In IEEE Symp. on Foundations of Comp. Sci. (FOCS 91). Copyright IEEE

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1 In IEEE Symp. on Foundations of Comp. Sci. (FOCS 91). Copyright IEEE A Linear Time Algorithm for Triconnectivity Augmentation 1 èextended Abstractè Tsansheng Hsu Vijaya Ramachandran Department of Computer Sciences, University oftexas at Austin, Austin, TX 7871 Abstract We consider the problem of ænding a smallest set of edges whose addition triconnects an undirected graph. This is a fundamental graphtheoretic problem that has applications in designing reliable networks and faulttolerant computing. We present a linear time sequential algorithm for the problem. This is a substantial improvement over the best previous algorithm for this problem, which runs in Oènèn + mè è time on a graph with n vertices and m edges. 1 Introduction The problem of augmenting a graph to reach a certain connectivity requirement by adding edges has important applications in network reliability ë5, 18ë and faulttolerant computing. One version of the augmentation problem is to augment the input graph to reach a given connectivity requirement by adding a smallest set of edges. We refer to this problem as the smallest augmentation problem. The following results are known for solving the smallest augmentation problem on an undirected graph to satisfy a vertex connectivity requirement. Eswaran & Tarjan ëë gave alower bound on the smallest number of edges for biconnectivity augmentation and proved that the lower bound can be achieved. Rosenthal & Goldner ë16ë developed a linear time sequential algorithm for ænding a smallest augmentation to biconnect a graph; however, the algorithm in ë16ë contains an error. Hsu & Ramachandran ë10ë gave an corrected linear time sequential algorithm. An Oèlog nè time parallel algorithm on an EREW PRAM using a linear number of processors for ænding a smallest augmentation to biconnect an undirected graph was also given in Hsu & Ramachandran ë10ë, where n is the numberofvertices in the input graph. Watanabe & Nakamura ë4, 6ë gave an This work was supported in part by NSF Grant CCR89 Oènèn + mè è time sequential algorithm for ænding a smallest augmentation to triconnect a graph with n vertices and m edges. There is no polynomial time algorithm known for ænding a smallest augmentation to kvertexconnect a graph, for ké3. Results on other versions of augmentation problems can be found in ë1,,4,6,7, 1, 14, 17, 1,, 3, 4, 5, 7ë. In this paper, we present a linear time sequential algorithm for ænding a smallest augmentation to triconnect a graph. The algorithm is divided into two stages. During the ærst stage, we biconnect the input graph. Then in the second stage, we triconnect the resulting biconnected graph using the smallest number of edges. We have tomake sure that the total number of edges added in these two stages is minimum. It turns out that we cannot use the algorithm in Hsu & Ramachandran ë10ë to implement stage 1, since there exists a graph G such that any smallest augmentation for biconnecting G does not lead to a smallest augmentation for triconnecting G. An example is shown in Figure??. Note that for edgeconnectivity, itis shown in ë14, ë that there exists a smallest augmentation to kedgeconnect a graph G such that it is included in a smallest augmentation to èk + 1èedgeconnect G, for an arbitrary k. Owing to space limitation, many proofs are omitted in this abstract. They can be found in the full paperë11ë. Deænitions block ë, 3, 16ë Given an undirected graph G with a set of vertices V, let V = fv i j1 ç i ç kg be a set of subsets of V such that ë k i=1 V i = V and two vertices u and w are in the same subset if and only if there is a simple cycle in G which contains u and w or u = w. The induced subgraph of G on each set of vertices V i,1ç i ç k, is a block. The union of all blocks includes all edges in G that are not cutedges. A block containing only one vertex is called a trivial blocks. A trivial block
2 of G is called a cutblock if it is a cutpoint. block graph Given an undirected graph G, we deæne its block graph, blkègè, as follows. Each cutpoint and block that is not a cutblock is represented by avertex in blkègè. The vertices of blkègè that represent blocks that are not blocks are called bvertices and those representing cutpoints are called cvertices. Foraver tex u in blkègè, let V u = fug if u is a cvertex and V u = fwjw is a vertex in the corresponding block represented by ug if u is a bvertex. The induced subgraph of G on V u is the corresponding subgraph of u. Two vertices u and w in blkègè are adjacent ifand only if any one of the following conditions is true: èiè jv u j =1,jV w j = 1 and the vertex in V u and the vertex in V w are adjacent ing; èiiè jv u j = 1 and V u ç V w ; èiiiè jv w j = 1 and V w ç V u. Itiswellknown that blkègè is a forest. If G is connected, blkègè isa tree. If blkègè is a tree, we refer to it as a block tree. Foravertex v in G, let d èvè be the degree of its corresponding cvertex in blkègè ifv is a cutpoint. If v is not a cutpoint, let d èvè = 1. For more on properties of blkègè, see ë10ë. An example of a graph and its blkègè is shown in Figure??. Tutte component The Tutte components of a biconnected graph are the triconnected components deæned in Tutte ë0ë èsee also Ramachandran ë15ëè. Each Tutte component can be a single vertex, a triconnected component, a simple cycle èa polygonè or a pair of vertices with at least three edges between them èa bondè. 