Low-Connectivity Network Design on Series-Parallel Graphs

Transcription

1 Low-Connectvty Network Desgn on Seres-Parallel Graphs S. Raghavan The Robert H. Smth School of Busness, Van Munchng Hall, Unversty of Maryland, College Park, Maryland Network survvablty s a crtcal ssue for modern fberoptc telecommuncaton networks. Networks wth alternate routes between pars of nodes permt users to communcate n the face of equpment falure. In ths paper, we consder the followng low-connectvty network desgn (LCND) problem: Gven a graph G (N, E) and a connectvty requrement d {0, 1, 2} for each node and edge costs c e for each edge e E, desgn a mnmum-cost network that contans at least d st mn{d s, d t } dsont paths between nodes s and t. We present lnear-tme algorthms for both node- and edge-connectvty versons of the problem on seres-parallel graphs. Due to the sparsty of telecommuncatons networks, ths algorthm can be appled to obtan partal solutons and decompostons that may be embedded n a heurstc soluton procedure as well as exact soluton algorthms for the problem on general graphs Wley Perodcals, Inc. Keywords: network desgn; seres-parallel graphs; connectvty 1. INTRODUCTION In ths paper, we consder the low-connectvty network desgn (LCND) problem that arses as a fundamental problem n the practcal desgn of telecommuncaton networks (see [7]). In ths problem, gven an undrected network G (N, E), wth node set N, edge set E, a connectvty requrement d {0, 1, 2} for each node N, and edge costs c e for each edge e E, we wsh to desgn a mnmum-cost network that has at least d st mn{d s, d t } dsont paths between each par of nodes s and t. We consder both edge- and node-connectvty versons of the problem.* To dstngush between edge-connectvty and * There are two edge-dsont paths (respectvely, node-dsont paths) between a par of nodes f the deleton of a sngle edge (respectvely, sngle node) n the network does not dsconnect them. A network s 2-edgeconnected (respectvely, 2-node-connected) f there are two edge-dsont paths (respectvely, node-dsont paths) between every par of nodes. Receved July 2001; accepted November 2003 Correspondence to: S. Raghavan; e-mal: raghavan@umd.edu Contract grant sponsor: Bell Laboratores Martn Farber Internshp 2004 Wley Perodcals, Inc. node-connectvty, we append N (e.g., 2N) to denote nodeconnectvty requrements and append E to denote edgeconnectvty requrements. The LCND s NP-hard on general graphs, snce t generalzes the Stener tree problem, and has rased consderable nterest among researchers. Grötschel et al. [12] descrbed a cuttng-plane approach for the problem on general graphs and were able to use ths approach to solve some problems arsng n local telephone companes. Magnant and Raghavan [15] descrbed a dual-ascent algorthm for the edge-connectvty verson of the problem. Ths algorthm generates both a heurstc soluton, as well as a lower bound on the optmal soluton value for the problem. Several other researchers studed the problem on general graphs. See Grötschel et al. [13] and Raghavan and Magnant [20] for recent surveys on ths problem. Telecommuncaton networks are typcally sparse and planar [18]. Further, n many cases, parts of the network are seres-parallel graphs. Consequently, researchers have studed the restrcton of some versons of the LCND to seresparallel graphs. Wald and Colbourn [25] descrbed a lneartme algorthm for the Stener tree problem (d s {0, 1}) on a seres-parallel graph. Wnter [26] descrbed lnear-tme algorthms for the LCND on seres-parallel graphs when d s {0, 2N } and d s {0, 2E} (.e., no node has a connectvty requrement of 1). Snce network desgn problems are often modeled as nteger programs, researchers have nvestgated the polyhedral structure of the LCND problem on seres-parallel graphs. Mahoub [16] provded a complete descrpton of the 2-edge-connected spannng subgraph polytope on seres-parallel graphs (.e., the case where d s 2E for all nodes). Baïou and Mahoub [3] consdered the case where d s {0, 2E} and provded a complete descrpton of the Stener 2-edge-connected polytope. Coullard et al. [8, 9] consdered the node-connectvty verson. In [8], they gave a complete descrpton of the 2-node-connected spannng subgraph polytope on seres- A complete descrpton s a lnear nequalty descrpton of the convex hull of nteger feasble solutons to the problem. NETWORKS, Vol. 43(3),

2 parallel graphs (.e., the case where d s 2N for all nodes), and n [9], they gave a complete descrpton of the Stener 2-node-connected polytope (.e., the case wth d s {0, 2N }). Seres-parallel graphs are defned by a recursve constructon process. Consequently, because of ths structure, many NP-hard problems are polynomally solvable on them. However, the dervaton of the polynomal-tme algorthms are nontrval and, ndeed, very problem-specfc. As we wll show, the edge- and node-connectvty verson of the LCND admt lnear-tme algorthms. The dentfcaton of a lnear-tme algorthm s mportant for several reasons: Frst, due to the sparsty of telecommuncaton networks, large portons of them are seres-parallel. As a result, these algorthms may be used to obtan partal solutons on the seres-parallel portons that could be used n a heurstc procedure for the problem. Second, decomposton usng 2-separators s used to solve the problem on general graphs (see [12]). Usng a decomposton procedure makes t computatonally vable to solve large-scale problems va an exact procedure. In the decomposton procedure, a 2-separator (a par of nodes whose deleton separates the graph nto two or more components) s dentfed. Then, a seres of smaller problems s solved on the components (n a partcular order and wth some mnor modfcatons to these components), whose solutons can be peced together to obtan a soluton to the orgnal problem. The algorthm for the LCND on seres-parallel graphs s also based on decomposton usng 2-separators. Thus, the decomposton can easly be appled to general graphs and combned wth an exact soluton approach (such as a cuttng-plane or branch-and-cut algorthm) could result n the exact soluton of large-scale LCND problems. Interestngly, as a consequence of our algorthm for the edge-connectvty verson of the LCND, we observe that a decomposton used by Grötschel et al. [12] s ncorrect. The organzaton of the rest of the paper s as follows: In Secton 2, we descrbe a well-known correspondence between seres-parallel graphs and partal 2-trees. In dong so, we lay the foundatons of our dynamc programmng algorthms. In Secton 3, we descrbe a lnear-tme algorthm for the node-connectvty verson of the LCND problem, whle n