Graphs with given connectivity properties

Graphs with Given Connectivity Properties

Serge Lawrencenko,1 Qiang Luo2

1 Sturwalnaya 7-1-45, Moscow 123363, Russia

2 Wuhan University of Hydrologic and Electronic Engineering, People’s Republic of China

Received 7 November 1996; accepted 7 June 1997

Abstract: A node of a graph G , thought of as representing a communication network, is said to beredundant provided that its removal does not diminish the connectivity. In constructing networks, werequire reliable connectedness in addition to the usual requirement of reliability ( i.e., the higher theconnectivity, the more reliable the network) . Two nodes are called reliably connected if they are joinedby a reliable path, i.e., a path whose internal nodes ( if any) are all redundant. For a given n , we constructa communication network on n nodes having a given value k of connectivity ( reliability condition) and asfew edges as possible (optimization condition) and in which any pair of nodes, with one exceptionalnode, are reliably connected (reliable connectedness condition) . We also studied the associated-with-G ‘‘strong redundancy graph’’ and a sequence of ‘‘weak redundancy graphs,’’ quotient graphs whichdisplay decompositions of G into connected subgraphs in accordance with the contributions of the nodesto the connectivity k(G ) . q 1997 John Wiley & Sons, Inc. Networks 30: 255–261, 1997

1. INTRODUCTION u-essential node cuts of G will be denoted by Cu(G) . Tosee that Cu(G) x M for any u √ V (G) , pick a node, u *,

The symbol G will always denote a finite simple con- adjacent to u ; clearly, then, V (G)" {u *} √ Cu(G) . Thenected nontrivial graph. By V (G) , E(G) , n(G) , and redundancy of a node u in the graph G is defined to bek(G) , we denote the node set, the edge set, the order,and the connectivity of graph G , respectively. A node cut rG(u) Å min{ÉAÉ 0 k(G) : A √ Cu(G)}. (1)of G is a subset A of V (G) such that the induced subgraphG[V 0 A] , also denoted by G 0 A , is disconnected or

A node u is called (connectivity-) essential if rG(u) Å 0,trivial. A node cut is strong if G 0 A is disconnected; it

and m-redundant if rG(u) ¢ m ¢ 1. By a redundantis weak if G 0 A is trivial. Thus, A is a node cut of G if

node, we shall mean a 1-redundant node.and only if k(G 0 A) Å 0. An ,-node cut is a node cut

Redundancy of a node is a measure ‘‘inverse’’ to theof size , ; in particular, a strong 1-node cut is called a

contribution of that node to the connectivity of G . Thiscut node. Our general graph-theoretic terminology and

measure is more subtle than is the previously studied [1]notation are essentially those of Bondy and Murty [2] .

cohesiveness of a node u in graph G , defined by cG(u)For u √ V (G) , let A be a node cut of a graph G Å k(G) 0 k(G 0 u) . Clearly, u is redundant (respec-

containing u . We will say that A is a u-essential node cuttively, essential) if and only if cG(u) ° 0 (respectively,

if A 0 u is no longer a node cut of G . The family of all ú0). A number of papers (see, e.g., [1, 4]) were devotedto the study of the number of redundant nodes in a givengraph.Correspondence to: S. Lawrencenko

q 1997 John Wiley & Sons, Inc. CCC 0028-3045/97/040255-07

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We shall think of G as representing a communication and also infinitely many graphs I satisfying Ià Rm(I) . Tocharacterize all such graphs I remains an open problem.network; its nodes correspond to the communication sta-

tions, and the edges to the communication links betweenthose stations. Usually, a network is regarded as reliableas the value of its connectivity (see [2]) . In addition to 2. PROPERTIES OF m-REDUNDANTthis characteristic, here we introduce the concept of reli- NODESable connectedness in a communication network. A (u ,£)-path is called a reliable (u , £)-path, for communica- Let a graph G represent a communication network. Bytion between nodes u and £, if it internally avoids essential choosing m-reliable paths for communication in the sys-nodes. Nodes u and £ are called reliably connected if tem (such choices are usually possible in an m-reliablythere is a reliable (u , £)-path. (Note that a breakdown of connected communication network), we gain an im-a communication station represented by an essential node portant advantage: namely, a blockage of a number (inwould decrease the value of connectivity of the network; total less than m) of their internal nodes at a time doestherefore, intensively using essential nodes may jeopar- not create new essential nodes (which are undesirable todize communication in the system.) More generally, use for communication) provided that the blocked nodesnodes u and £ are called m-reliably connected (m ¢ 1) constitute a subset of V (G) stable in the sense that itsif there is an m-reliable (u , £)-path, i.e., a (u , £)-path removal does not alter the value of connectivity (bywhose internal nodes (if any) are all m-redundant. Adja- Lemma 2.3, this condition is normally satisfied).cent nodes are m-reliably connected for any m . We now state the result formally: Let m ¢ 2. A

