Simpler Testing for Two-page Book Embedding of Partitioned Graphs

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1 Simpler Testing for Two-page Book Embedding of Partitioned Graphs Seok-Hee Hong 1 Hiroshi Nagamochi 2 1 School of Information Technologies, Uniersity of Sydney, seokhee.hong@sydney.ed.a 2 Department of Applied Mathematics and Physics, Kyoto Uniersity, nag@amp.i.kyoto-.ac.jp Abstract: A 2-page book embedding of a graph is to place the ertices linearly on a spine (a line segment) and the edges on the two pages (two half planes sharing the spine) so that each edge is embedded in one of the pages withot edge crossings. Testing whether a gien graph admits a 2-page book embedding is known to be NP-complete. In this paper, we stdy the problem of testing whether a gien graph admits a 2-page book embedding with a fixed edge partition. We first show that finding a 2-page book embedding of a gien graph can be redced to the planarity testing of a graph, which yields a simple linear-time algorithm for soling the problem. We also characterize the graphs that do not admit 2-page book embeddings ia forbidden sbgraphs, and gie a linear-time algorithm for detecting the forbidden sbgraph of a gien graph. 1 Introdction For an integer k 1, a k-page book embedding (or a k-stack layot) of a graph is to place the ertices linearly on a spine (a line segment) and the edges on k pages (k half planes sharing the spine) so that each edge is embedded in one of the pages withot generating edge-crossings. The book embedding problem has applications in the roting of mltilayer printed circit boards and in the design of falt-tolerant processor arrays [3, 19]. See [11] for nmeros applications of book embeddings. Graphs with 1-page book embeddings are the oterplanar graphs. Yannakakis proed that eery planar graph admits a 4-page book embedding [21]. Bernhart and Kainen [2] show that a planar graph has a 2-page book embedding if and only if it is sb-hamiltonian. A planar graph is sb-hamiltonian if and only if it is Hamiltonian or can be made Hamiltonian by inserting additional edges withot iolating planarity. The problem of testing sb-hamiltonicity is NP-complete [22]. Hence the problem of determining whether a gien planar graph G = (V, E) is a 2-page book embedding or not is NP-complete. See [11, 12] for a srey on book embeddings and graph linear layots. 1 Technical report , Febrary 12,

2 The 2-page book embedding problem contains two combinatorial aspects. One is how to partition an edge set in two edge sbsets, each corresponds to one of the two sides along the spine. The other is how to decide an ordering of the ertices on the spine. Note that if an ordering π of all the ertices along the spine is fixed, then we can test whether a gien graph admits a 2-page book embedding with π or not in linear time; the problem can be conerted into a planarity testing problem by adding edges between eery two consectie ertices in π (where the last ertex is connected by the first one). Howeer, it is not known whether the problem remains NP-complete or can be soled in polynomial time if a partition of the edge set is prescribed. In this paper, we consider the problem of testing whether a gien graph admits a 2-page book embedding for a fixed edge partition. Based on strctral properties of biconnected planar graphs, we show that the problem of finding a 2-page book embedding of a graph with a partitioned edge set can be soled in linear time. A preliminary ersion of this paper appeared as a report [16], which has been recently sed to sole seeral other graph drawing problems sch as simltaneos drawing of two graphs. In [16], we characterize 2-page book embeddings as splitter-free and disjnctie plane embeddings (see Section 6 for the definitions), and show that sch an embedding (if any) can be constrcted in linear time by designing three procedres, one for detecting rigid splitters, a special type of splitters, one for compting splitter-free plane embeddings and the other for compting disjnctie plane embeddings. Recently Angelini et al. [1] implemented the algorithm in in [16] after they simplified the procedre for compting disjnctie plane embeddings redcing the hidden constant factor in the time bond. In