Secton 4, we descrbe a lnear-tme algorthm for the edge-connectvty verson of the LCND. Fnally, n Secton 5, we dscuss how our results may be appled as a decomposton procedure to general graphs, thus provdng a way to possbly solve large LCND problems on general graphs. Notaton: We use standard graph theory terms and notaton as n the text by Bondy and Murty [5]. For clarty, we elaborate on the followng termnology where we dffer from [5]. A tral s a path that does not repeat edges. A smple path s a tral that does not repeat nodes. Let G 1 (N 1, E 1 ) and G 2 (N 2, E 2 ) be two graphs. The subgraph nduced by G 1 on G 2 has edges E 1 E 2 and nodes as the end ponts of the edges E 1 E 2.Anode cut s a subset N of N such that G N s dsconnected. A k-node cut s a node cut of k elements. A 2-separator s a 2-node cut. 2. SERIES-PARALLEL GRAPHS AND 2-TREES: ALGORITHM DESIGN OUTLINE A graph G (N, E) s sad to be seres-parallel f and only f t contans no subgraph homeomorphc to K 4 (the complete graph on four nodes). An alternate, constructve, defnton s as follows: A graph s seres-parallel f each of ts 2-node connected components can be constructed, startng wth an edge, by the repeated applcaton of the followng two operatons: Seres constructon: Replace an edge e (s, t) by a par of edges (s, u) and (u, t). Parallel constructon: Replace an edge e (s, t) by two parallel edges e (s, t) and e (s, t). A closely related graph to a seres-parallel graph s a 2-tree. It s defned va the followng recursve constructon procedure: K 3, the complete graph on three nodes s a 2-tree. Gven a 2-tree and any edge (, k) on the 2-tree, the graph obtaned by addng a new node and connectng t to nodes and k va new edges (, ) and (, k) s a 2-tree. Wald and Colbourn [25] characterzed those networks, partal 2-trees, that can be completed to 2-trees by addng new edges. They showed that a network s a partal 2-tree f and only f t has no subgraph homeomorphc to K 4. Robertson and Seymour [22, 23] ntroduced a closely connected concept of treewdth for graphs. Van Leeuwen [24] showed that partal 2-trees are dentcal to graphs wth treewdth at most 2. (Ths statement remans true when 2 s replaced by a general nteger parameter k 0,.e., partal k-trees are dentcal to graphs wth treewdth at most k, as proved n [24].) Thus, partal 2-trees are exactly seres-parallel graphs, and they are dentcal to graphs wth treewdth at most 2. We now brefly sketch a lnear-tme procedure developed by Wald and Colbourn [25] and adapted by Wnter [26] to complete a partal 2-tree to a 2-tree. In the process of completng a partal 2-tree to a 2-tree, we wll make the cost of added edges L a suffcently large number (settng L 1 e E c e, where E s the set of edges n the orgnal graph, s suffcent). By dong so, t s suffcent to focus our attenton to solvng the LCND on 2-trees. For example, f the cost of the soluton obtaned s L or greater, t mples that an edge not present n the orgnal graph s n the soluton and so the problem s nfeasble. Wthout loss of generalty, we assume that the graph G s connected. Otherwse, the problem s ether nfeasble (f A 2-node-connected component of a graph G s a maxmal subgraph of G that s 2-node-connected. 164 NETWORKS 2004

3 nodes wth connectvty requrements,.e., nodes wth d s 1, are n dfferent connected components) or we can dscard all components that do not contan nodes wth connectvty requrements. Checkng whether the graph s connected and dentfyng f all nodes wth connectvty requrements le n the same connected component can be done n lnear tme usng depth frst search (DFS) (see [10]). The procedure to complete a partal 2-tree to a 2-tree (from [25, 26]) also dentfes when the orgnal graph s not a partal 2-tree. It frst converts the connected graph to a 2-node connected graph. Ths s done usng DFS n such a way that the partal 2-tree structure s mantaned (f the orgnal graph s a partal 2-tree). The procedure then converts the 2-node connected partal 2-tree to a 2-tree by addng edges to t. To determne the edges to be added, the nodes of G are scanned sequentally and those of degree 2 are placed on a stack. The followng steps are then repeated upon a copy of G untl the copy s reduced to K 3 or the stack s empty: Let H G be a copy of graph G. (1) Remove the top node, from the stack. If the stack s empty, then G s not a partal 2-tree. STOP. (2) Determne edges (, ) and (, k) ncdent to n H. (3) If there s no edge (, k), add t to both H and to G wth cost c k L. (4) Delete node. H H. IfH K 3, G s a 2-tree. STOP. (5) If the degree of node or k n H s 2, place them on the stack. The DFS procedure to transform a connected partal 2-tree to a 2-node connected partal 2-tree takes lnear tme. The procedure of convertng a 2-node connected partal 2-tree to a 2-tree also takes lnear tme snce each step deletes a node n the graph and the number of operatons n each step s constant. Consequently, the procedure to complete a partal 2-tree to a 2-tree takes lnear tme. The motvaton for our dynamc programmng algorthms comes from the recursve constructon process of a 2-tree. We reverse the constructon process and sequentally contract the graph. At each stage, we repeatedly elmnate nodes of degree 2 untl the graph obtaned s an edge. At any stage n the contracton process (ncludng ntally pror to contracton), let G denote the graph represented by edge (, ). In other words, G represents the subgraph n the orgnal 2-tree G that has been contracted onto edge (, ) (Fg. 1 provdes an example). Durng the contracton process, we keep track of state nformaton for the subgraph G represented by each edge (, ). The state nformaton descrbes solutons to certan problems (whch can be dfferent from the orgnal problem) restrcted to the subgraph G. To use the contracton process to devse lnear-tme algorthms, there are three requrements: Frst, the number of states that we assocate wth each subgraph s fnte and ndependent of the number of nodes. Second, suppose that durng the contracton process node has degree 2 and s FIG. 1. Example of subgraph s of the orgnal graph G represented by an edge n the contracton process. Suppose that the graph s contracted to (, ), (, k), (, k). Then, G {(, ), (, f ), (, f ), (d, f ), (, d), (d, e), (e, f )}, G k {(, k), (, c), (k, c), (, b), (b, c), (a, b), (a, c)}, and G k {(, k), (, h), (h, k), (, g), (g, h)}. connected to nodes and k va edges (, ) and (k, ) pror to elmnaton of. Then, the new state nformaton for G k (.e., after the elmnaton of ) must be computable solely from the state nformaton for G, G k, and G k. Fnally, we should be able to deduce the