It can be easily seen that if two nodes are m-reliably nonempty subset B of V (G ) is called an m-redundantconnected, then they cannot be separated by any subset subset if rG (u ) ¢ m ¢ 2 for all u √ B ; B is calledof V (G) of size less than m / k(G) . Here, we include stable if k(G ) Å k(G 0 B ) . Let Rm (G ) denote thean infinite series of examples showing that the converse is family of all m-redundant stable subsets of V (G ) hav-not necessarily true. Take an even cycle C Å £1£2rrr£2p£1 ing size less than m .(with p ¢ 2) and p disjoint copies K2( i) of K2 ( i Å 1,. . . , p) . Identify one node of K2( i) with the node £2i of Theorem 2.1. Assume that m ¢ 2 and B √ Rm(G) .C for each i and denote the resulting graph by Xp . Now, Then, the removal of B from G does not create new essen-let u and £ be some two nodes of Xp both having degree tial nodes, i.e., rG0B(£) Å 0 implies that rG(£) Å 0 .two. We have that Xp is a graph with k(Xp) Å 1 in whichno node can separate u and £, but u and £ are not 1- We shall prove this result at the end of this section,reliably connected. but first we develop three lemmas.

A communication network G is said to be reliably (m-reliably) connected if any pair of its nodes, possibly with Lemma 2.2. Let u be a node of G such that k(G) Å k(Gone exceptional node, are joined by a reliable (m-reliable) 0 u) . Then, rG(£)° rG0u(£)/ 1 for any £√ V (G)" {u} .path. When G x Kn , G has to have a strong node cut and

Proof. Let £√ V (G0 u) . By the connectivity hypoth-has to have an exceptional node, u0 ; furthermore, G hasesis, G 0 u is not trivial, and, therefore, C

£(G 0 u) xMa unique minimum strong node cut which is the neighbor

(see the Introduction). Pick A in C£(G 0 u) so thatset of u0 . A good communication network is to be de-

signed so that the exceptional node represents a ‘‘smallnonterminal station.’’ ÉAÉ 0 k(G 0 u) Å rG0u(£) (2)

In Section 2, we continue to discuss the advantages ofm-reliable connectedness in a network. In Section 3, we [see Eq. (1)] . Then, A < {u} √ C

£(G) . Therefore, by

solve the problem of constructing reliably (i.e., 1-reli- Eqs. (1) and (2),ably) connected communication networks; the generalproblem (for an arbitrary m) is still open. In Section 4, rG(£) ° ÉA < {u}É 0 k(G)we consider decompositions of graphs into parts so that

Å rG0u(£) / k(G 0 u) / 1 0 k(G) (3)each part is a maximum connected subgraph having onlynodes of the same redundancy or ‘‘close’’ redundancies. Å rG0u(£) / 1.To display such decompositions, we introduce special

jquotient graphs associated with a given graph G , namely,the strong redundancy graph R(G) and a sequence of

The following is a result of [1, 4] . Recall that weweak redundancy graphs Rm(G) , and characterize thoseassume G to be a connected nontrivial graph.graphs under certain connectivity assumptions. In Section

5, we construct infinitely many graphs I with the propertythat I and R(I) are isomorphic graphs, written I à R(I) , Lemma 2.3. A node u of G is essential if k(G) ú k(G

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GRAPHS WITH GIVEN CONNECTIVITY PROPERTIES 257

0 u) and redundant otherwise. Furthermore, G may have operations creates new essential nodes. This completesthe proof. jat most one node u with k(G) õ k(G 0 u) , which is the

case if and only if G can be obtained from a k *-connectedgraph G*, for some k* ¢ 2 , by adding a node, u, andjoining it to a set of k* 0 1 nodes in G*. j 3. RELIABLY CONNECTED

COMMUNICATION NETWORKSLet u be a 2-redundant node of G . In case k(G)õ k(G

The following is an optimization problem that emerges0 u) , one can easily construct examples to show that Gfrom constructing reliably connected communication net-0 u may have more essential nodes than has G . However,works.if k(G) Å k(G 0 u) , no such an example is possible.