3block graph Given a block H of G, we deæne the 3block graph, 3blkèHè, as follows. The 3block graph contains three sets of vertices: ævertices, çvertices and çvertices. For every Tutte component ofh that is a single vertex or a triconnected component, we create a ævertex in 3blkèHè. A Tutte component that consists of only one vertex is a trivial Tutte component. For every polygon Q of H,we create a çvertex; if w is a vertex in Q with degree in H, we create a ævertex b w for w, and we call w the corresponding Tutte component represented by b w. Let z 1 and z be the two vertices in Q that are adjacent tow. We create a çvertex for the pair of vertices z 1 and z. For every pair of vertices a 1 and a in H, we create a çvertex for èa 1,a èifwehave performed a Tutte split with respect to a 1 and a in H. An example of a çvertex, ævertex and çvertex is shown in Figure??. Vertices u and v in 3blkèHè are adjacent ifany one of the following conditions is true: èiè u is a æ vertex corresponding to a Tutte component H u that is a triconnected component, v is a çvertex and H u contains the pair of vertices corresponding to v; èiiè u is a ævertex corresponding to a degree vertex w in H and v is the çvertex corresponding to the pair of vertices in H that are adjacent tow; èiiiè u is a çvertex corresponding to a polygon Q in H, v is ç vertex corresponding to a pair of vertices z 1 and z in Q; èivè interchanging u and v in any one of the previous three conditions. We call the pairs of vertices that correspond to çvertices Tutte pairs. It is known that the number of Tutte pairs in an nnode biconnected graph is Oènè ë13, 15ë. éfrom ë9, 15, 0ë, we know that 3blkèHè is a tree if H is a block. We call this tree a 3block tree. We call the set of trees corresponding to blocks in G the 3block graph of G or 3blkèGè. Each Tutte component that corresponds to a ævertex in the 3block graph is a 3block of G. Given a graph G with n vertices and m edges, blkègè and 3blkèGè can be computed in Oèn + mè time using procedures in ë9, 15ë. Examples of 3block graphs are shown in Figures?? &??. The implied path in the block graph Let G be an undirected graph and let u and v be two distinct vertices in G. Let G 0 be the graph obtained from G by adding the edge èu,vè and let b u and b v be the two vertices in blkègè whose corresponding subgraphs contain u and v, respectively. We denote the path P between b u and b v in blkègè the implied path between u and v in blkègè èwe let P =ëb u ëif there is no such pathè. Blocksequence, separatingsequence and cutsequence Let Y be the set of all bvertices in P and those c vertices in P whose corresponding cutpoints are cutblocks. Let jy j = r and y i be the ith vertex in Y encountered when we traverse P starting form b u to b v. We construct a separatingsequence W =ëw 1 = u; w ; æææ;w r,1 ;w r = vë for vertices u and v such that for 1 éiér, w i,1 and w i are the two cutpoints adjacent toy i if y i is a bvertex; w i,1 and w i are the two vertices adjacent to y i in G if y i is a cutpoint; w and w k,1 are cutpoints adjacent to y 1 and y r, respectively. Note that w i and w i+1 can be equal, for 1 ç iér; they are diæerent if and only if any there is at least one cvertex in fy i,y i+1 g.acutsequence C = ëu; c 1 ; æææ;c x ;vë for vertices u and v is a sequence of vertices in G such that c i is the cutpoint corresponding to the ith cvertex encountered when we traverse P from b u to b v and x is the number of cvertices in P. An example is shown in Figure??. The collection of implied paths For 1 ç i ç r, let H i be the corresponding block
3 represented by y i if y i is bvertex; let H i be the corresponding cutpoint represented by y i if y i is a cvertex. For 1 ç i ç r, let Q i =ëh i ëify i is a cvertex; otherwise, let Q i be the path in 3blkèH i èbetween the two ævertices that correspond to Tutte components containing w i,1 and w i. Each Q i,1çiçr, ischosen such that both w i,1 and w i are each contained in exactly one of the Tutte components corresponding to ævertices in Q i if y i is a bvertex. Note that w i and w i+1 can be equal, for 1 ç iér; they are diæerent if and only if there are a sequence of more than one cutpoint between y i and y i+1 in P. If there is more than one Tutte component that contains w i,1çiçr, we choose an arbitrary one. We call Q = fq 1 ; æææ;q r g the collection of implied paths between u and v in 3 blkègè. If G is biconnected, r =1;Q 1 is also called the implied path between u and v in 3blkèGè. An example of these deænitions is shown in Figure??. Separating degree Given a çvertex s, let èa 1,a è be its corresponding Tutte pair. We deæne d 3 èèa 1 ;a èè or d 3 èsè to be the degree of z in the 3block graph. We know that d 3 èèa 1 ;a èè ç for any Tutte pair èa 1,a è. Recall that for a vertex v in G, d èvè is the degree of v in blkègè if v is a cutpoint; d èvè is1if v is not a cutpoint. Given a graph with h connected components, the separating P degree ofatutte pair èa 1,a è, sdèèa 1 ;a èè, is i=1 d èa i è+d 3 èèa 1 ;a èè + h,4. We will show èclaim 1 in Section 3è that the separating degree of a Tutte pair is equal to the smallest number of edges needed to connect the graph obtained from G by removing the Tutte pair. The separating degree for the corresponding çvertex s, sdèsè, is equal to sdèèa 1 ;a èè. The triconnectivity augmentation number Given a graph G, the triconnectivity augmentation number of G is the smallest number of edges needed to triconnect G. 3 A Lower Bound for the Triconnectivity Augmentation Number In this section, we state a lower bound for the triconnectivity augmentation number. This lower bound turns out to be the exact triconnectivity augmentation number. Separating degrees of çvertices We state a claim to justify the way the separating degree of a Tutte pair in a 3block graph is deæned. Claim 1 The separating degree of a Tutte pair is equal to the smallest number of edges needed to connect the graph obtained from G by removing the Tutte pair. 3block leaf and isolated 3block vertex We identify ævertices in a 3block graph whose corresponding Tutte components must contain a new incoming edge if we want to triconnect the input graph. Deænition 1 Given a ævertex b, let H b be its corresponding Tutte component of G. A degree1 vertex b is a 3block leaf if any one of the following conditions is true: èiè H b consists of only one vertex u and u is not a cutpoint; èiiè ifh b contains any cutpoint c of G, c is in a Tutte pair of G contained inh b. ènote that èiè holds if and only if u is in a polygon.