soluton to the problem from the state nformaton of the edge that s left at the end of the contracton procedure. Although t s easy to state the algorthm desgn phlosophy, and ths has been formalzed n several dfferent ways (see [2, 4, 6]), t s a nontrval task to determne the state nformaton requred for a gven problem (and s very problem-specfc). In the next few sectons, we wll develop lnear-tme algorthms for dfferent versons of LCND. The basc algorthm can be descrbed as follows: The state nformaton s ntalzed for each edge of the graph G. The nodes of G are scanned sequentally and those of degree 2 are placed on a stack. The followng steps are repeated untl G s reduced to an edge: (1) Remove the top node from the stack. (2) Determne edges (, ) and (, k) ncdent to n G. (3) Snce G s a 2-tree, edge (, k) exsts. Compute the new state nformaton for the graph G k G k G G k. (4) Delete node from G by settng G G. Update the state nformaton assocated wth edge (, k) n G (.e., for graph G k ) by settng the state nformaton for G k equal to the state nformaton for graph G k computed n Step 3. (5) If the degree of node or k n G s 2, place them on the stack. NETWORKS

4 3. NODE-CONNECTIVITY REQUIREMENTS In the case of node-connectvty requrements, let Y 1 be the set of nodes wth d s 1 and let Y 2N be the set of nodes wth d s 2N. We now motvate the graphcal structures (states) that we compute n the course of the algorthm (.e., the graphcal structures we wll compute n Step 3 of the algorthm descrbed at the end of the prevous secton). At the end of the contracton process, an edge, say (, k), represents the graph. The mnmum-cost soluton to the problem ether ncludes both nodes and k, or ncludes node but not k, or ncludes node k but not, or excludes both nodes and k. Thus, at the mnmum, we must keep track of graphcal structures correspondng to these four forms. In our notaton, the captal letters denote the graphcal structure, and the small letters, ther costs. For ease of exposton, we wll use (nstead of L) to denote the cost of edges that were added to complete the partal 2-tree to a 2-tree, as well as to denote the cost of nfeasble solutons. However, n a computer mplementaton, as we have ndcated earler, may be replaced by a suffcently large number L. The graphcal structures, S k, T k, T k, and U k, correspondng to the four possble cases dentfed above are descrbed below: S k mnmum-cost network on G k that satsfes all the connectvty requrements on G k. It must nclude both nodes and k. T k mnmum-cost network on G k that satsfes all the connectvty requrements on G k and must nclude node and must exclude node k. Ifk s requred (.e., d k 1), then such a structure s nfeasble, and by conventon, t k. U k mnmum-cost network on G k that satsfes all the connectvty requrements on G k but excludes nodes and k. If ether or k are requred, then, by conventon, u k (snce U k s not feasble f ether or k are requred). Notce that the graphcal structures are varants of the orgnal problem restrcted to the graph G k. In addton, the followng graphcal structures are necessary for the computatons: P k mnmum-cost network on G k contanng both nodes and k and such that for every Y 2N node there s a smple path (.e., a path that does not repeat a node) from to k through t and every Y 1 node s connected to and to k. Q k mnmum-cost network on G k comprsng of two dsont trees that nclude all nodes wth nonzero connectvty requrements. One tree ncludes node and the other tree ncludes node k. IfG k {, k} has a Y 2N node, then, by conventon, q k. R k mnmum-cost network on G k comprsng of two dsont networks that nclude all nodes wth nonzero connectvty requrements. One subnetwork ncludes node and the other subnetwork ncludes node k. Further, all the Y 2N nodes belong to only one of the two dsont subnetworks, and there must be two node-dsont paths between every par of Y 2N nodes. If ths s not the case, for example, f and k Y 2N, then r k. To motvate the need for these structures, consder S k. Suppose that n Step 3 of the basc algorthm we need to compute S k for the graph G k G k G G k. Consder the possble graphcal structures nduced by S k on G k, G, and G k, respectvely. To compute S k, we need to have avalable the costs of these graphcal structures (.e., the possble ones S k nduces on G k, G, and G k, respectvely). S k may nduce a connected graph on G k, G, and G k, respectvely. In ths case, the graphcal structures S k nduces on G k, G, and G k are P k, P, and P k, respectvely (see Lemma 2), motvatng the need for P k.ifs k nduces a graph that s not connected on any one of G k, G, and G k, then, as we wll show, all Y 2N nodes n G k are n exactly one of G k, G, and G k. Consequently, the dsconnected subgraph nduced by S k ether contans all Y 2N nodes, motvatng R k, or t does not contan any Y 2N node (other than one of the two endponts, say or, f the graphcal structure s nduced on G ), motvatng Q k. Fnally, the followng varables provde logcal nformaton to enable us determne whether a partcular network confguraton s feasble: a k ndcates whether G k {} contans at least one Y 2N node. If G k {} contans at least one Y 2N node, then a k and s zero otherwse. m k ndcates whether any of the nodes n G k have a connectvty requrement. m k s f one of the nodes on G k has a connectvty requrement and s zero otherwse. Notce that, except for T k and a k, all graphcal structures and varables are symmetrcal n the sense that P k P k. Intally, before the frst contracton, the costs for the graphcal structures on each edge (, k) E are ntalzed as follows: u k s k t k c k f Y 2N and k Y 2N otherwse f k Y 1 Y 2N 0 otherwse f ether or k Y 1 Y 2N 0 otherwse p k c k We consder the soluton to be defned by the edges n the network. Thus, the soluton does not contan nodes wth degree 0. q k NETWORKS 2004