This fact is the content of the following lemma whichstrengthens Lemma 2.2 in the particular case rG0u(£) Å 0

The General Problem[under the additional restriction rG(u) ¢ 2].GIVEN:

Lemma 2.4. Let u be a 2-redundant node of a graph G• A nontrivial edge-weighted graph L of order n whosesuch that k(G) Å k(G 0 u) . Then, the removal of u from

nodes are denoted by u0 , u1 , . . . , un01 , andG does not create new essential nodes, i.e., rG0u(£) Å 0implies that rG(£) Å 0 . • two positive integers k and m satisfying

Proof. Assume that £ is an essential node of G 0 u .k / m õ n 0 1. (5)Pick A in C

£(G 0 u) satisfying ÉAÉ 0 k(G 0 u)

Å rG0u(£) Å 0. Then, A < {u} is a node cut of G . ButA < {u} √/ Cu(G) , for otherwise, rG(u) ° ÉA < {u}É

FIND: A spanning subgraph S Å Sk ,n ,m of L satisfying the0 k(G) Å k(G 0 u) / 1 0 k(G) Å 1, which contradictsfollowing three conditions:the 2-redundancy of u . Hence, (A < {u})" {u} Å A is

also a node cut of G . This node cut is necessarily mini-(C1) k(S) Å k (reliability condition),mum because ÉAÉ Å k(G 0 u) Å k(G) . Hence, A(C2) S be an m-reliably connected communication net-√ C

£(G) , and therefore rG(£) ° ÉAÉ 0 k(G) Å 0. The

work with u0 as the exceptional node (reliable con-proof is complete. jnectedness condition), and

(C3) S be of minimum possible weight (optimizationUnder the hypotheses of Lemma 2.4, assume now that condition).

u is m-redundant (in G) with m ¢ 3. Then, it may be thatrG0u(£) ° m 0 2, while rG(£) ú m 0 2. To show this,

Observe that Eq. (1) gives m° n0 10 k with equalityhere we include an example for m Å 4. Consider K4 with

possible only if S Å Kn . On the other hand, Kn cannotone edge deleted. Denote by £ and x its two nodes of degree

satisfy this inequality for any positive m . Thus, Inequalitythree. Add two new nodes, u and y , and join them to x .

(5) is not really restrictive. So, S is not Kn and has atTake the resulting graph as G . We have k(G) Å k(G 0 u)

least one strong node cut (whose nodes are all essential,Å 1, rG(u) Å 4, rG0u(£) Å 2, but rG(£) Å 3.of course) . So, we have to admit one exceptional node;we assume that u0 is such a node.Proof of Theorem 2.1. By induction on m . When m

The general problem with (C2) omitted is known asÅ 2, the theorem is equivalent to Lemma 2.4. Let M ¢ 3.the problem of construction of reliable communicationSuppose that the theorem is true for all m õ M , and letnetworks. It appears to be very difficult and presently isB √ RM(G) with ÉBÉ ¢ 2. Then, by Lemma 2.3, thereunsolved in such a general setting (see [2]) . So, it isexists a node u in B such that k(G) Å k(G 0 u) . Let £reasonable to impose additional restrictions. Here, we√ B" {u}. By Lemma 2.2, rG0u(£) ¢ M 0 1. Further-shall solve the general problem in one such a particularmore,setting, namely, we shall refer to it as the ‘‘restrictedproblem’’ if the following restrictions hold:k((G 0 u) 0 (B" {u})) Å k(G 0 B) Å k(G)

Å k(G 0 u) .(4)

Restrictions

(R1) L Å Kn , a complete graph in which every edge isTherefore, B" {u} √ RM01(G 0 u) . The removal of Bfrom G is equivalent to the removal of u from G followed assigned unit weight,

(R2) k ¢ 2,by the removal of B" {u} from G 0 u . By Lemma 2.4and the induction hypothesis, neither of the two latter (R3) m Å 1,

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258 LAWRENCENKO AND LUO

(R4) n ¢ 6. Theorem 3.1. The graph Q Å Qk,n with u0 (Åz0) as theexceptional node is a solution of the restricted problem.