this paper, we first show that a gien instance can be conerted into another instance with a special strctre, called a canonical instance (see Section 6 for the definition). We then proe that finding a 2-page book embedding of a gien graph can be redced to the planarity testing of a modified graph, which yields a simple linear-time algorithm for soling the problem withot sing any of the three procedres in [16]. We also characterize the graphs that do not admit 2-page book embeddings ia forbidden sbgraphs, and gie a linear-time algorithm for detecting the forbidden sbgraph of a gien graph. This algorithm is obtained by significantly simplifying the first procedre for detecting rigid splitters tilizing the restricted strctre of canonical instances. The paper is organized as follows. In Section 2, we reiew basic terminologies on graph connectiity and oterplanar graphs In Section 3, we show how to conert a gien instance into a canonical instance, an instance with a special strctre. In Section 4, we characterize a 2-page book embedding as a plane embedding which satisfies two conditions: it has no splitter and it is disjnctie. In Section 5, we show that the 2-page book embedding of a canonical instance can be redced to the planarity testing problem on a modified graph. In Section 6, after we reiew SPQR tree representation of triconnected components, we design a linear-time algorithm for detecting a forbidden sbgraph of an infeasible instance. 2 Preliminaries Let G = (V, E) be a graph. The set of edges incident to a ertex V is denoted by E(; G). A path with end ertices and is called a, -path. The degree of a ertex in G is denoted by deg(; G). For a sbset X E (resp., X V ), G X denotes 2

3 the graph obtained from G by remoing the edges in X (resp., the ertices in X together with the edges in X E(; G)). Sbdiiding an edge e = (, ) is to replace the edge with a, -path, w 1, w 2,..., k, for some k 1. A graph H is a sbdiision of G if H is obtained by sbdiiding some edges in G. A block (biconnected component) of a graph is a maximal biconnected sbgraph (which possibly consists of a single ertex or a single edge). A graph each of whose blocks is a simple cycle or a single edge is called a cacts, i.e., a graphs in which any two distinct cycles share at most one ertex. We see that a graph is a cacts if and only if no two ertices are joined by three ertex-disjoint paths. A planar graph G = (V, E) with a fixed embedding F of G is called a plane graph. The set of ertices, set of edges and set of facial cycles of a plane graph H may be denoted by V (H), E(H) and F (H), respectiely. We say that a cycle Q in a plane embedding of a graph separates a ertex/edge a 1 and a ertex/edge a 2 (which are not elements in Q) if a 1 and a 2 are respectiely contained in the two regions R 1 and R 2 obtained by diiding the plane by Q. A planar graph is called oterplanar if it admits a plane embedding sch that all the ertices appear along the oter bondary, which we call an oterplane embedding. We see that each block B of a simple oterplanar graph is a single edge or a cycle Q of length at least 3 possibly with some chords (see Fig. 1(a) for an example of oterplanar graphs). Note that sch a cycle Q for the block B is niqely determined, and we call Q the frame of B (we let Q = B if B is a single edge). We obsere the next. Lemma 1 Let B i, i = 1,..., p be the blocks of a simple oterplanar graph G, and Q i be the bondary of B i. Then a plane embedding Γ G of G is an oterplane embedding of G if and only if each frame Q i appears on the oter bondary of Γ G. B 4 C 4 B 2 B 3 B 1 C 2 C 3 C 1 (a) oterplanar graph (b) the cats of frames Figre 1: (a) An oterplanar graph; (b) The cacts obtained from the oterplanar graph in (a) by deleting the chords, where C i denote the frame of each block B i of the graph in (a). Throghot the paper, we denote an instance of two-page book embedding problem by a graph G = (V, E 1 E 2 ) with a partition of E into E 1 and E 2 (i.e., E 1 E 2 = E and E 1 E 2 = ), where two ertices may be joined by two edges e E 1 and e E 2. We call the edges in E 1 (resp., E 2 ) red edges (resp., ble edges). A sbgraph H in G is called red (resp., ble) if E(H) consists of only red (resp., ble) edges. A ertex to which only red (resp., ble) edges are incident is called an r-ertex (resp., b-ertex). A ertex to which both red and ble edges are incident is called a br-ertex. A 2-page book embedding 3

