)

Statements ( b ) are proved along the same lines as above. rn

As a direct consequence of the above theorem, we get the following results, to be compared with ( 5 ) and ( 6 ) :

( a . I ) K = 6 / / 'a = 2 and D I g - I ; ( a . 2 ) K 2 min { 6 , 4 } 11'6 2 3 und D 4 g - 2 . ( h . I ) X = d (6 = 2 urid D I g; ( b . 2 ) X 2 m i n ( 6 , 4 ) ~ / ' 6 s 3 u i i c l D - ( g - I . rn

In the case 6 2 5. we can improve upon the above results by using the conditional diameter D 3 .

We have some particular examples showing that Cor- ollary 2. I 1 ( a ) is the best possible. Figure I shows a graph with minimum degree 5. girth g = 4. and connectivity K

= 4. such that D = g - 2 = 2 and D3 = g - 2 = 2. On the other hand, this result is independent of the one given in ( 5 ) . Indeed. for a given minimum degree d 2 5 , the graph constructed as follows fulfills the hypothesis in the above corollary. but does not satisfy the condition in ( 5 ) . Consider the complete bipartite graphs K,, I .b and Then, take all the vertices of Ks+l .n except one in the partite set of cardinality d + 1 and add all the edges from these vertices to the vertices of K6.6.

Recalling that the minimum number of vertices of a graph with even girth g = 21 + 2 and minimum degree 6 2 3 is n ( 6 , g ) = 2 [ ( 6 / + ' - I ) / ( 6 - 2 ) ] . Corollary 2.1 I leads to the following result, whose proof is similar to that of Corollary 2.4:

(0) K = 6 $D I g - 2 and D.$ 5 g - 3 j i ) r sotnc S

( 6 ) X = 6 l / ' D I g - 1 and Ds I g - 2 for sonic S C C.(G), I SI = n(2 . I ) - 1;

c 1'( G), I SJ = n ( 2 , I) - I .

Fig. 1. A graph with d = 5, D = D3 = 2, K = 4.

3. CONNECTIVITY OF BIPARTITE DIGRAPHS

In this section. we study the case of bipartite (di)graphs. It seems natural to ask whether the results of the previous section can be improved with additional information about the structure of the considered digraphs. In particular, the works of Plesnik and Znim [ 231. Volkmann [ 26, 771, and the authors [ 3, 121 suggest that this is the case when the digraphs are bipartite. Thus. in this section, we elaborate upon the previous work to derive some new results for such a case.

Since between any two vertices of a bipartite digraph there are no two paths whose lengths differ by one, we only need to consider the parameter I= with K = 0. Then, the following simplified definition of 1 = lo holds:

Using this parameter. the following sufficient conditions for a digraph to have maximum connectivity or edge- connectivity have been given in [ 12):

Note that. in this case. the upper bounds on the diameter are one greater than the bounds given in (3) and ( 3 ) for general digraphs and K = 0. In this section, we will show that. by using again some conditional diameters, such conditions can be relaxed without affecting their consequences.

Given a bipartite digraph G with vertex set I,'( G ) = Lro U (I,, let D,,.,, p . 4 E { 0, I }, denote the conditional diameter where 'P is the property of G I and Gz being single vertices belonging to the (not necessarily different) partite sets V,, and U,, respectively. Thus. D,,,, measures the

CONNECTIVITY I DIAMETER OF GRAPHS AND DIGRAPHS 103

maximum distance between the vertices of the corresponding stable sets. Of course, D,,,, is even (respectively. odd) iff p = q (respectively, p # q). Moreover, it is easy to realize that the diameter of G is always upper-bounded by mi~/,,qE!o.l: DI1.Y + 2.

Proposition 3.2. Let G he a hipurtite digruph Mitli min- imirrn degree d, purumerrir I , and c*onnectivi(ies K atid A. Then.