By dG(u) and NG(u) , we shall denote the degree of Proof. First, the graph Q indeed satisfies conditionnode u and its neighbor set in graph G , respectively. To (C1) because it contains as a subgraph the graph H Å Hk ,nsolve the restricted problem is actually to find a graph S known [2, 3] to be of connectivity k . Furthermore, byof connectivity k on n nodes, having as few edges as construction,possible to satisfy the condition that any two nodes, otherthan the exceptional node u0 , be 1-reliably connected. ÉE(Q)É Å ÉE(H)É / n /2 Å kn /2 / n /2 . (8)Thus, any solution S of the restricted problem has a uniquestrong k-node cut ( i.e., a unique minimum strong node Thus, Q also satisfies condition (C3) thanks to Inequalitycut) , namely, NS(u0) . Thus, we can estimate the number (7) . It remains to prove that Q satisfies condition (C2).of edges in S as follows: Observe that dQ(u0) Å k , while the other nodes have

higher degrees. Thus, it is enough to prove that NQ(u0)is a unique strong k-node cut of Q ( i.e., a unique minimum

2ÉE(S)É Å ∑n01

iÅ0

dS(ui ) ¢ k / (n 0 1)(k / 1), (6) strong node cut) .We shall first examine the structure of minimum strong

node cuts of H (ÅHk ,n) . By a continuous p-set, we willand, hence, mean a set of p nodes of H which occur consecutively

around the circle {z : ÉzÉ Å 1} in the complex plane. Itcan be easily verified that after the removal of any set ofÉE(S)É ¢ (kn / n 0 1)/2 Å kn /2 / n /2 . (7)k nodes which does not contain a continuous k /2-set wecan certainly identify a Hamilton cycle using appropriateWe shall first exhibit a graph, Qk ,n , having exactly kn /edges of H in Case 1 and of Hk01,n in Cases 2 and 3. It2 / n /2 edges. Then, we shall verify that Qk ,n is afollows that in Case 1 any strong k-node cut of H can besolution of the restricted problem.expressed as W1 < W2 , a disjoint union of two continuousWe start our construction with a special graph Hk ,n , kk /2-sets such that the union itself does not form a¢ 2, of order n and connectivity k , having exactly kn /continuous k-set. In Cases 2 and 3, ÉW1 < W2É Å k 0 1,2 edges. This graph was originally constructed by Hararyso that any strong k-node cut of H can be expressed as[3] . The construction can be also found in [2] , in which

it is actually shown that Hk ,n is a solution of the problemW Å W1 < W2 < {u} (9)of constructing reliable communication networks [i.e., the

restricted problem with (C2) omitted] . Here, for the sakefor some u √ V ( H) . Thanks to the edges that we addedof completeness, we include a reproduction of that con-to Hk01,n in Cases 2 and 3 to obtain H , the set W is astruction. Our reproduction differs from the presentationstrong k-node cut if and only if the two conditions areof [2]: We shall exploit the complex plane. The nodessatisfied:ur of Hk ,n are associated with the n points zr Å exp(2pri /

n) (r Å 0, . . . , n 0 1) in the complex plane. The proce-( i) W1 and W2 be separated by a single node—more

dure of adding edges splits into three cases:precisely, there should be a node £ of H such thatW1 < W2 < {£} is a continuous k-set, andCASE 1. k even. To obtain Hk ,n , join each node zr to all

nodes zs that can be expressed as zrexp(2pti /n) , where (ii ) dH(£) Å k .t √ {{1, {2, . . . , {k /2}.

Necessarily, then, u in Eq. (9) is the sole member ofCASE 2. k odd, n even. We obtain Hk ,n from Hk01,n by NH(£)"(W1 < W2) .

joining zr to zs Å 0zr for r Å 0, 1, . . . , (n 0 2)/2. Now we turn to the graph Q . Except the case k Å 3,n Å 6 (which is settled straightforwardly) , the graph H

CASE 3. k odd, n odd. We obtain Hk ,n from Hk01,n by 0 W has precisely two components, one of which is ajoining zr to zs Å 0zrexp(pi /n) for r Å 1, 2, . . . , (n single node £. Unless £ Å u0 , we have dQ(£) ú dH(£) ,/ 1)/2. so an edge in E(Q)"E(H) incident to £ joins the two

components of H 0 W , and so Q 0 W is connected.Therefore, in Cases 2 and 3, the set W is the only strongNow, to construct Qk ,n , we add n /2 more edges to