4 (2PB-embedding, for short) π of a graph G = (V, E 1 E 2 ) is a linear ordering of the ertices sch that all ertices are placed in this order on a spine and all red edges are drawn aboe the spine and all ble edges are drawn below the spine withot any edge-crossings. See Fig. 2(a) and (b) for examples of 2-page book embeddings of the same graph G. In a 2PB-embedding π of a graph G = (V, E 1 E 2 ), if we join the first and last ertices on the spine with a new cre so that the spine together with the cre forms a simple closed cre which encloses all red edges bt no ble edges (see Fig. 2(a)). Ths, a 2PBembedding π can be regarded as a plane embedding Γ of G in which a simple closed cre λ isits each ertex exactly once withot intersecting any edge and encloses all red edges bt no ble edges. We call sch a cre λ a separating cre of the embedding Γ. Note that the first and last ertices appear on the oter facial cycle in the plane embedding Γ. Howeer by choosing a new oter face, any ertex can appear along the oter facial cycle. This does not change the combinatorial embedding, and thereby the ertex can appear as the first ertex on the spine in the 2PB-embedding obtained from the reslting plane embedding Γ. See Fig. 2(b), where the ertices 3 and 2 are chosen as the first and last ertices on the spine. E E (a) π1 (b) π2 Figre 2: (a) A 2PB-embedding π 1 of a planar graph G = (V, E 1 E 2 ); (b) A 2PBembedding π 2 of G obtained from π 1 by choosing the ertices 3 and 2 as the first and last ertices on the spine. 3 Canonical Instances In this sbsection, we gie a linear-time algorithm for conerting a gien instance into another instance with a special strctre, called a canonical instance, withot changing the 2PB-embdedability. Clearly an instance G = (V, E 1 E 2 ) admits a 2PB-embedding only when G is planar, and each E i indces an oterplanar graph. If E 1 = or E 2 =, then G has a 2PBembedding if and only if G is oterplanar. In what follows, we assme that E 1 = E 2. Also we assme that G is not a simple cycle (otherwise it always admits a 2PB-embedding). An instance G = (V, E 1 E 2 ) is canonical if (i) G is a simple, biconnected and planar graph, bt G is not a simple cycle; (ii) Each E i indces a cacts (V, E i ); and (iii) Each br-ertex of G is of degree 2. We first show how G can be assmed to be biconnected. 4

5 Lemma 2 A planar graph G = (V, E 1 E 2 ) admits a 2PB-embedding π with partition E 1 and E 2 if and only if each block H of G has a 2PB-embedding π H for the partition E(H) E 1 and E(H) E 2 of E(H). Proof. Since the only if part is triial, we consider the if part. Let H 1, H 2,..., H p be the blocks of G indexed so that each H i, i = 2, 3,..., p has a common ertex Hi with some block H j with j < i, where sch Hi is niqe to H i and is called the parent of H i. Assme that each block H i of G has a 2PB-embedding π Hi with the partition E(H i ) E 1 and E(H i ) E 2. As we hae obsered, we can regard each embedding π Hi as a plane embedding of H i, and by choosing the oter face so that the parent Hi appears on the bondary, we can obtain a 2PB-embedding π H i of H i in which Hi appears as the first ertex on the spine. Let π := π H 1. For each i = 2, 3,..., p, we place the 2PB-embedding π H i in the space between ertex Hi and the ertex next to Hi in the crrent 2PB-embedding π, letting π H i share the ertex Hi with π. In this way, we can constrct a desired 2PB-embedding π of G. Now we can assme that an instance G = (V, E 1 E 2 ) is biconnected. We next show how each indced graph (V, E i ) from G is assmed to be a cacts. Let Γ i be an oterplane embedding of (V, E i ) (where all ertices in V appear along the oter bondary). In Γ, each block of (V, E i ) is either a single edge or a cycle Q with some chords, where each chord is drawn within the cycle. By Lemma 1, in any sch embedding of (V, E i ), chords are drawn inside the corresponding cycle and no other edges/ertices are inclded within the cycle. On the other hand, in any 2PB-embedding of G, sch a cycle (frame) C with E(Q) E i is drawn withot