5 r k f Y 2N and k Y 2N 0 otherwse m k a k f k Y 2N 0 otherwse f or k Y 1 Y 2N 0 otherwse. As the algorthm proceeds, let node be the node of degree two beng elmnated and let nodes and k be the nodes adacent to t. Then, the state nformaton on G k G k G G k s updated as descrbed n the followng recursve equatons: The Recursve Equatons (Node-connectvty Case) (1) s k mn p p k p k, t t k mn s k a a k, p k a k a k, p k a k a, s k p q k a a k, p k s q k a k a k, p k p r k a a k, s k p k q a a k, p k p k r a k a k, p k s k q a a k, r k p p k a a k, q k s p k a k a k, q k p s k a a k t k mn t k t k mn p a a k, s a k a k, p a a k, m k t k t mn a, a k ũ k mn m m k u k, m k m k u, m m k u k, t t k m k mn a k, a p k mn p p k p k, p k t t k a a k, p p k q k, p k p q k a, p k p k q a k q k mn q k a a k mn t t k, p q k, p k q r k mn t t k mn r k a a k, q k a k a k, q k a k a, r k p q k a a k, q k s q k a k a k, q k p r k a a k, r k p k q a a k, q k p k r a k a k, q k s k q a a k ã k a a k a k m k m m k m k. Once we have reduced the graph to an edge [say (, k)], the cost of the soluton s mn{s k, t k, t k, u k }. We now establsh the correctness of the above equatons: Theorem 1. The recursve equatons (1) correctly compute the costs of the graphcal structures for the nodeconnectvty case. Proof. The proof requres enumeraton of all possble cases. We descrbe the cases for structure S k n detal. The other cases are dentfed n the Appendx. Once we have dentfed all possble cases, t s easy to obtan the recursve equatons. We wll consder the graphcal structure S k. In dong so, we wll also motvate the need for the structures P k, Q k, and R k. The followng result, whch follows mmedately from the well-known Mengers theorem [17], s useful n our dscusson: Lemma 2. Let {, } be a 2-separator, separatng two nodes s and t Y 2N. If there are two node-dsont paths between s and t, then one of the paths must nclude node and the other path node. We restrct our attenton to the graph G k G G k G k. Suppose that all the Y 2N nodes are not contaned entrely n one of the subgraphs G, G k, and G k. In that case, there ether exst two nodes s and t n Y 2N that have one of {, }, {, k}, {, k} as 2-separators separatng them or,, k are the only nodes n G k that belong to Y 2N.In both cases, usng Lemma 2, t follows that each of the graphs nduced by S k on G k, G k, and G s a connected graph that ncludes nodes and k, nodes and k, and nodes and, respectvely. From ths, t follows that the unon of P, P k, and P k provdes us wth a soluton that satsfes the requrements of S k at mnmum cost. Notce that, n ths case, s k p p k p k. Now suppose that all the Y 2N nodes are contaned entrely n one of the subgraphs G, G k, and G k. Then, ether each of the graphs nduced by S k on G k, G k, and G s a connected graph that ncludes nodes and k, nodes and k, and nodes and, respectvely, n whch case P P k P k provdes us wth a soluton that satsfes the requrements of S k at mnmum cost. Or there are 12 possble cases: (1) s not n S k. (a) All Y 2N nodes are contaned n G k. Then, the unon of S k, T, and T k gves S k. In addton, a and a k should be zero; otherwse, all Y 2N nodes are not contaned n G k. In ths case, s k t t k s k a a k. (b) All Y 2N nodes are contaned n G. Then, the unon of P k, T, and T k gves S k. Note that f G k contans no Y 2N node then P k gves the optmal Stener tree on the node set Y 1 {, k} on G k. In addton, a k and a k must be zero; otherwse, all Y 2N nodes are not contaned n G. In ths case, s k t t k p k a k a k. (c) All Y 2N nodes are contaned n G k. Then, the unon of P k, T, and T k gves S k. In addton, a k and a should be zero; otherwse, all Y 2N nodes NETWORKS

6 are not contaned n G k. In ths case, s k t t k p k a k a. (2) s n S k, and and k are not connected n the graph nduced on G k. # The graph nduced on G k s ether R k or Q k (defned earler) based on whether Y 2N nodes are contaned n G k. (a) All Y 2N nodes are contaned n G k. Then, the unon of S k, P, and Q k gves S k. To ensure feasblty, a and a k Thus, s k s k p q k a a k. (b) All Y 2N nodes are contaned n G. Then, the unon of P k, S, and Q k gves S k. To ensure feasblty, a k and a k Thus, s k p k s q k a k a k. (c) All Y 2N nodes are contaned n G k. Then, the unon of P k, P, and R k gves S k. To ensure feasblty, a and a k Thus, s k p k p r k a a k. (3) s n S k, and and are not connected n the graph nduced on G. These cases are symmetrcal to cases 2(a) 2(c). (4) s n S k, and and k are not connected n the graph nduced on G k. These cases are symmetrcal to cases 2(a) 2(c). S k s obtaned by computng the structure wth the lowest cost out of these 13 cases. Substtutng the equatons for these 13 cases gves the equaton for s k showng that s k s computed correctly. t k, ũ k, p k, q k, and r k are computed correctly as shown by the cases n the Appendx. ã k s computed correctly. ã k s f any node n G k s a Y 2N node. Thus, t s f any node n G k, G, org k s a Y 2N node. The equaton ã k a k a a k expresses ths condton. m k s computed correctly. m k s f any node n G k has a connectvty requrement. Thus, t s f any node n G, G k,org k has a connectvty requrement. The equaton m k m m k m k expresses ths condton. Theorem 3. The node-connectvty verson of the LCND problem on a seres-parallel graph can be solved n lnear tme. Proof. At each step, the algorthm performs a fxed number of operatons. The number of steps s lnear n the number of nodes. Thus, based on our precedng dscusson, t follows that the soluton to the node-connectvty verson of the LCND problem can be computed n lnear tme on a seres-parallel graph. In our dscusson, we restrcted ourselves to obtanng the cost of the soluton. It should be clear that by keepng track # In the rest of ths paper, especally n the Appendx, we wll nterchangeably use the term the graph nduced on G k s dsconnected to refer to the stuaton where nodes and k are not connected n the graph nduced on G k. of the assocated graphs for each structure the optmal network can also be obtaned n lnear tme. 4. EDGE-CONNECTIVITY REQUIREMENTS In the case of edge-connectvty requrements, let Y 1 be the set of nodes wth d s 1 and let Y 2E be the set of nodes wth d s 2E. The methodology s smlar to that for the node-connectvty case, except that the graphcal structures that we need to keep track of are more complcated. At the end of the contracton process, let the edge remanng, whch represents the graph, be (, k). Then, the mnmumcost soluton to the problem ether ncludes both nodes and k, or ncludes node but not node k, or ncludes node k but not node, or excludes both node and k. These possble structures are denoted as follows: S k mnmum-cost network on G k that satsfes the requrements of all nodes on G k and must nclude nodes and k. T k mnmum-cost network on G k that satsfes the connectvty requrements on G k and must nclude and exclude k. Ifk s requred, then t k. U k mnmum-cost network on G k that satsfes the connectvty requrements on G k and excludes nodes and k. If ether or k have connectvty requrements, then, by conventon, u k. One of the most mportant dfferences between the two problems les n the fact that the edge-connectvty verson of Lemma 2 s not true. In other words, suppose that {, } s a 2-separator separatng two nodes s, t Y 2E and there are two (or more) edge-dsont paths between nodes s and t. Then, t s