Hk ,n ; namely, in Cases 1 and 2, we join zr for r Å 1, 2, k-node cut of Q . This node cut, in fact, coincides withNQ(u0) . Similarly, one can obtain the same conclusion. . . , n /2 to zs Å 0zrexp(02pi /n) if n is even and to

zs Å 0zrexp(0pi /n) if n is odd. In Case 3, we join zr to in Case 1; for this, take into account the edges that weadded to H in order to obtain Q .zs Å 0zrexp(0pi /n) for r Å 2, 3, . . . , (n 0 1)/2, and

also join z (n/1) /2 to zn01 . The proof is complete. j

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GRAPHS WITH GIVEN CONNECTIVITY PROPERTIES 259

4. THE REDUNDANCY QUOTIENT GRAPHS i.e., the graph obtained from H[K2 1 Kp] by deleting theedges satisfying the conditions specified above in Eq. (12).

Observe first that for any connected H and any p ¢ 2In this section, we develop another point of view on thethe graph GÅ Sp(H) has connectivity p and entirely consistsconcepts of the preceding section.of essential nodes. Hence, R(G) à Rm(G) à K1 for eachLet < Pi be a partition of V (G) such that for each im . Thus, the case of H trivial is settled by G Å Sk(K1), andPi is a nonempty subset of V (G) and the induced sub-in what follows, we shall assume that H à/ K1 .graph G[Pi ] is connected. The quotient graph associated

The following is another obvious but useful observation:with a given pair (G , < Pi ) is denoted by G /< Pi anddefined to be the graph whose nodes correspond to the

Lemma 4.1. If u is a cut node of a graph G of connectiv-parts Pi and two nodes Pi and Pj are adjacent wheneverity one, then in each component of G 0 u there is a nodei x j and a member of Pi is adjacent to a member of Pj£ with rG(£) ¢ 1 . j(as nodes of G) .

Consider the partition < Ri of V (G) , where each Ri In the following constructions, we shall process a suit-is the node set of a largest connected subgraph of G with

able superstructure on H by judiciously contracting itsthe property that any two of its nodes u and £ satisfy

edges. A contraction of an edge u£ is the operation ob-rG(u) Å rG(£) . The quotient graph G /< Ri is denoted

tained by the removal of u and £ and the addition of aby R(G) and called the strong redundancy graph of G .

new node adjacent to the nodes formerly adjacent to u orFor a given integer m satisfying 0 õ m õ n(G) 0 k(G) ,

£. Especially, when u Å (a , b , c1) and £ Å (a , b , c2) ,the (weak) m-redundancy graph is defined to be

the new node will be denoted by (a , b , min(c1 , c2)) .(Note that under the operation of edge contraction as

Rm(G) Å G / (<i Rmi ) < (<jE

mj ) , (10) defined in [2] there may arise pairs of multiple edges;

however, omitting a duplicated edge in each such pair,where each Rm

i (respectively, Emj ) is the node set of a we obtain the result of the contraction in our sense.)

largest connected subgraph of G having only nodes £ withrG(£) ¢ m (respectively, õm) . Clearly, each Rm(G) is Theorem 4.2. Let k be a given positive integer. A non-a connected bipartite graph (unless it is trivial) having trivial graph H is the strong redundancy graph of some

graph G of connectivity k if and only if H is connected<i Rmi and <j Em

j as two parts of the bipartition.and, for the case k Å 1 , has a cut node.Roughly speaking, the smaller the size of the part <i

Rmi , the more pairs of nodes are m-reliably connected. In Proof. The necessity follows from Lemma 4.1. To

particular, any solution S of the restricted problem of the prove sufficiency, let H be a connected graph with V ( H)preceding section satisfies Å {0, 1, . . . , n 0 1}. Especially in the case k Å 1, we

may assume that 0 is a cut node (as matter of notation).R 1(S) à Kc

2 Û Kck , (11) We shall obtain graph G as desired by processing the

superstructure Sn/k(H) . First, only in the case k Å 1,where the right-hand part denotes the join of the comple- contract all the edges of the form (0, b1 , c1)(0, b2 , c2)ments of (disjoint) K2 and Kk , with É<i R 1

i É Å ÉKc2É into a single node and designate that node by (0, 0, 0) .