srronding any edge in E j (j i) in its interior. This means that we can remoe all the chords in the embedding Γ i (i = 1, 2) in the sense that we can pt back them into any 2PB-embedding of the reslting instance withot creating any edge crossings. Hence we can assme that each (V, E i ) is a simple cacts. We transform a biconnected graph G = (V, E = E 1 E 2 ) into a canonical instance of 2PB-embedding problem as follows. Definition 1. Step 1. For each oterplanar graph (V, E i ), i = 1, 2, remoe the chords of each block B to obtain a cacts (V, E i ) in the reslting instance G = (V, E 1 E 2 ) (see Fig. 3(a) and (b)). Step 2. Let V br be the set of br-ertices V of degree 3 in G. (i) Replace each V br of degree 3 with three ertices 1, w and 2 joined by a new red edge ( 1, w ) and a new ble edge ( 2, w ); and (ii) For each i = 1, 2, change the end ertices of each edge (, ) E i to ( i, i ) (see Fig. 3(b) and (c)). Let G = (V, E ) be the reslting graph. 1 E 2 Lemma 3 Let G be the canonical instance obtained from a biconnected instance G = (V, E 1 E 2 ) by Definition 1. Then G admits a 2PB-embedding if and only if G admits a 2PB-embedding. Frthermore a 2PB-embedding of G can be conerted into a 2PBembedding of G in linear time. 5

6 C 1 2 w (a) G (b) G (c) G Figre 3: (a) a biconnected instance G = (V, E 1 E 2 ); into a canonical instance G in (d) by ; (b) a biconnected instance G = (V, E 1 E 2 ) with cacts (V, E i ), i = 1, 2; and (c) a canonical instance G = (V, E ) with cacts (V, E ), i = 1, 2; 1 E 2 i Proof. The correctness of Step 1 of Definition 1 has been discssed. We show the correctness of Step 2. Only if part: Assme that G admits a 2PB-embedding Γ. In Γ, we replace each ertex with three ertices 1, w and 2 in this order on the spine, adding a red edge ( 1, w ) on the red edge page and a ble edge ( 2, w ) on the ble edge page, and changing the end ertices of each edge (, ) E i to ( i, i ). By taking a sfficiently small space for the replacement for each ertex, we can modify Γ into a 2PB-embedding of G. If part: Assme that G admits a 2PB-embedding Γ. See Fig. 4(a). We modify Γ into another a 2PB-embedding of G so that the three ertices 1, w and 2 for each V appear consectiely (possibly in a different order) in Γ. Then the reslting embedding can be conerted into a 2PB-embedding of G by contacting the three ertices 1, w and 2 for each V into a single ertex. Sppose that there is some other ertex x ( 2 ) between 1 and w on the spine. Then we change the position of 1 to a new position between the crrent positions of 1 and w sch that no other ertex x ( 2 ) appears between 1 and w. Since 1 and w are joined by a red edge before the change, the red edge page has a space for the rest of red edges incident to 1 to be drawn withot creating any crossing with other edges. See Fig. 4(b). We can change the position of ertex 2 analogosly. By applying the method for all V, we can obtain a desired embedding for G. Note that the modification can be exected jst by tracing the red edge ( 1, w ) (resp., ( 2, w )) and the other red (resp., ble) edges incident to 1 for each V, and it can be implemented to rn in linear time. A canonical instance G is simple, since G is not a cycle and it has no pair of ble and red mltiple edges. Also, if there are a red, -path and a ble, -path for some ertices, V, then and are br-ertices of degree 2. In the following two sections, a gien instance is assmed to be canonical. 6

7 1 (a) w (b) 1 w Figre 4: (a) A 2PB-embedding Γ of G ; (b) A modified 2PB-embedding so that 1 is placed next to w. 4 Disjnctie and Splitter-free Plane Embeddings In a plane embedding Γ of G, a red (resp., ble) cycle Q of G is called a splitter if each of the two regions obtained by the cycle Q contains a ertex V V (Q) or a ble (resp., red) edge. In Γ, a ertex is called disjnctie if for each i = 1, 2, all the edges in E i (; G) appear consectiely arond. We call Γ disjnctie if all ertices in V are disjnctie. Theorem 4 Let G = (V, E 1 E 2 ) be a biconnected planar graph with E 1 = E 2 (not necessarily canonical). Then G admits a 2PB-embedding π if and only if G admits a disjnctie and splitter-free plane embedding Γ. Moreoer, a 2PB-embedding π of G can be obtained from a disjnctie and splitter-free