possble that all the edge-dsont paths pass through node but not node or pass through node but not node. In the former case, node must have at least two edge-dsont paths to node s and at least two edge-dsont paths to node t. Smlarly, n the latter case, node must have at least two edge-dsont paths to node s and at least two edge-dsont paths to node t. Consequently, we may observe the followng: If all paths between two nodes s, t Y 2E n a feasble network satsfyng all the connectvty requrements pass through a node u, node u must have two edge-dsont paths to node s, two edge-dsont paths to node t, and, thus, two edge-dsont paths to all nodes n Y 2E. It s as f node u hasa2e connectvty requrement mposed on t. Ths stuaton drastcally ncreases the number of structures that we need to consder. Unlke the node-connectvty case, we need to consder structures where the endponts have 2E connectvty requrements mposed on them (ths corresponds to cases where two edge-dsont paths pass through the same node). Consequently, the followng S and T structures are also necessary: S k k mnmum-cost network on G k that satsfes the requrements of all nodes on G k and, furthermore, a 2E requrement s mposed on nodes and k. 168 NETWORKS 2004

7 S k mnmum-cost network on G k that satsfes the requrements of all nodes on G k and, furthermore, a 2E requrement s mposed on node and node k s requred to be connected. k S k mnmum-cost network on G k that satsfes the requrements of all nodes on G k and, furthermore, a 2E requrement s mposed on node k and node s requred to be connected. T k mnmum-cost network on G k that satsfes the connectvty requrements on G k, excludes node k, and mposes a 2E requrement on node. Ifk s requred to be connected, then t k. In addton, the followng structures are necessary for the computatons: P k mnmum-cost network on G k such that, for every Y 2E node n G k, there s a tral (.e., a path that does not repeat an edge) from to k through t and every Y 1 node s connected to and k. Q k k mnmum-cost network on G k comprsng of two dsont networks on G k that nclude all nodes wth nonzero connectvty requrements. One subnetwork ncludes node and the other subnetwork ncludes node k, both of whch have 2E requrements mposed on them. Further, the connectvty requrements wthn each dsont network are satsfed. Q k mnmum-cost network on G k comprsng of two dsont networks on G k that nclude all nodes wth nonzero connectvty requrements. One subnetwork ncludes node and the other subnetwork ncludes node k. Node hasa2e requrement mposed on t, and all Y 2E nodes n G k {, k} are n the subnetwork contanng node. Further, the connectvty requrements wthn each dsont network are satsfed. Note that f the subnetwork of Q k contanng node k has a node wth a 2E requrement other than node k then q k. Ths mples that the subnetwork of Q k contanng node k s a tree or ust node k. k Q k mnmum-cost network on G k comprsng of two dsont networks on G k that nclude all nodes wth nonzero connectvty requrements. One subnetwork ncludes node and the other subnetwork ncludes node k. Node k hasa2e requrement mposed on t, and all Y 2E nodes n G k {, k} are n the subnetwork contanng node k. Further, the connectvty requrements wthn each dsont subnetwork are satsfed. Q k mnmum-cost network on G k comprsng of two dsont trees that nclude all nodes wth nonzero connectvty requrements. One tree ncludes node and the other tree ncludes node k. IfG k {, k} has an Y 2E node, then, by conventon, q k. R k mnmum-cost network on G k comprsng of two dsont networks that nclude all nodes wth nonzero connectvty requrements. One subnetwork ncludes node and the other subnetwork ncludes node k. Further, all the Y 2E nodes belong to only one of the dsont subnetworks, and there must be two edge-dsont paths between every par of Y 2E nodes. If ths s not the case, for example, f and k Y 2E, then r k. Fnally, the followng varables provde nformaton on feasblty: b k ndcates whether G k {} contans at least one Y 2E node. If G k {} contans at least one Y 2E node, then b k and s zero otherwse. m k ndcates whether any of the nodes n G k have a connectvty requrement. m k s f any node on G k has a connectvty requrement and s zero otherwse. w k ndcates whether node k Y 2E. w k s f k Y 2E and s zero otherwse. Notce that except for T k, T k, and b k all graphcal structures and varables are symmetrcal n the sense that P k P k (S k S k, Q k Q k, and so on). We ntalze the costs as follows: u k s k s k s k k c k p k c k s k k c k c k f k Y 2E otherwse f Y 2E otherwse f Y 2E and Y 2E otherwse f ether or k Y 1 Y 2E 0 otherwse r k t k t k f k Y 1 Y 2E 0 otherwse f k Y 1 Y 2E 0 otherwse q k k 0 q k 0 q k k 0 q k 0 f Y 2E and k Y 2E 0 otherwse b k f k Y 2E 0 otherwse NETWORKS

8 m k f or k Y 1 Y 2E 0 otherwse. As the algorthm proceeds, the state nformaton on G k G k G G k s updated as descrbed n the followng recursve equatons: q k mn q k t k b k t, q k p k b k q, q k s q k, q k s q k q k k mn q k k t t k k, q k k s q k k, q k k s q k k, q k k s k k q, q k k s k k q The Recursve Equatons (Edge-connectvty Case) (2) s k mn p p k p k, t t k mn s k b b k, p k b k b k, p k b b k, s k t b k t k, s k k k t k b t, s k k t t k k, s k p q k b b k, p k s q k b k b k, p k p r k b b k, s k s q k, s k k p b q k k, s p k q k b k, s k s q k, s k k s q k k, s k k s q k k, s k p k q b b k, p k s k q b b k, p k p k r b k b k, s k k s k k q, s k p k b k q, s k p k b k q, s k k s k k q, s k k s k k q, s k k s k k q, p s k q k b b k, s p k q k b k b k, p p k r k b b k, s k k p b q k k, s p k b k q k, s s k q k, s s k q k, s s k k q k k, s s k k q k k s k k mn p k p p k, s k k t t k k, s k k s q k k k, s k s q k k, s k k s k k q, s k k s k k q, s s k k q k k s k mn p p k p k, s k b k t k t, s k k k t k t, s k s q k, s k s q k, s k k s q k k, s k k s q k k, s k p k b k q, s k k s k k q, s k k s k k q, s p k b k q k, s s k q k, s s k k q k k t k mn t k t m k mn b, b k, t k t m k, t k t k mn p b b k, s b k b k, p b b k,t k s t k b k, t k s t k b k, t k s t k t k mn t k t m k, t k s t k b k, t k s t k ũ k mn u k m m k, u m k m k, u k m m k, t t k m k mn b k, b, t p k mn p p k p k, p k t t k m k t k k, p k s q k k, p k s q k k, p k s k k q, p k s k k q, p p k q k k q k mn q k b b k mn t t k, p q k, p k q r k mn t t k mn r k b b k, q k b k b k, q k b b k, q k t t k b k, q k k t k k t b, r k q p k b b k, q k r p k b k b k, q k q s k b b k, q k b, q k q q s k p k b k b k, q k k k q s k b k, q k k q s k k w, r k p q k b b k, q k s q k b k b k, q k p r k b b k, q k k p q k k b b, q k s q k b k, q k s q k b