Å 2, and the symbol ‘‘à’’ denotes the isomorphy of the Second, for any k , repeatedly contract the edges of thegraphs. form (a , b , c1)(a , b , c2) satisfying min(c1 , c2) ¢ a / k

In this section, we shall obtain criteria for a given 0 1. When no such edges are left, we have a graph whichgraph H to be the redundancy graph of some graph G we take as G . Observe that G has the property that everywith a prescribed value k of connectivity. By n Å n(H) , subgraph induced by the set of nodes (a , b , c) with awe shall denote the order of H , and by 0, 1, . . . , n 0 1, fixed a is isomorphic to the graph K2 1 Ka/k , with theits nodes. Let V ( Kp) be {0, 1, . . . , p 0 1}. The composi- obvious exception of a Å 0, k Å 1. It remains to provetion H[K2 1 Kp] has as its node set the product V ( H) that G satisfies R(G) à H and k(G) Å k . For this, let A1 V ( K2) 1 V ( Kp) . The nodes of the graph H[K2 1 Kp] be a node cut of G attaining the minimum in Eq. (1) forwill be denoted by (a , b , c) , where a √ {0, 1, . . . , n a given node u Å (a(u) , b(u) , c(u)) √ V (G) ; in other0 1}, b √ {0, 1}, and c √ {0, 1, . . . , p 0 1}, and its words, A is a member of Cu(G) of minimum size. It canedges by (a1 , b1 , c1)(a2 , b2 , c2) . The p th superstructure be easily seen that the projection of A to V ( H) collapseson graph H is defined to be into a single node a(u) of H and that ÉAÉ Å a(u) / k ,

so that k(G) Å k . For instance, one such node cut isgiven bySp(H) Å H[K2 1 Kp]

0 {(a1 , b1 , c1)(a2 , b2 , c2) : A Å {(a(u) , b(u) , c(u))} <

{(a , b , c) : a Å a(u) , b x b(u) , c x c(u)}.(13)

a1 x a2 and b1b2 Å 0},

(12)

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260 LAWRENCENKO AND LUO

Therefore, we have rG((a , b , c)) Å a for each node (a , problem to characterize strongly irreducible (or weaklym-irreducible) graphs with a given value k of connectivityb , c) √ V (G) . The proof is complete. j

(with given values of m and k) .The following is a construction for generating infinitelyThe next theorem is a criterion for a graph to be the

many strongly irreducible graphs with a prescribed valueweak m-redundancy graph of some graph with a givenk , k ¢ 1, of connectivity. Let T be a nontrivial tree, notvalue k of connectivity. This splits into two cases:K2 . Denote its node set by V (ÅV (T )) , and by V1 , the

CASE 1. mk ¢ 2. set of its nodes of degree one. We have V1 x M andV "V1 x M. For z in V1 , its neighbor set NT(z) consistsCASE 2. mk Å 1.of a single node which will be denoted by N(z) . Assignto the nodes y √ V positive integers n(y) so thatTheorem 4.3. Let m and k be given positive integers. A

nontrivial graph H is isomorphic to Rm(G) for somegraph G of connectivity k if and only if H is a connected ( i ) min{n(y) : y √ V "V1} Å k ,bipartite graph in Case 1 , or H is a connected bipartite ( ii ) n(y) x n(x) whenever yx √ E(T ) and y , x √/ V1 ,graph with one part, X, of the bipartition entirely con- andsisting of cut nodes in Case 2 .

( iii ) (yxN (z ) n(y) x n(N(z)) for each z √ V1 .Proof. The necessity is obvious in Case 1 and follows

from Lemma 4.1 in Case 2. To establish sufficiency, let Then, replace each node y √ V by the graph Y ( y)H be a bipartite graph with parts X Å {0, 1, . . . , l 0 1} à Kc

n (y ) [ i.e., the edgeless graph on n(y) nodes] , andand Y Å {l, l / 1, . . . , n 0 1}. To construct G in Case join each node in Y ( y) to each node in Y ( x) whenever2, it is enough to contract, for each a √ {0, 1, . . . , l yx √ E(T ) . Denote the resulting graph by Y (T ) .0 1}, all the edges in S2(H) that join the nodes havingthe same a-component into a single node, £(a) . Let us