è A degree0 ævertex b is an isolated 3block vertex if any one of the following conditions is true: èiè H b contains at most cutpoints; èiiè H b consists of only one vertex u and the degree ofu in G is at most. Figure?? illustrates a 3block leaves. Note that if G is biconnected, then a degree1 vertex in 3blkèGè must be a 3block leaf. Demanding vertex Deænition Given a 3block leaf or an isolated 3 block vertex b in 3blkèGè, let H b be its corresponding Tutte component. A vertex u in H b is a demanding vertex of b if any one of the following conditions is true: èiè u is not a cutpoint or in any Tutte pair of G contained inh b ;èiiè b is an isolated 3block vertex and H b consists of only the vertex u. The vertex u is also called a demanding vertex in G. Claim Let b be a 3block leaf or an isolated 3block vertex in 3blkèGè. Then there is at least one demanding vertex of b. When we specify the 3block graph of a graph G, we will include a demanding vertex for each 3block leaf and each isolated 3block vertex in 3blkèGè. The weight of a graph We now deæne the weight of a graph, which we will relate later to the number of edges needed to triconnect the graph. Deænition 3 Let H be a block in a graph G whose corresponding vertex in blkègè is r H. Let l 3 èhè be the number of 3block leaves in H. We deæne the weight of the graph G, wègè, to be P 8 blocks H maxf3, d èr H è;l 3 èhèg, if G is not biconnected. Otherwise, let wègè =l 3 ègè. Massive, critical and balanced For an undirected graph G with the weight wègè, a Tutte pair z or its corresponding çvertex is massive if sdèzè é d wègè e. ATutte pair z or its corresponding ç vertex is critical if sdèzè =d wègè e. IfnoTutte pair in G is massive, then G and its 3block graph are called
4 balanced. Given a çvertex v in 3blkèGè, let 3blkèGè, v be the graph obtained from 3blkèGè by removing v. Avchain is a component of3blkègè, v which contains only one 3block leaf in 3blkèGè. The 3block leaf of 3blkèGè inavchain is called the vchain leaf. Alower bound of the triconnectivity augmentation number We now state a lower bound for the triconnectivity augmentation number for a general undirected graph. Lemma 1 Given an undirected graph G, weneed at least maxfd,d wègè eg edges to triconnect G, where d is the largest separating degree among all çvertices in 3blkèGè. Proof: The ærst component of the lower bound comes from Claim 1. For the other component of the lower bound, suppose that G 0 is triconnected and is obtained from G by adding a smallest set of edges. For each  block H, wemust have at least 3,d èr H è new incoming edges in G 0. For each 3block leaf in 3blkèHè, the induced subgraph on vertices corresponding to it must contain at least one new incoming edge in G 0 maxf3,dèrh è;l3èhèg. Hence we need at least d e new edges for each block H. The total number of new edges needed is thus d wègè e. Hence we need at least maxfd,d wègè eg edges in G 0. This proves the lemma. 4 Finding a Smallest Augmentation to Triconnect a Biconnected Graph In this section, we consider the problem of ænding a smallest set of edges to triconnect a biconnected graph. We show that the lower bound given in Lemma 1 can be achieved, and we give a linear time algorithm to ænd a smallest augmentation to triconnect the graph. 4.1 Properties of the 3Block Tree for a Biconnected Graph In this section, we explore properties of 3blkèGè that will be used in the following sections, where G is a biconnected graph. First we give bounds on the number of massive and critical çvertices in 3blkèGè. They are similar to the bounds on the number of massive and critical c vertices in blkègè given in ë10ë. Proofs in ë10ë can be easily modiæed to prove the following claim. Claim 3 Let s 1 beaçvertex with the largest separating degree among all çvertices in 3blkèGè and let s i be açvertex with the largest separating degree among all çvertices other than s 1 ; æææ;s i,1, for ç i ç 3. Then the following æve properties hold. P 3 èiè i=1 d 3ès i è,4 ç l 3 ègè, where l 3 ègè is the number of degree1 vertices in 3blkèGè. èiiè If blkègè has more than two çvertices, then d 3 ès 3 è ç l3ègè+4. 3 èiiiè There can be at most one massive çvertex in blkègè. èivè If there is a massive çvertex in blkègè, then there is no critical çvertex in blkègè. èvè There can be at most two critical çvertices in blkègè if l 3 ègè é. Given an input biconnected graph G, its 3block graph and a graph G 0 obtained from G by adding an edge between two distinct vertices u and v, wenow describe a method to obtain 3blkèG 0 è from 3blkèGè by local updating operations on 3blkèGè instead of computing it directly from G 0. Let P 3 be the implied path between u and v in 3blkèGè. We deæne the implied graph G P3 of G on the path P 3 as follows. G P3 contains vertices and edges that are in Tutte components of G corresponding to ævertices in P 3.For every çvertex q in P 3, let s q a and s q b be the two çvertices connected to q in P 3. Let èa q 1,aqè and èb q 1,bqè be the Tutte pairs represented by sq a and sq b, respectively. If s q a is adjacent toaævertex in P 3 that corresponds to a trivial Tutte component fwg, then let a q = 1 w and let aq = w. The same procedure is applied on s q b, bq and 1 bq.weassume that if we traverse clockwise around the simple cycle C q represented by q starting from a q 1, the sequence of vertices visited in fa q,bq 1,bq g is b q, 1 bq and aq.we add the edge èaq 1,bqèto 1 G P3 if a q 1 6= b q and we add the edge 1 èaq,bq ètog P 3 if a q 6= b q. An example is shown in Figure??. Claim 