k, q k b k b b k b k m k m k m m k. s q k w k As n the node-connectvty case, once we have reduced the graph to an edge, the cost of the soluton s mn{s k, t k, t k, u k }. We now prove the correctness of the above equatons: Theorem 4. The recursve equatons (2) correctly compute the costs of the graphcal structures for the edgeconnectvty case. Proof. The proof s a process of enumeratng all 108 possble cases. We descrbe the cases for one graphcal structure S k n detal. The others are dscussed n the Appendx. As before, we restrct our attenton to the graph G k G G k G k. There are 34 possble cases: (1) Nodes and are connected n the network nduced by S k on G, nodes and k are connected n the network nduced by S k on G k, and nodes and k are connected n the network nduced by S k on G k (.e., none of the networks nduced on G, G k, and G k by S k are dsconnected). Then, t follows that S k P k P P k. In ths case, s k p p k p k. (2) Node s not n S k. There are sx subcases: (a) All the Y 2E nodes are contaned wthn G k. Then, S k S k T T k. For feasblty, we need b and b k to be zero. Thus, s k t t k s k b b k. (b) All the Y 2E nodes are contaned wthn G. Then, S k P k T T k. For feasblty, we need b k and b k to be zero. Thus, s k t t k p k b k b k. (c) All the Y 2E nodes are contaned wthn G k. Then, S k P k T T k. For feasblty, we requre 170 NETWORKS 2004

9 that b and b k to be zero. Thus, s k t t k p k b b k. (d) All the Y 2E nodes are contaned wthn G G k, and there s at least one Y 2E node wthn G and at least one Y 2E node wthn G k, both dstnct from. Ths forces a 2E requrement on. Thus, S k S k T T k. In addton, for feasblty, we requre that b k be zero. Thus, s k s k t t k b k. (e) All the Y 2E nodes are contaned wthn G k G k, and there s at least one Y 2E node wthn G k and at least one Y 2E node wthn G k, both dstnct from k. Ths forces a 2E requrement on k. Thus, S k S k k T T k k. For feasblty, we requre that b be zero. Thus, s k s k k t t k k b. (f) G contans at least one Y 2E node and G k contans at least one Y 2E node. Ths forces a 2E requrement on and k. Thus, S k S k k T T k k, and s k s k k t t k k. (3) Node s n S k, and nodes and k are not connected n the graph nduced on G k. There are nne subcases: (a) All the Y 2E nodes are contaned wthn G k. Then, S k S k P Q k. In addton, for feasblty, we requre that b and b k be zero. Thus, s k s k p q k b b k. (b) All the Y 2E nodes are contaned wthn G. Then, S k P k S Q k. For feasblty, we requre that b k and b k be zero. Thus, s k p k s q k b k b k. (c) All the Y 2E nodes are contaned wthn G k. Then, S k P k P R k. For feasblty, we requre that b and b k be zero. Thus, s k p k p r k b b k. (d) All the Y 2E nodes are contaned wthn G G k, and there s at least one Y 2E node wthn G and at least one Y 2E node wthn G k, both dstnct from. Ths forces a 2E requrement on. Thus, S k S k S Q k. Note that no addtonal feasblty varables are requred here snce q k f any nodes n G k {, k} are n Y 2E. Thus, s k s k s q k. (e) All the Y 2E nodes are contaned wthn G k and the component contanng node k n the network nduced by S k on G k and each contans at least one Y 2E node dstnct from k. Ths forces a 2E requrement on k. Then, S k S k P Q k k. For k feasblty, we requre that b be zero. Thus, s k s k k p q k k b. (f) All the Y 2E nodes are contaned wthn G and the component contanng node n the network nduced by S k on G k, and each contans at least one Y 2E node dstnct from. Ths forces a 2E requrement on. Then, S k P k S Q k. For feasblty, we requre that b k be zero. Thus, s k p k s q k b k. (g) G k contans at least one Y 2E node, the component contanng node n the network nduced by S k on G k contans at least one Y 2E node, and the component contanng node k n the network nduced by S k on G k does not contan any Y 2E node. Ths forces a 2E requrement on and. Then, S k S k S Q k. Thus, s k s k s q k. (h) G contans at least one Y 2E node, the component contanng node k n the network nduced by S k on G k contans at least one Y 2E node, and the component contanng node n the network nduced by S k on G k does not contan any Y 2E node. Ths forces a 2E requrement on and k. Then, S k S k k S Q k k. Thus, s k s k k s q k k. () Both the component contanng node and the component contanng node k n the dsont network nduced by S k on G k contan at least one Y 2E node. Ths forces a 2E requrement on, k, and. Then, S k S k k S Q k k. Thus, s k s k k s q k k. (4) Node s n S k, and nodes and are not connected n the graph nduced on G. There are nne subcases symmetrcal to subcases 3(a) 3(). (5) Node s n S k and nodes and k are not connected n the graph nduced on G k. There are nne subcases symmetrcal to subcases 3(a) 3(). s k k, s k, t k, t k, ũ k, p k, q k k, q k, q k, and r k are computed correctly (see Appendx). b k s computed correctly. Proof smlar to ã k n Theorem 2. m k s computed correctly. Proof as n Theorem 2. Theorem 5. The edge-connectvty verson of the LCND problem on a seres-parallel graph can be solved n lnear tme. Proof. Smlar to Theorem DECOMPOSITION PROCEDURE FOR GENERAL GRAPHS We now turn our attenton to more general graphs and dscuss how some of the deas n ths paper may be appled to them. We frst consder the node-connectvty case. Usng Lemma 2, we observe that f we fnd a 2-separator {, } n a graph, whch separates two nodes n Y 2N, then we may decompose the orgnal problem nto two smaller problems as follows: Consder the two connected graphs G 1 (N 1, E 1 ) and G 2 (N 2, E 2 ), each contanng at least one node n Y 2N dstnct from and, wth E 1 E 2 E, E 1 E 2, N 1 N 2 N, and N 1 N 2 {, }. [These can easly be determned by deletng nodes and (the nodes n the 2-separator) from the graph.] Create G 1 G 1 (, ) and G 2 G 2 (, ). Gve nodes and a nodeconnectvty requrement of 2, and set c 0, n both G 1 and G 2. Solve the LCND problem on each of the smaller graphs, and denote the solutons as G* 1 and G* 2, respectvely. The soluton to the LCND problem on the orgnal graph s obtaned as G* 1 G* 2 (, ). Ths procedure can be generalzed as follows: Consder the connected graphs G t (N t, E t ), t 1,..., q, each contanng at least one node n Y 2N dstnct from the nodes and of the 2-separator, wth t 1,...,q E t E, t 1,...,q N t N, E r NETWORKS