( a ) K = 6 i/'D,,q I 21- 1 jbrsomep. q E {O. l } ,p # q;

( b ) X = 6 $ D ,,,/, I 21 .for so~no p E { 0. I },

Proq/.' We will only prove ( a ) , case ( b ) being similar. Suppose that G has K s 6 - I . Then, by Theorem 2.1 (a ) , there exist two induced subdigraphs GI. G2 such that d ' (G, ) 2 I , 6 - ( G 2 ) 2 1 and d ( G , , Gz) 2 21. This means that both digraphs. GI and G2 contain at least one vertex of each stable set, say, .v,.,~ E V ( G, ) f l U,,, i = 1, 2. p = 0, 1 . But. then, the distances d(xl . ( , . s ~ . ~ ) and XI.^, .u2.0)

must be at least 21 + 1 (since they concern vertices in different partite sets). Then. Dl,,Y 2 21 + 1 for any p # 4, a contradiction.

Note that the conditions on the diameter in ( 7 ) and ( 8 ) imply that Dl,.q d 21 - I for any p # 4 and DIJ,/' I 21 for any p , respectively (but not conversely). So. the above proposition generalizes the above-mentioned re-

As in the general case, the connectivity or edge-connectivity of a bipartite digraph is maximum. even if there exist some vertices with eccentricity greater than or equal to 21 + 1 or 21 + 2, respectively. To prove this, we consider again the conditional diameter D J , i.e., the P-diameter, where P is the property of being a vertex in V ( G)\S.

sults of [ 121.

Corollary 3.3. Let G he u hipurlire cligruph with tninitnurn degree 6, cotinec'tiritic~s K und A. and girth g. Then,

( u ) ~ = d i J D . ~ s 2 I / i ) r r o r t t e S C V ( G ) , IS1 = g - I ; ( h ) A = d i f D , 1 2 1 + I . f o r w t n c S C V ( G ) , IS1 =

g - 1 .

Proof' Suppose again that K s 6 - I and let GI, G2 be two induced subdigraphs of G with &+(GI ) 2 1, 6-(G2) 2 1. such that d( GI, G2) 2 2 / . Since G has even girth g, we have that I V(G,) fl Ul>1 2 ( g / 2 ) , for i = 1. 2 , p = 0, 1 . Let x,.,, E V ( G I ) r l U,. J;.,~ E L'( G?) r l U,,, for I I ; s ( g / 2 ) , p = 0. I . Hence. d ( ~ , , . ~ , y,, I ) 2 21 + 1 and d(x,. ,, y,.,)) 2 21 + I for any;. From the hypothesis, we deduce that at least one vertex of each pair must belong to S, which is a contradiction, since I SI = g - 1 . As a consequence, D , s 21 - 1 and Corollary 2.2( a ) applies. Case (b ) is proved analogously by using Corollary 2.2( b). rn

Instead of merely applying Theorem 2.1, we can take advantage of the special features of bipartite digraphs to go further. First, we will prove a new theorem specially concerned with such a case.

Theorem 3.4. Let G be a hiparrite digraph with minimtun degree 6 = { A + , 6 - 1. parameter I , diumeler D , and con- t i c w i v i r i c v K and A.

( a ) I/'K < 6, rhcn citlier D 2 21 + 2 or D = 21 + 1 and i h m erisr (MY) indtrced sirbdigruphs GI, G2 C G , wiih 6 ' ( GI) 2 6' - K + 1 and 6-( G2) 2 6 - - K + I , such that

( 6 ) ?/'A < 6. then cither D L 21 + 3 or D = 21 + 2 and I h c w erist f HW indirwd subdigraphs GI , G2 C G, with

~ l ( GI, Gz) 2 21;

6 + ( G , ) 2 6' - X + 1 atIdd-(Gz) 2 6 - - X + 1 , S U C ~

t h ~ ~ d( GI, G2) 2 21 + 1 .