4 Given a biconnected graph G and two demanding vertices u and v in G where u and v are in diæerent Tutte components, let P 3 be the implied path in 3blkèGè between u and v. The implied graph on P 3, G P3,isbiconnected if and only if u and v are not both in trivial Tutte components or there ismore than one çvertex in P 3. We now deæne the crack operation which is useful in describing the relation between 3blkèGè and the 3block graph of G after adding an edge. Deænition 4 Let G be a biconnected graph and let G 0 be the graph obtained from G by adding an edge between two demanding vertices u and v in G, where u and v are in diæerent Tutte components. Let P 3,q,s q a,s q b,aq 1,aq,bq and 1 bq be the path and vertices deæned above. The crack operation on q with respect to P 3 consists of the following procedures. èiè We add the edge èa q 1,bqè to the simple cycle 1 C q corresponding
5 to q if a q 1 6= b q and 1 èaq 1,bqè is not an edge in 1 C q. The edge èa q,bqè is added under the same condition for aq and b q.for each new simple cycle C created by adding these edges, we create a new çvertex if C contains a Tutte pair in C q èexcluding èa q 1,aqè and èbq 1,bqèè. èiiè After performing operations given in part i, let q 1 and q be the two çvertices corresponding to simple cycles created inpart i that contain only èa q 1,bqè and 1 only èa q,bqè, respectively. New çvertices s 1 ècorresponds to new Tutte pair èa q 1,bqèè and 1 s ècorresponds to new Tutte pair èa q,bq èè are added and connected to q 1 and q,respectively. èiiiè After performing operations given in parts i and ii, let s be açvertex in 3blkèGè such that s 6= s q a and s 6= s q b. Anedge ès,qè in 3blkèGè is changed toès,q 1 èorès,q è depending on whether the Tutte pair represented bys is on the simple cycle represented byq 1 or by q. Note that the crack operation creates at most two ç vertices. If there are two çvertices created by performing the crack operation on a çvertex q with respect to a path P 3, then P 3 is nonadjacent on q. Otherwise, P 3 is adjacent on q. An example is shown in Figure??. Lemma states the relations between 3blkèGè and 3blkèG 0 è, where G 0 is the graph obtained from G by adding an edge. Lemma Let G be an input biconnected graph and let G 0 be the graph obtained from G by adding an edge between two vertices u and v in G, where u and v are in diæerent Tutte components. Let P 3 be the implied path between u and v in 3blkèGè. Wecan obtain 3blkèG 0 è from 3blkèGè by applying the following operations. èiè Edges in P 3 are eliminated. Vertices and edges in 3blkèGè that are not in P 3 remain in 3blkèG 0 è.èiiè The ævertices in P 3 shrink into a new ævertex b P3 in 3blkèG 0 è if u and v are not both in trivial Tutte components or there ismore than one ç vertex in P 3. The corresponding Tutte component of b P3 is G P3 ë èu; vè, where G P3 is the implied graph on P 3. Otherwise, ævertices in P 3 are eliminated. èiiiè A vertex s in P 3 with d 3 èsè =is eliminated in3 blkèg 0 è if s is a çvertex or a çvertex. A çvertex s in P 3 with d 3 èsè ç 3 is adjacent to b P3 if b P3 exists. èivè A çvertex q in P 3 with d 3 èqè ç 3 is cracked èdeænition 4è and new çvertices are connected tob P3 if b P3 exists. If b P3 does not exist, we create a new ç vertex q 0 and the two çvertices created by the crack operation are connected toq 0. Proof: Parts i and iii are trivial. We only prove parts ii and iv. G P3 is biconnected èclaim 4è if u and v are not both in trivial Tutte components or there is more than one çvertex in P 3.éFrom the deænition of G P3,we know that the removal of any Tutte pair in G P3 will result in a graph with two connected components which contain u and v, respectively. Thus the graph G 0 P 3 obtained from G P3 by adding èu,vè is triconnected. It is also true that G 0 P 3 is maximal èno vertex can be added such that the resulting graph is still triconnectedè. This proves part ii. We know that pairs of endpoints between edges added in cracking çvertices correspond to new Tutte pairs. Every new Tutte pair created is contained in G 0 P 3. This proves part iv. An example of updating the 3block graph is shown in Figure??. The following is a corollary of Lemma. Corollary 1 Let G 0 be the graph obtained from G by adding an edge between any two demanding vertices u and v in G. The degree ofaçvertex s in P 3, the implied path in 3blkèGè between u and v, with degree é in 3blkèGè is decreased by1in3blkèg 0 è. Deænition 5 ëthe leafconnecting condition for triconnecting a biconnected graphë Let G be a biconnected graph and let u and v be two demanding vertices in G. Let P 3 be the implied path between u and v in 3blkèGè. The pair of vertices u and v satisæes the leafconnecting condition if and only if any one of the following conditions is true: èiè the path P 3 contains a ævertex of degree atleast 4; èiiè the path P 3 contains two vertices of degree atleast 3; èiiiè there exists a çvertex in P 3 such that P 3 is nonadjacent on it. Lemma 3 Let G be a biconnected graph and G 0 be the graph obtained from G by adding a new edge èu,vè. Let l 3 ègè and l 3 èg 0 è be the numbers of degree1 vertices in 3blkèGè and 3blkèG 0 è, respectively. If u and v satisfy the leafconnecting condition èdeænition 5è and l 3 ègè é 3, then l 3 èg 0 è=l 3 ègè,. Proof: We know that two degree1 vertices in 3blkèGè that correspond to Tutte components containing u and v are eliminated. If u and v satisfy the leafconnecting condition, the only ævertex created has a degree at least. Since we do not create any degree1 vertex, the lemma holds. 4. An Algorithm for Triconnecting a Biconnected Graph Using the Smallest Number of Edges We will show after the description of the algorithm that by using algorithm augto3 given below, the lower bound given in Lemma 1 can be reduced by 1 each time we add a new edge. We now describe the algorithm.