Proposition 5.1. The graph Y (T ) is a strongly irreduc-now construct G in Case 1 to satisfy k(G)Å k and Rm(G)ible graph of connectivity k.à H . For this, repeatedly contract in Sm/k(H) the edges

of the form (a , b , c1)(a , b , c2) satisfying both conditions: Proof. Consider the projection p : V (Y (T )) r V de-fined by p(u) Å y , where y is such that u √ Y ( y) . For

(i) a √ {0, 1, . . . , l 0 1}, and u √ V (Y (T )) , let A be a member of Cu(Y (T )) of mini-( ii ) min(c1 , c2) ¢ k 0 1; mum size. Since NY (T ) (u) Å NY (T ) (£) whenever p(u)

Å p(£) , it follows that Y (p(u)) , A whenever u √ A ,in the case k Å 1, condition (ii ) is replaced by and, therefore, p(A) is a member of Cp(u ) (T ) of minimum(ii *) min(c1 , c2) ¢ 1. size. If p(u) √/ V1 , then {p(u)} is the only member of

Cp(u ) (T ) of minimum size. If p(u) √ V1 , the only suchFinally, only in the case k Å 1 (necessarily, then, m member is V " {N(p(u))}. Therefore, by condition (i) ,

¢ 2 as we are now considering Case 1), to make the k(Y (T )) Å k . Furthermore,resulting graph (which is to be taken as G) be of connec-tivity one, we adjoin a cycle of length five to the node

rY (T ) (u)(0, 0, 0) . Then, except the node (0, 0, 0) having rG((0,0, 0)) Å 0, the nodes of that cycle have redundancy oneas well as the nodes of the form (0, b , c) . Hence, all the Å

n(p(u)) 0 k if p(u) √ V "V1

∑yxN (p(u ) )

n(y) 0 k if p(u) √ V1(14)

named nodes are in the same set Emj of Eq. (10). Proceed-

ing as in the proof of the preceding theorem, one canverify that the so-constructed graph G is as desired. The

Now, by conditions (ii ) and (iii ) , the proof is complete.proof is complete. jj

5. IRREDUCIBLE GRAPHS A similar construction applies to obtain infinitely manyweakly m-irreducible graphs with a given positive valuek of connectivity. Let m be a positive integer. Let T be aGiven a connected nontrivial graph, by repeatedly con-

tracting the edges having both ends of the same redun- nontrivial tree, not K2 , in which the distance between anytwo nodes of degree one is even. Then, V splits into twodancy, we finally obtain a graph I with the property R(I)

à I . We will say that such a graph I is a strongly irreduc- classes: (1) nodes for which the distance to any node ofdegree one is even, and (2) nodes for which that distanceible graph. Similarly, a weakly m-irreducible graph is

defined to be a graph I satisfying Rm(I) à I . It is an open is odd. For y in class 1, assign to y any integer n(y)

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GRAPHS WITH GIVEN CONNECTIVITY PROPERTIES 261

satisfying n(y) ¢ m / k . For y in class 2, y is assigned The authors are grateful to the referees and personally toProfessor Frank Boesch for their helpful comments in improv-n(y) Å k . Denote the resulting graph by Ym(T ) .ing the presentation.

Proposition 5.2. Ym(T ) is a weakly m-irreducible graphof connectivity k. REFERENCES

Proof. This proceeds similarly to the proof of the pre-[1] J. Akiyama, F. Boesch, H. Era, F. Harary, and R. Tindell,ceding proposition. j

The cohesiveness of a point of a graph. Networks 11(1981) 65–68.

However, in this way, we do not obtain all irreducible [2] J. A. Bondy and U. S. R. Murty, Graph Theory with Appli-graphs. For example, the graph Xp of connectivity one cations. MacMillan, London (1976).constructed in the Introduction is strongly irreducible and [3] F. Harary, The maximum connectivity of a graph. Proc.weakly 1-irreducible for each p . But, unless p Å 2, no Nat. Acad. Sci. U.S.A. 48 (1962) 1142–1146.pair of nodes of Xp have the same neighbor set, and, [4] S. Lawrencenko and J. Mao, A sharp lower bound for thetherefore, Xp cannot be obtained from a tree as described number of connectivity-redundant nodes. Ars Combin., to

appear.above.

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