Pro($ We only prove (a). Let V ( G) = Uo U U , . Using the same notation as in Theorem 2.1. we have p r 1. Assume first that (p ' 2 ) p 2 I + I , then D 2 2p L 21 + 2. Suppose now that p = I and p' z I + 1. Let x E V / , then r + (s) rl l'; # 0. Otherwise, let x, E J/,- I , I 5 i 5 6, be 6 of its out-neighbors, and let/; E F be the vertex at minimum distance from ,xi, 1 I i I 6. As 1 FI < A, we should have j,' = 1; for some i f j . Then, there would be two different paths of length I from x to j ; , a contradiction. Take now that xI E V/ fl I ' + ( x ) . Then, for any vertex y E P';+l , we must h a v e d ( . u , y ) . d ( x l , ~ ) ) ~ 2 1 + I.There- fore. one of these distances must be at least 21 + 2, since .x and xI belong to different partite sets. We conclude that D 2 21 + 2. Finally, suppose that p = p' = I and D = 21 + 1. Let us consider the (nonempty) set V / . We claim that every vertex x of the subdigraph GI = ( P'l) has out-degree 6 t , (s) z 6 + - K + 1. (The reasoning on G2 = ( V ; ) is similar.) By contradiction, if 6 i . , ( s ) = I I " ( x ) rl V/I < 6 : . ( ~ - ) - K + I , vertex x has at least K

out-neighbors in V,-l , say xI , x2, . . . , x.. Besides, let x,,~ E I '+ (x ) fl V/ . Now. for each xi. 1 s i 5 K + 1, let/; E F be the vertex at minimum distance from xi. Hence, j ; = = j ; for some 1 5 i I K , which gives two paths from x to,f, one of length I and the other of length I + I , which is impossible in a bipartite digraph. Finally, since any path from V - to V'. goes through F, we have d( G I , Gz) 2 d ( V ( GI), F ) + d ( F , V ( G 1 ) ) 2 21.

From the proof of the above theorem, we can deduce a lower bound on the connectivities of a bipartite digraph.

Corollary 3.5. Let G he u hipurtite digraph wiih minitniitn d q y w 6 2 2 , puratneter 1. diameter D, and connectivities K and A. Then.

104 BALBUENA ET AL.

These conditions are the best possible. To show this, consider two copies of the symmetric digraph of a cycle of even length 21 + 2. and identify one vertex of each copy. The digraph thus obtained has diameter D = 21 + 2. parameter I , minimum degree 6 = 2. and connectivity K

= 1 . For the edge case, it suffices to connect both copies by a digon joining two vertices in different partite sets.

For bipartite digraphs. let D,, 0 I u I 6, be the conditional diameter defined as in Section 2.

Corollary 3.6. Let G he u bipartitc. digraph wirli minimuni dcgri.c. 6. purumerer 1. und c ~ o n n ~ ~ ~ i v i t i e . ~ K and A. Then.

( a ) u = b i / ' D 1 2 1 + 1 a n d D 2 s 2 1 - I ;

Prouf.' We only prove ( a ) . I f D s 21, the result follows from (7) . so that we can assume that D = 21 + 1. Suppose that K I 6 - I . Then, by Theorem 3.4(a), there exist two induced subdigraphs G I , Gz such that 6 + ( G I ) 2 6' - K

+ I 2 2. 6 - ( G2) 2 2, and d ( G , , G2) 2 21, which contra-

( h ) X = 6 If'D 5 21 + 2 and D2 I 21.

diets the hypothesis.

Consider now a subset of vertices. S ( p . q ) C U,, U U4 for p , q E { 0, 1 } and let DSu,,,) be the 7'-diameter, where P is the property of ( G I , G2) being a vertex in IY,~\S(~, q ) and b',\S(p, q ) , respectively. Then. we get the following corollary:

Corollary 3.7. Let G he a hipurrire digruph wirh rninimtrni degreo 6, parumc'tc~r I , girth g, diamc>rer D, and contiec'riv- ities K und A. Lc~t S ( p , q ) denote u .siib.wt qf'gJ2 verticw. Then.