6 fæ The algorithm ænds a smallest augmentation to triconnect G, where G is biconnected with ç 4vertices. æg graph function augto3ègraph Gè; fæ The algorithmic notation used here is from Tarjan ë19ë. æg T := 3blkèGè; let l 3èGè be the number of degree1 vertices in T ; do l 3èGè ç! let s 1 be a çvertex with the largest degree in T ; 1. if s 1 is massive! v := s 1; w := s 1 j s 1 is not massive! let s be a çvertex with the largest degree in T other than s 1; let d 3ès è=0ifs does not exist; let b 1 be a nonçvertex with the largest degree in T ; if d 3ès 1è é!. v := s 1; 3. if d 3ès è é! w := s j d 3ès è ç and d 3èb 1è é! w := b 1 j d 3ès è ç and d 3èb 1è ç! w := v æ j d 3ès 1è ç! let b be a nonçvertex with the largest degree in T other than b 1; 4. v := b 1; æ; æ if d 3èb è é! v := b j d 3èb è ç! w := v æ if v = w and v is a çvertex and d 3èvè é 3! 5. ænd two degree1 vertices y and z such that the path between them in T is nonadjacent on v j v 6= w or v is not a çvertex or d 3èvè ç 3! 6. if s 1 is massive! ænd two s 1chain leaves y and z j s 1 is not massive! ænd two degree1 vertices y and z such that the path between them passes through v and æ w æ; ænd demanding vertices u 1 and u of y and z, respectively; fæ Claim shows that this is always possible. æg add an edge between u 1 and u ; update the 3block graph T ; if l 3èGè 6= 3! l 3èGè :=l 3èGè, j l 3èGè =3! l 3èGè :=l 3èGè, 1 æ od; return G end augto3; Before the proof of correctness for algorithm augto3, we state a claim for an input graph G that is unbalanced. The proof of this claim is similar to a proof in ë10, 16ë. Claim 5 Let G be an unbalanced biconnected graph with at least 4 vertices. Let s be the massive çvertex in 3blkèGè and let æ = d 3 èsè, 1,d l3ègè e. There are æ + schains in 3blkèGè. Let M be the set of schain leaves. By adding k; k ç æ, edges to connect k +1 vertices of M, wereduce both d 3 èsè and the number of leaves in the 3block tree by k. Claim 6 Let d be the largest separating degree among all çvertices in 3blkèGè and let l 3 ègè be the number of degree1 vertices in 3blkèGè. IfG is biconnected with at least 4 vertices, we can triconnect G by adding maxfd,d l3ègè eg edges using algorithm augto3. Proof: If 3blkèGè contains a massive vertex s 1, then the two s 1 chain leaves are found in steps 1 and 6. If 3blkèGè is balanced, all critical vertices are found in steps and 3. Thus if d çd l3ègè e, then d is decreased by 1 ècorollary 1è. The pair of vertices u 1 and u also satisæes part ii of the leafconnecting condition èdeænition 5è by steps, 3 and 4 if possible. Otherwise, it satisæes part i of the leafconnecting condition by step 4 or part iii of the leafconnecting condition by steps 4 and 5. Thus if déd l3ègè e, l 3 ègè decreases by each time algorithm augto3 adds an edge. The algorithm guarantees that maxfd,d l3ègè eg decreases by 1 each time we add an edge. We know that maxfd,d l3ègè eg = 0 implies that 3blkèGè is a single ævertex. Thus G is triconnected if G contains at least 4 vertices. Hence the claim is true. Theorem 1 The triconnectivity augmentation number for a biconnected graph G equals maxfd,d l3ègè eg, where d+1 is the largest degree among all çvertices in 3blkèGè and l 3 ègè is the number of degree1 vertices in 3blkèGè. Claim 7 Algorithm augto3 runs in Oèn + mè time onagraph with n vertices and m edges. 5 Finding a Smallest Augmentation to Triconnect a Graph In this section, we consider the problem of ænding a smallest augmentation to triconnect any undirected graph. We show that the lower bound given in Lemma 1 can be always achieved by giving an algorithm for it. Finally, we describe a linear time implementation for the algorithm.
7 5.1 Properties of the 3Block Graph In this section, we study properties of the 3block graph that are used in the next section for designing an algorithm to ænd a smallest augmentation to triconnect a graph. We ærst study the changes made on the 3block graph after adding an edge into the original graph in Deænition 6 and Lemma 4. Then in Claim 8, Claim 9, Lemma 5 and Corollary 3, we show that we can always decrease the lower bound given in Lemma 1 by 1by adding an edge. Let G 0 be the graph obtained from G by adding an edge between two demanding vertices u and v in G. We describe the updating operation performed on 3blkèGè after adding an edge. Deænition 6 Let G 0 be the graph obtained from G by adding an edge between two demanding vertices u and v in G, where u and v are in diæerent Tutte components. Given fq 1 ; æææ;q r g, the collection of implied paths in 3blkèGè between u and v, the block sequence ëy 1 ; æææ;y r ë, the separatingsequence ëw 1 ; æææ;w r ë and the cutsequence ëu,c 1 ; æææ;c x,vë, we can obtain 3 blkèg 0 è from 3blkèGè by performing the following operations. èiè For each Q i, 1 ç i ç r, weperform the operation given in Lemma on the 3block tree that contains Q i if Q i contains more than 1 vertex. èiiè A new çvertex is created with its corresponding simple cycle ëu,c 1 ; æææ;c x,v,uë. Each pair of vertices w i,1 and w i, 1 ç i ç r, isatutte pair. The corresponding çvertex for Tutte pair èw i,1,w i è is incident on the new çvertex created. èiiiè For each cvertex y i, 1 ç i ç r, wecreate a ævertex and connect it to the çvertex corresponds to Tutte pair èw i,1,w i è. èivè For each bvertex y i, 1 ç i ç r, the çvertex corresponding to the Tutte pair èw i,1,w i è is incident on the ævertex z whose corresponding Tutte component contains w i,1 and w i if z exists. èvè Wemerge ç vertices corresponds to the same Tutte pair and delete degree1 çvertices. The following lemma