10 E s, and N r N s {, } for any r, s {1,..., q} and r s. Agan, these may be determned by deletng nodes and n the 2-separator. For t 1,..., q, create G t G t (, ), gve nodes and a node-connectvty requrement of 2, and set c 0. Solve the LCND problem on each G t and denote the soluton as G* t. The soluton to the LCND problem on the orgnal graph s obtaned as t 1,...,q G* t (, ). By repeatng ths decomposton procedure, ether two problems at a tme or multple subproblems at a tme, t s possble to decompose the orgnal LCND problem nto a seres of smaller LCND problems. Grötschel et al. [12] took ths approach (consderng multple subproblems at a tme) to solve to optmalty several problems that arse n telecommuncatons practce. Note that 2-separators n a graph may be found n lnear tme usng an algorthm based on depth frst search (see [14]). We now turn our attenton to the edge-connectvty case. Decompostons for ths case are somewhat more nvolved snce we do not have a counterpart to Lemma 2. Consequently, the decomposton procedures are more complex, requrng the soluton of multple smaller problems, and may not be as easy to apply (except for the smplest cases) n general graphs. For completeness, we dscuss the decomposton: As n the node-connectvty case, let {, } be a 2-separator separatng two nodes n Y 2E, and G 1 (N 1, E 1 ) and G 2 (N 2, E 2 ), the two connected graphs, each contanng at least one node n Y 2E dstnct from and, wth E 1 E 2 E, E 1 E 2, N 1 N 2 N, and N 1 N 2 {, }. There are nne dfferent cases to consder. We use the notaton developed n Secton 4, wth superscrpts 1 and 2, to denote the structure belongs to graphs G 1 or G 2 respectvely: (1) The networks nduced on G 1 and G 2 by the soluton are connected. Then, the soluton s P 1 P 2. (2) Nodes and are not connected n the network nduced on G 2, and the network nduced on G 1 s connected. There are three subcases: (a) Both the component contanng node and the component contanng node n the network nduced on G 2 contan Y 2E nodes. Then, the soluton s S 1 Q 2. (b) The component contanng node n the network nduced on G 2 does not contan Y 2E nodes. Then, the soluton s S 1 Q 2. (c) The component contanng node n the network nduced on G 2 does not contan Y 2E nodes. Then, the soluton s S 1 Q 2. (3) Nodes and are not connected n the network nduced on G 1, and the network nduced on G 2 s connected. There are three subcases: (a) Both the component contanng node and the Smlar decompostons to the node-connectvty case may be appled by fndng two edges whose removal dsconnects two nodes n Y 2E. component contanng node n the network nduced on G 1 contan Y 2E nodes. Then, the soluton s S 2 Q 1. (b) The component contanng node n the network nduced on G 1 does not contan Y 2E nodes. Then, the soluton s S 2 Q 1. (c) The component contanng node n the network nduced on G 1 does not contan Y 2E nodes. Then, the soluton s S 2 Q 1. (4) Node s not n the soluton. Then, the soluton s T 1 T 2. (5) Node s not n the soluton. Then, the soluton s T 1 T 2. We consdered the decomposton procedure when 2-separators separate two nodes n Y 2E. In a smlar fashon, t s easy to derve the decomposton procedure when a 2-separator separates a node n Y 1 from a node n Y 2E, but does not separate two nodes n Y 2E. In [12], Grötschel et al. descrbed one such decomposton procedure that they clam can be appled to the edge-connectvty case of the LCND problem. We now show that ths decomposton procedure s ncorrect. In ths decomposton, a 2-separator {, } separates a node n Y 1 from a node n Y 2E, but does not separate two nodes n Y 2E. Let G 1 (N 1, E 1 ) and G 2 (N 2, E 2 ) be the two graphs wth E 1 E 2 E, E 1 E 2, N 1 N 2 N, and N 1 N 2 {, }, wth all Y 2E nodes n N 2 and wth N 1 {, } contanng at least one node n Y 1 and no Y 2E nodes. Grötschel et al. clamed that the network nduced on G 1 by the soluton to the LCND problem can only take one of the four followng forms: (1) Two dsont trees. One tree ncludes and the other tree ncludes. (2) A tree that does not nclude node. (3) A tree that does not nclude node. (4) A tree that ncludes both nodes and. They neglect the case where nodes and are not connected n the graph nduced on G 2, forcng both edgedsont paths between nodes n Y 2E to go through G 1. Thus, the decomposton procedure dscussed n Grötschel et al. s ncorrect. CONCLUDING REMARKS In ths paper, we have descrbed lnear-tme algorthms for the low-connectvty network desgn problem on seresparallel graphs. Our research generalzes earler work [25, 26] by consderng 0, 1, and 2 connectvty requrements together (.e., d s {0, 1, 2}). Further, we show how these deas may be appled to obtan decomposton procedures that can be appled to general graphs. For many problems, the lnear-tme algorthms on partal 2-trees generalze to partal k-trees [1, 2, 4, 6]. In partcular, Arnborg et al. [1] showed that all propertes defnable n monadc second-order logc (MSOL) wth quantfcaton over vertex and edge sets can be decded n lnear tme for 172 NETWORKS 2004

11 partal k-trees gven the tree decomposton. They also consdered problems nvolvng countng or summng evaluatons over sets defnable n MSOL and showed that a large class of these problems can be solved n lnear tme for partal k-trees. Bore et al. [6] descrbed a predcate calculus n whch many graph problems can be expressed. They showed that any problem that can be expressed usng ths predcate calculus can be solved n lnear tme on any recursvely constructed graph, once the decomposton tree needed to construct the recursvely constructed graph s known. We are not aware of a way of expressng the LCND problem n ether of these two formats. Thus, n the case of the LCND problem, extendng these results even to 3-trees seems partcularly challengng. One of the prme dffcultes n generalzng the approach (descrbed n ths paper) to 3-trees s the number of states, that s, graphcal structures, and cases grow qute rapdly and substantally. Further, t s concevable that the LCND problem tself s NP-hard on a k-tree (for some k 3). An nterestng and recent result by Nshzek et al. [19] showed that the edge-dsont paths problem s NP-complete on 2-trees (whle t s polynomally solvable on 1-trees and outerplanar graphs). Another nterestng result n an earler paper by Rchey and Parker [21] showed that a partcular generalzaton of the Stener tree problem that they call the multple Stener subgraph problem (ths s actually a specal case of the weghted Stener tree packng problem ntroduced n [11]) s polynomally solvable on trees, but s NP-complete on seres-parallel graphs. APPENDIX Hand-drawn fgures for all cases are avalable from the author. Lstng of Cases for Theorem 1 Cases for T k. There are fve cases for T k. (1) s ncluded n T k. (a) All Y 2N nodes are n G k. Then, T k T k P T k. In addton, for feasblty, we requre that a and a k be zero. (b) All Y 2N nodes are n G. Then, T k T k S T k. For feasblty, a k and a k (c) All