( a ) K = 6 I fD 5 21 + 1 and Ds,,,.4) I 21 - 1 for .some

( b ) X = 6 I f ' D I 21 + 2 and Ds0,.,,, 5 21ji)r .sonic p

Proof.' To prove (a) , suppose that the hypothesis holds. for instance, for p = 0 and q = 1 . If D I 21, the result follows from ( 7 ). so that we can assume that D = 21 + 1. Assume that A' I 6 - I , and let G I and Gz be two induced subdigraphs such that 6 + ( G I ) 2 2 . 6 - ( G) 2 2 and d ( G l , G,) 2 21, then I l ' ( G j ) rl U,]l 2 g / 2 + 1 for i = I , 2, p = 0, I , since G has girth g. Let s,, J',, 1 I j I ( g / 2 ) + I , be different vertices of V ( G , ) fl C;, and I,'( G2) fl U I , respectively. Then, d(x , , v,) 2 21 + I , since x,, y, belong to different partite sets. Therefore, at least one of each pair must belong to S ( p , q ) , which leads to the contradiction 1 S ( p , q ) ] 2 ( g / 2 ) + I . Then, Corollary 3.6(a)

P . q 9 P + 4;

= 0. 1 .

applies and K = 6.

Considering the conditional diameter Ds, we get the following result whose proof is analogous to the pre- ceding one:

Corollary 3.8. Ler G he u bipartite digraph wiih minimirm degree 6, parameter I, <girth g. diumcter D. and connec'tivitic.s K and A. Then.

( u ) K = 6 1f.D I 21+ I and D S s 2l.forsomeSC C,(G).

( b ) X = 6 1j'D I 21 + 2 and D.s. I 21 + I Jiw some S IS( = g + I ;

C V ( G ) , IS1 = g + 1 .

Next, we are going to consider undirected bipartite graphs. Since such graphs have even girth, the results stated at the end of Section 2 apply. However, we can do better if, as in the case of general digraphs, we consider the corollaries obtained from the results in this section. We men- tion only two of them. For instance, Proposition 3.2 allows us to state the following result, to be compared with Cor- ollary 2. I I .

Corollary 3.9. Let G he a hi~>artirc. graph MYlh ininirnirm d q r w 6. girrli g, und conntr.rivirie.s K arid A. Then.

( a ) K = 6 If'D,,,, I g - 3 $)r .soin(' p , q E { 0, 1 ;, p

( b ) X = 6 ifD,,.,, I g - 2.fi)r sonw p E { 0, I } . f 4;

Since g 2 4 for any bipartite graph 13, we have the following consequence of Corollary 3.9( b) : 11:f or some .stahkc SOI L! tlic ~1i.istrinc~c h e t ~ . c ~ n wrticm g / ' U is ti01 greurcr fhun 2, rhcn G is ma.i-imullj7 ciLi~L'-c.ontiec,r~~d. This was already given by Plesnik and Zn6m [ 23. Th. 3.1 ( i i ) ] . As a corollary. they also pointed out that uiiy hipurlire graph \i.irli di~iinctor D 5 3 is niu.i-irnal1j9 i~dge-conn~~cted [a result that can also be obtained as a consequence of( 7) . ] Finally, Corollary 3.7 leads to the following result:

Corollary 3.10. Lc.1 G he u hipurrilc grupli nirh minimirm ckigri~c 6, girrli gs und c'onnc'c'livitiev K and A. T h m ,

( u ) K = 6 i f ' D I g - 1 arid d ( s , y ) I g - 3. .i- E U,,\ { u } , J' E U,, p # q, ji)r some u E U,,;

( 6 ) X = 6 $D I g utid d ( s , 1') I g - 2 , .I-. J'E [I,,\ { u ,

j i ~ sotno u E U,,.

This work was supported by the Spanish Research Coun- cil (Comisibn Interministerial de Ciencia y Tecnologia. CI- CYT) under Projects TIC 92-1228-E. TIC 94-0592. the EU- HCM program ERBCHRX-CT920049. and the Generalitat- E.T.S.E.C.C.P.B.

REFERENCES

[ I ] J. N. Ayoub and 1. T. Frisch. On the smallest-branch cuts in directed graphs. IEEEI T r u m Circi/i/ Thewry c1'-I7 ( 1970) 249-250.