can be easily derived from Lemma, the deænition of Tutte split and the deænition of the 3block graph. An example is shown in Figure??. Lemma 4 Let G 0 be the graph obtained from G by adding an edge between two demanding vertices u and v, where u and v are in diæerent Tutte components. Given fq 1 ; æææ;q r g, the collection of implied paths between u and v in 3blkèGè, and P, the implied path in blkègè between u and v, the 3block graph for G 0 can be obtained from 3blkèGè by performing the updating operations deæned in Deænition 6 in OèjP j + P r i=1 jq i jè time. We know that if G is biconnected, there can be at most two critical çvertices in 3blkèGè èclaim 3è. But this is not true for G is not biconnected èa similar claim is also given in ë6ëè. We now state a claim about the set of all critical çvertices in 3blkèGè for a general graph G. In general, if there are more than critical çvertices in 3blkèGè, we can partition the set of critical çvertices into at most two subsets such that either a subset has one çvertex or Tutte pairs corresponding to critical çvertices in a subset share a common cutpoint. Claim 8 Let G be aconnected graph with wègè é. Let S = fs 1 ; æææ;s r g be the set of critical çvertices in 3blkèGè. Given S, let = 1 = fc j c is a cutpoint that is shared by more than one Tutte pair represented by members of Sg. Let = = fs j s Sand 69c = 1 such that the Tutte pair represented by s contains cg. Then j= 1 j + j= jç. Proof: The proof uses 3 propositions and is given in ë11ë. All propositions are proved by ærst ænding separating degrees of çvertices in S and a lower bound value for wègè in all possible cases and then deriving a contradiction by the fact that the separating degree of any critical çvertex must be equal to d wègè e. Corollary Let G be a connected graph with wègè é. We can ænd two demanding vertices u 1 and u in G such that each critical çvertex is either in one of the paths in the collection of implied paths between u 1 and u in 3blkèGè or its corresponding Tutte pair contains a cutpoint in the implied path between them in blkègè. Claim 9 states some conditions under which separating degrees of certain çvertices decrease by 1. Claim 9 Let G be an undirected graph and let u and v be two demanding vertices in G. Let G 0 be the graph obtained from G by adding the edge èu,vè and let s be a çvertex in 3blkèGè with sdèsè ç 3. Then sdèsè decreases by 1 in G 0 if any one of the following conditions is true: èiè s is in one of the path in the collection of implied paths between u and v in 3blkèGè; èiiè the corresponding Tutte pair of s contains a cutpoint in the implied path between u and v in blkègè. Lemma 5 states a condition under which the weight of the graph decreases by after adding an edge. Lemma 5 Let G 0 be the graph obtained from a graph G by adding an edge between two distinct demanding vertices u and v. Then wèg 0 è=wègè, if u and v are in diæerent connected components or in diæerent blocks.
8 By Lemma 5 and Corollary, we now show that in any biconnected graph, we can always decrease the lower bound given in Lemma 1 by1by adding an edge. Corollary 3 Let G be aconnected graph that is not biconnected. We can ænd two demanding vertices u 1 and u in G such that they satisfy the condition given in Corollary and wèg ëfèu 1 ;u ègè =wègè,. Proof: If we can ænd two demanding vertices not in the same block that satisfy the condition given in Corollary or any two critical çvertices whose Tutte pairs share a common cutpoint, then Lemma 5 shows wègëfèu 1 ;u ègè =wègè,. Let u 1 and u be in the block H; every critical çvertex is also in H and they do not share any cutpoint. éfrom Claim 8, we know that there can be at most two critical çvertices whose Tutte pairs do not share cutpoint. If there is only one critical çvertex in H, we can easily substitute one of u 1 and u for a demanding vertex in a block that is not H such that u 1 and u satisfy both the conditions given in Corollary and Lemma 5. Thus wèg ëfèu 1 ;u ègè =wègè,. For the case of having two critical çvertex in H, their degrees in 3blkèGè must be greater than since G is not biconnected and they are the only two critical çvertices in 3blkèGè. By Lemma 3, the number of 3block leaves decreases by. Thus the weight ofg decreases by after adding èu 1,u è. 5. An Algorithm for Triconnecting an Undirected Graph Using the Smallest Number of Edges In the following algorithm, aug3, we ænd a pair of demanding vertices u and v in G such that the collection of implied paths Q in 3blkèGè between u and v passes through the massive çvertex or all critical çvertices if G is connected. If G is not connected, u and v are demanding vertices in G that are in diæerent connected components. We also have to make sure that wègè is reduced by if 3blkèGè is balanced by satisfying the conditions given in Lemma 5. fæ The graph G has at least 4 vertices; the algorithm ænds a smallest augmentation to triconnect G. æg graph function aug3ègraph Gè; T := blkègè; T 3 := 3blkèGè; let wègè be the weight ofg; fæ Deænition 3. æg do wègè ç and 9 a cvertex in T! if G is not connected! 