Y 2N nodes are n G k. Then, T k T k P T k. For feasblty, a and a k (2) s not n T k. (a) All Y 2N nodes are n G k. Then, T k T k T. For feasblty, m k and a (b) All Y 2N nodes are n G. Then, T k T k T. For feasblty, m k and a k (b) All Y 2N nodes are n G. Then, Ũ k U. For feasblty, m k and m k (c) All Y 2N nodes are n G k. Then, Ũ k U k. For feasblty, m and m k (2) s ncluded n Ũ k. (a) All Y 2N nodes are n G. Then, Ũ k T T k. For feasblty, m k and a k (b) All Y 2N nodes are n G k. Then, Ũ k T T k. For feasblty, m k and a Cases for P k. There are fve cases for P k. (1) None of the graphs nduced on G, G k, and G k s dsconnected. Then, P k P k P P k. (2) s not ncluded n P k. Then, P k P k T T k. For feasblty, a and a k (3) s ncluded and the graph nduced on G k by P k s dsconnected. Then, P k P P k Q k. (4) s ncluded and the graph nduced on G k by P k s dsconnected. Then, P k P k P Q k. For feasblty, a (5) s ncluded and the graph nduced on G by P k s dsconnected. Then, P k P k P k Q. For feasblty, a k Cases for Q k. There are three cases for Q k. (1) s not ncluded n Q k. Then, Q k Q k T T k. For feasblty, a and a k (2) s ncluded n Q k and the graph nduced on G k s dsconnected. Then, Q k Q k P Q k. For feasblty, a and a k (3) s ncluded n Q k and the graph nduced on G s dsconnected. Then, Q k Q k Q P k. For feasblty, a and a k Cases for R k. There are nne cases for R k. (1) s not n R k. (a) All Y 2N nodes are n G k. Then, R k R k T T k. For feasblty, a and a k (b) All Y 2N nodes are n G. Then, R k Q k T T k. For feasblty, a k and a k (c) All Y 2N nodes are n G k. Then, R k Q k T T k. For feasblty, a and a k (2) s n R k and the graph nduced on G k s dsconnected. (a) All Y 2N nodes are n G k. Then, R k R k P Q k. For feasblty, a and a k (b) All Y 2N nodes are n G. Then, R k Q k S Q k. For feasblty, a k and a k (c) All Y 2N nodes are n G k. Then, R k Q k P R k. For feasblty, a and a k (3) s n R k and the graph nduced on G s dsconnected. These cases are symmetrcal to subcases 2(a) 2(c) for R k. Lstng of Cases for Theorem 4 Cases for Ũ k. There are fve cases for Ũ k. Cases for S k k. There are seven cases for S k k. (1) s not n Ũ k. (a) All Y 2N nodes are n G k. Then, Ũ k U k. For feasblty, m and m k k (1) None of the graphs nduced on G, G k, and G k by S k are dsconnected. Then, S k k P k P P k. (2) Node s not n S k k. Then, S k k S k k T T k k. NETWORKS

12 (3) s n S k k and the graph nduced on G k s dsconnected. (a) The component contanng node n the network nduced by S k k on G k does not contan Y 2E nodes. Then, S k k S k k S Q k k. (b) The component contanng node n the network nduced by S k k on G k contans Y 2E nodes. Then, S k k S k k S Q k k. (4) s n S k k and the graph nduced on G s dsconnected. These cases are symmetrcal to subcases 3(a) and 3(b) for S k k. (5) s n S k k and the graph nduced on G k s dsconnected. The only way for and k to be two-edge connected s through node. Then, S k k S S k k Q k k. Cases for S k. There are 13 cases for S k. (1) None of the graphs nduced on G, G k, and G k by S k are dsconnected. Then, S k P k P P k. (2) Node s not n S k. (a) G k does not contan Y 2E nodes. Then, S k S k T T k. In addton, for feasblty, we requre that b k be zero. (b) G k contans Y 2E nodes. Then, S k S k k T T k k. (3) s n S k and the graph nduced on G k s dsconnected. (a) G k does not contan Y 2E nodes. Then, S k S k S Q k. (b) The component contanng node, but not the component contanng node k, n the network nduced by S k on G k contans Y 2E nodes. Then, S k S k S Q k. (c) The component contanng node k, but not the component contanng node, n the network nduced by S k on G k contans Y 2E nodes. Then, S k S k k S Q k k. (d) Both the component contanng node and the component contanng node k n the network nduced by S k on G k contan Y 2E nodes. Then, S k S k k S Q k k. (4) s n S k and the graph nduced on G s dsconnected. (a) G k and the component contanng node n the network nduced by S k on G do not contan Y 2E nodes. Then, S k S k P k Q. For feasblty, we requre that b k be zero. (b) G k contans Y 2E nodes, and the component contanng node n the network nduced by S k on G does not contan Y 2E nodes. Then, S k S k k k S k Q. (c) The component contanng node n the network nduced by S k on G contans Y 2E nodes. Then, S k S k k S k k Q. (5) s n S k and the graph nduced on G k s dsconnected. These cases are symmetrcal to subcases 4(a) 4(c) for S k. Cases for T k. There are nne cases for T k. (1) s not ncluded n T k. (a) All the Y 2E nodes are contaned wthn G k. Then, T k T k T. For feasblty, m k and b must be zero. (b) All the Y 2E nodes are contaned wthn G. Then, T k T k T. For feasblty, m k and b k must be zero. (c) All the Y 2E nodes are contaned wthn G G k, and there s at least one Y 2E node wthn G and at least one Y 2E node wthn G k, both dstnct from. Then, T k T k T. For feasblty, m k must be zero. (2) s ncluded n T k. (a) All the Y 2E nodes are contaned wthn G k. Then, T k T k P T k. For feasblty, b and b k (b) All the Y 2E nodes are contaned wthn G. Then, T k T k S T k. For feasblty, b k and b k (c) All the Y 2E nodes are contaned wthn G k. Then, T k T k P T k. For feasblty, b and b k (d) All the Y 2E nodes are contaned wthn G G k, and there s at least one Y 2E node wthn G and at least one Y 2E node wthn G k, both dstnct from. Then, T k T k S T k. For feasblty, b k (e) All the Y 2E nodes are contaned wthn G G k, and there s at least one Y 2E node wthn G and at least one Y 2E node wthn G k, both dstnct from. Then, T k T k S T k. For feasblty, b k (f) Both G k and G k contan Y 2E nodes. Then, T k T k S T k. Cases for T k. There are three cases for T k. (1) s not ncluded n T k. Then, T k T k feasblty, m k (2) s ncluded n T k. T. For (a) G k does not contan Y 2E nodes. Then, T k T k S T k. For feasblty, b k (b) G k contans Y 2E nodes. Then, T k T k S T k. Cases for Ũ k. There are sx cases for Ũ k. (1) s not n Ũ k. (a) All Y 2E nodes are contaned wthn G k. Then, Ũ k U k. For feasblty, m and m k (b) All Y 2E nodes are contaned wthn G. Then, Ũ k U. For feasblty, m k and m k (c) All Y 2E nodes are contaned wthn G k. Then, Ũ k U k. For feasblty, m and m k (2) s n Ũ k. (a) All Y 2E nodes are contaned wthn G. Then, Ũ k T T k. For feasblty, m k and b k must be zero. (b) All Y 2E nodes are contaned wthn G k. Then, Ũ k T T k. For feasblty, m k and b must be zero. (c) All Y 2E nodes are contaned wthn G G k, and there s at least one Y 2E node wthn G and at least one Y 2E node wthn G k, both dstnct from. Then, Ũ k T T k. For feasblty, m k must be zero. 174 NETWORKS 2004