M. C . Balbuena. A. Carmona. and M. A. Fiol. Distance connectivity in graphs and digraphs. J. Gruph Theory, to appear. M. C. Balbuena, A. Carmona. J. Fibrega, and M. A. Fiol. Connectivity of large bipartite digraphs and graphs. Discr Math., to appear. M. C. Balbuena. A. Carmona, J. Fibrega. and M. A. Fiol. On the order and size of s-geodetic digraphs with given connectivity. Discr. Math., to appear. J.-C. Bermond, N. Homobono. and C. Peyrat. Large fault- tolerant interconnection networks. Graphs Comb. 5 (1989) 107-123. J. Bond and C. Delorme, New large bipartite graphs with given degree and diameter. Ars Coinh. 25C ( 1988) 123- 132. L. Cacceta and R. Haggkvist. On minimal digraphs of given girth. P r o c d i t g s of the Ninth Soirthiw.rtiw Con- J 2 r m v on Cornbinatoric:s. Gruph Theory and Cornpiiting. Utilitus Marhc~muticu, Winnipeg ( 1978) 181-187. G. Chartrand, A graph-theoretic approach to a commu- nications problem. SIAM J. ,4pp/, Muth. 14 ( 1066) 778- 781. G. Chartrand and L. Lesniak. Graphs and Digraphs. Wadsworth. Monterey ( 1986). A. H. Esfahanian. Lower-bounds on the connectivities of a graph. J. Gruph T h i w y 9 ( I985 ) 503-5 I I . J. Fibrega and M. A. Fiol. Maximally connected digraphs. J. Gruph Throrj* 13 ( 1989) 657-668.

J. Fibrega and M. A. Fiol, Bipartite graphs and digraphs with maximum connectivity. Discr. Appl. Mu//?., to appear. M. A. Fiol, The connectivity of large digraphs and graphs. J. Graph Theory 17 ( I993 ) 3 1-45. M. A. Fiol, J. Fibrega. and M. Escudero. Short paths and connectivity in graphs and digraphs. Ars C'ornh. 29B

D. Geller and F. Harary, Connectivity in digraphs. L ~ ~ i r r c . No te$ on Mutheinatics 186, Springer. Berlin ( 1970) 105- 114.

(1990) 17-31.

D. L. Goldsmith and R. C. Entringer. A sufficient condition for equality of edge-connectivity and minimum degree of a graph. J. Gruph T/in)ry3 ( 1979) 251-255. D. L. Goldsmith and A. T. White, On graphs with equal edge-connectivity and minimum degree. Discr. Math. 23

Y. 0. Hamidoune, An application ofconnectivity theory in graphs to factorization of elements in groups. Eirr. J. C'omh. 2 ( 1981 ) 349-355. M. Imase. T. Soneoka. and K. Okada. Connectivity of regular directed graphs with small diameter. IEEE Trans.

J. L. Jolivet. Sur la connexiti des graphes orientis. C.R. .-tcud. Sci. Puri.s 274A ( 1972) 148-150. L. Lesniak. Results on the edge-connectivity of graphs.

J. Plesnik. Critical graphs of given diameter. Acta Fac. Ri~riir?~ NLIYIIK Univ. Cornmian. Muth. 30 ( 1975 ) 7 1-93. J. Plesnik and S . Znim. On equality ofedge-connectivity and minimum degree of a graph. Arch. Math. (Brno) 25

T. Soneoka. H. Nakada. and M. Imase. Sufficient conditions for dense graphs to be maximally connected. Proc.

T. Soneoka. H. Nakada, M. Imase. and C. Peyrat. Suf- ficient conditions for maximally connected dense graphs.

L. Vol kmann. Bernerkungen zum p-fachen Kantenzus- ammenhang von Graphen [Remarks on the p-fold edge connectivity of graphs]. An. L'niv. Bircwi. Mar. 37 ( 1988) 75-79. L. Volkmann. Edge-connectivity in p-partite graphs. J. Gruph Tlrc,or:l* 13 (1989) 1-6.

(1978) 31-36.

CvO/?7f)t1/ . C-34 ( 1985) 267-273.

D i . ~ . r . , % l ~ t h . 8 ( 1974) 351-354.

( 1989) 10-26.

IS<'.-ISX.F (1985) 81 1-814.

Di.~c'r. i Z U t I I . 63 ( 1987 ) 53-66.

Received September 6. I995 Accepted March 2, 1996

on the connectivity and the conditional diameter of graphs and digraphs

Documents