1. ænd y and z that are 3block leaves or isolated 3block vertices in diæerent trees of T 3 j G is connected! let d be the largest separating degree among all çvertices in T 3; if déd wègè e!fæthere exists an unique massive çvertex in T 3 èclaim 10è. æg. let s be the massive çvertex in T 3; 3. ænd two schain leaves y and z in T 3 j d çd wègè e! 4. ænd y and z that are 3íblock leaves or isolated 3block vertices in T 3 such that demanding vertices u and v of y and z, respectively, satisfy the condition given in Corollary 3 fæ Corollary 3 shows that this is always possible. æg æ æ; 5. ænd demanding vertices u 1 and u of y and z, respectively; fæ Claim shows that this is always possible. æg add an edge between u 1 and u ; update the block graph T ; update the 3block graph T 3; if G is connected and déd wègè e! wègè :=wègè, 1 j G is not connected or d çd wègè e! wègè :=wègè, æ od; fæ G is biconnected. æg return augto3ègè end aug3; Claim 10 If there exists a massive çvertex s 1 in 3 blkègè, then a çvertex s with the largest separating degree among çvertices other than s 1 is not massive or critical. Let æ = sdès 1 è,d wègè e. There areat least æ + s 1 chains in 3blkèGè. Let M be the set of s 1 chain leaves. By adding k; k ç æ, edges to connect k +1 vertices of M, wereduce both sdès 1 è and the weight of the graph by k. Claim 11 Let G be an undirected graph with at least 4 vertices. We can triconnect G by adding maxfd,d wègè eg edges using algorithm aug3, where d is the largest separating degree among all çvertices in 3blkèGè and wègè is the weight of G. Proof: If G is not connected, algorithm aug3 ænds u 1 and u in diæerent connected components at steps 1 and 5. éfrom Lemma 5, we know that d wègè e is decreased by 1. éfrom the deænition of separating de gree, d is decreased by 1 because the number of connected components is decreased by 1. éfrom Claim 10, we know that there can be at most one massive vertex s if déd wègè e. Algorithm aug3 ænds s at step. We reduce maxfd,d wègè eg by 1by adding a new
9 edge between two schain leaves èstep 3è if G is connected and unbalanced. If G is balanced, algorithm aug3 adds an edge èu 1,u è such that u 1 and u are two demanding vertices satisfy the condition given in Corollary 3. Thus d wègè e is decreased by 1 and G remains balanced. Algorithm aug3 keeps doing this un til G is biconnected. Notice that if G is biconnected, wègè = l 3 ègè. At this point, algorithm aug3 calls algorithm augto3 on the current biconnected graph. Hence by Claim 6, the claim is true. Theorem The triconnectivity augmentation number for a graph G equals maxfd,d wègè eg, where d is the largest separating degree among all çvertices in 3blkèGè and wègè is the weight of G. 5.3 A Linear Time Implementation Let s G be the sum of the separating degrees of all Tutte pairs in the 3block graph of a graph G with n vertices and m edges. By using techniques similar to those described in ë10ë and the proof of Claim 7, algorithm aug3 can be implemented to run in Oèn+m+s G è time. Since s G could be æèn è for a graph with n vertices, we do not have a linear time implementation for algorithm aug3 if m is not æèn è. However, Lemma 6 èë6ë stated a similar claim which is part of the lemmaè tells us of a possible method to obtain a linear time implementation. Lemma 6 Given a critical Tutte pair s 1 and another Tutte pair s, s 1 6= s,inaconnected graph G, let s 1 and s share acommon cutpoint c. Let s 1 =èc; a 1 è and let s =èc; a è. Then d 3 ès 1 è ç d 3 ès è+èd èa è, 1è. Ifs is also critical, then d èa 1 è=1and d èa è= 1. Lemma 6 says that for a set of more than 1 critical çvertex whose corresponding Tutte pairs share a common cutpoint c, the other vertices in these Tutte pairs are not cutpoints. Let S c be the set of Tutte pairs share c and let J c be the subset of S c with the largest degree in 3blkèGè among all Tutte pairs in S c. ATutte pair s in S c is a candidate if and only if any one of the following conditions is true: èiè J c = fsg; èiiè s J c and only the vertex c in each Tutte pair in J c is a cutpoint; èiiiè s J c and both vertices in s are cutpoints. We can ænd a candidate for any S c. Lemma 6 also states that if js c jç and any one of the candidates is not critical, then none of the Tutte pairs in S c is critical. Let n be the numberofvertices and m be the number of edges in the input graph. Using Lemma 6 and the sorted table data structure given in ë16ë, we can implement algorithm aug3 in Oèn + mè time. Given a set of p nodes V with a key in each node whose value is Oèpè, the sorted table is an array L of Oèpè entries. The ith entry of L points to a doubly linked list which contains all nodes in V with the key value i. For Tutte pairs that do not share any cutpoint with any other Tutte pairs, we maintain a sorted table using their separating degrees as keys. Entries are updated whenever their separating degrees are changed. The sum of the separating degrees of all çvertices in the sorted table is Oènè. Given a sorted table L, an equivalent sorted list is a doubly linked list on entries of L that does not point to an empty list. The relative order of entries in L is preserved. For each cutpoint c that is shared by Tutte pairs S c = fèc; a 1 è; æææ; èc; a r èg, where r ç, we maintain the following information: y c, a candidate for S c èy c is updated only if the current candidate no longer satisæes the condition of being a candidateè and Q c, a sorted list on the set of çvertices corresponding to Tutte pairs in S c with their degrees in the 3block graph as their key values. Note that we must rechoose y c, where c is a cutpoint, if and only if a degree in 3blkèGè foratutte pair is changed or a degree in  blkègè for a cutpoint becomes 1. Thus we only have to rechoose Oènè times. Thus the above information can be maintained in totally Oènè time. For a cutpoint c, let its y c be èc,a c è. We maintain an entry with key value sdèy c è in the sorted table if and only if any one of the following conditions is true: èiè a c is not a cutpoint; èiiè js ac jç1; èiiiè js ac j é 1 and y c = y ac. This makes sure that if we update èdecreaseè the degree of a cutpoint, only a constant number of entries in the sorted table are updated. Note that if we have tochange y c for a cutpoint c, we only have to get a constant number of entries in and out the sorted table. Every entry in the sorted table is updated whenever its separating degree is changed. During the augmentation using algorithm aug3, we create new çvertices by creating new polygons. The sum of the degrees in the 3block graph of all created çvertices is Oènè. Since the sum of the degrees of all çvertices in the 3block graph, the sum of the degrees of all cvertices in the block graph and the sum of the separating degree of all çvertices in 3blkèGè are all Oènè, the total time to update the sorted table is Oènè. Using the above method, we can implement algorithm aug3 in linear time. Claim 1 Algorithm aug3 runs in linear time. 6 Conclusion In this paper, we have presented a linear time se
10 quential algorithm for ænding a smallest augmentation to triconnect an undirected graph. The algorithm is divided into two stages. During the ærst stage, we biconnect the input graph. Then we triconnect the resulting biconnected graph using the smallest number of edges in the second stage. We have to make sure that the total number of edges added in these two stages is minimum. We ærst described a linear time sequential algorithm to triconnect a biconnected graph using the smallest number of edges. Then we presented a linear time algorithm to biconnect a graph such that the set of edges added is a subset of edges needed to triconnect the graph using the smallest number of edges. References ë1ë GuoRay Cai& YuGeng Sun, ë The minimum augmentation of any graph to a kedgeconnected graph," Networks, Vol. 19, 1989, pp ëë Kapali P. Eswaran & R. Endre Tarjan, ëaugmentation problems," SIAM J. Comput. Vol 5, December 1976, pp ë3ë S. Even, Graph Algorithms, Computer Science Press, Rockville, MD, ë4ë Andrças Frank, ëaugmenting graphs to meet edgeconnectivity requirements," Proc. 31th Annual IEEE Symp. on Foundations of Comp. Sci., 1990, pp ë5ë H. Frank & W. Chou, ëconnectivity considerations in the design of survivable networks," IEEE Trans. on Circuit Theory, Vol. CT17, November 1970, pp ë6ë Greg N. Frederickson & Joseph Ja'Ja', ëapproximation algorithms for several graph augmentation problems," SIAM J. Comput., Vol. 10, May 1981, pp ë7ë D. Gusæeld, ëoptimal mixed graph augmentation," SIAM J. Comput., Vol. 16, August 1987, pp ë8ë D. Harel & R. E. Tarjan, ëfast algorithms for ænding nearest common ancestors," SIAM J. Comput., Vol. 13, 1984, pp ë9ë J. E. Hopcroft & R. E. Tarjan, ëdividing a graph into triconnected components," SIAM J. Comput., Vol., September 1973, pp ë10ë Tsansheng Hsu & Vijaya Ramachandran, ëon ænding a smallest augmentation to biconnect a graph," Tech. Rep. TR911, U.T. Austin, ë11ë Tsansheng Hsu & Vijaya Ramachandran, ëa linear time algorithm for triconnectivity augmentation," Manuscript, U.T. Austin, ë1ë Yoji Kajitani & Shuichi Ueno, ëthe minimum augmentation of a directed tree to a kedgeconnected directed graph," Networks, Vol. 16, 1986, pp ë13ë Arkady Kanevsky, ëvertex connectivity of graphs: algorithms and bounds," Tech. Rep. ACT97 èph.d. thesisè, Coordinated Science lab., Univ. of Illinois, Urbana, ë14ë D. Naor, D. Gusæeld & C. Martel, ëa fast algorithm for optimally increasing the edgeconnectivity," Proc. 31th Annual IEEE Symp. on Foundations of Comp. Sci., 1990, pp ë15ë Vijaya Ramachandran, ëparallel open ear decomposition with applications to graph biconnectivity and triconnectivity," invited chapter for Synthesis of Parallel Algorithms, J. H. Reif, editor, MorganKaufmann. ë16ë Arnie Rosenthal & Anita Goldner, ësmallest augmentations to biconnect a graph," SIAM J. Comput., Vol. 6, March 1977, pp ë17ë Danny Soroker, ëfast parallel strong orientation of mixed graphs and related augmentation problems," Journal of Algorithms, 9, 1988, pp ë18ë Kenneth Steiglitz, Peter Weiner & D. J. Kleitman, ëthe design of minimumcost survivable networks," IEEE Trans. on Circuit Theory, Vol. CT16, November 1969, pp ë19ë R. E. Tarjan, Data Structures and Network Algorithms, SIAM Press, Philadelphia, PA, ë0ë W. T. Tutte, Connectivity in Graphs, University of Toronto Press, ë1ë Shuichi Ueno, Yoji Kajitani & Hajime Wada, ëminimum augmentation of a tree to a kedgeconnected graph," Networks, Vol. 18, 1988, pp ëë T. Watanabe, ëan eæcient way for edgeconnectivity augmentation," Tech. Rep. ACT76, Coordinated Science lab., Univ. of Illinois, Urbana, ë3ë T. Watanabe, Y. Higashi & A. Nakamura, ëgraph augmentation problems for a speciæed set of vertices," Proc. ærst Ann. Intl. Sym. on Algorithms, 1990, pp ë4ë T. Watanabe & A. Nakamura, ëon a smallest augmentation to triconnect a graph," Tech. Rep. C18, Department of Applied Mathematics, Hiroshima University, Japan è1983, revised 1987è. ë5ë T. Watanabe & A. Nakamura, ëedgeconnectivity augmentation problems," J. Comput. System Sci., 35è1987è, pp ë6ë T. Watanabe & A. Nakamura, ë3connectivity augmentation problems," Proc IEEE Intl. Sym. on Circuits and Systems, 1988, pp ë7ë T. Watanabe, T. Narita & A. Nakamura, ë3edge Connectivity augmentation problems," Proc IEEE Intl. Sym. on Circuits and Systems, 1989, pp
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[25] T. Watanabe, An ecient way for edgeconnectivity augmentation, Tech. Rep. ACT 76UILUENG872221, Coordinated Science lab., University of Illinois, Urbana, IL, 1987. [26] T. Watanabe and A. Nakamura,
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