plane embedding Γ of G in linear time. Proof. As obsered aboe, a 2PB-embedding π of G can be regarded as a plane embedding which admits a separating cre λ, where we treat λ as an oriented cre so that the red edges appear on or left hand side when we traerse λ along its orientation. Conersely, if a plane embedding admits sch a separating cre λ, then we can obtain a 2PB-embedding π of G. Only if part: Assme that G has a 2PB-embedding π. Let λ be a separating cre of π. The cre λ isits each ertex exactly once and it encloses only red edges. Hence the plane embedding π is disjnctie at any ertex. If the plane embedding π has a red splitter, then λ cannot isit a ertex (or a ble edge) enclosed by the splitter and a ertex (or a ble edge) otside the splitter, becase λ cannot intersect the splitter de to the disjnctie condition on the ertices in the splitter. Hence π has no red splitter. Similarly, π has no ble splitter. If part: Assme that G has a disjnctie and splitter-free plane embedding Γ. It sffices to show that Γ admits a separating cre λ. For this, we constrct an Elerian plane digraph G and its Elerian trail λ by the following procedre. Since G is biconnected, the facial cycle of each face in Γ is a simple cycle. Step 1. Let F br be the set of faces of Γ whose facial cycles contain both red and ble edges. Place a new ertex f inside each face f in F br. Step 2. Let V br be the set of br-ertices of G. For each br-ertex V br, there exists a niqe pair of faces f and f sharing the ertex sch that all the red (resp., ble) edges incident to appear in the clockwise order after isiting face f (resp., f ) arond, becase Γ is disjnctie. We join sch two ertices f and f ia a new directed edge ( f, ) and (, f ) (see Fig. 5(b)). 7

8 Step 3. We call all the new edges introdced in Steps 2 green edges. Let G = (V br V, E g ) be the Elerian plane bipartite digraph that consists of all the green edges and their end ertices, where V denotes the set of the end ertices f of all green edges; G is a plane, bipartite and Elerian digraph which has no isolated ertices. Step 4. Conert G into a simple directed cycle λ consisting of all the green edges withot self-intersection in the plane, by splitting each ertex f V into deg( f ; G )/2 ertices of indegree 1 and otdegree 1. Sch a cycle λ br isits each ertex V br exactly once and encloses all the red edges bt no ble edges (see Fig. 5(c)). Step 5. Each b-ertex V V br is on the facial cycle of a face f F br, becase otherwise the nion of the faces that contain wold inclde a ble splitter Q that separates from a ertex or a red edge otside the region Q. Symmetrically, each r-ertex V V br is on the facial cycle of a face f F br. We let the cycle λ br isit each b-, or r-ertex V V br while λ br passes throgh the face f F br. The reslting cycle λ br will be a separating cycle of Γ (see Fig. 5(d)). To complete the proof, it sffices to show that Step 4 can be exected. For this, we first show that the plane Elerian digraph G is connected. By the constrction of the plane digraph G, for each ertex f V, the incoming and otgoing green edges incident to f appear alternately arond f. Hence we can traerse the bondary of G following the edge directions. Ths the bondary of each component of G is a directed cycle. To derie a contradiction, assme that G has two components H and H (as shown in Fig. 5(a)), and let B and B denote the bondary of H and H, respectiely, where H is not contained in the interior of B withot loss of generality. Let the direction along B be the clockwise order (the other case can be treated symmetrically). By the constrction of G, only red edges are adjacent to B otside B (as shown in Fig. 5(a)), and each ertex f in B has no ble edges on the corresponding facial cycle f F br. This implies that the nion of these facial cycles contain a red cycle Q of G that separates a ble edge in E(H) and a ertex V (H ). Hence Q is a splitter of Γ, contradicting that G is splitter-free. Therefore G is connected (see Fig. 5(b)). Since G is a connected Elerian digraph, it has an Elerian trail (a simple directed cycle consisting of all the green edges). We show that there is an Elerian trail which has no self-intersection in the plane. Based on this obseration, we partition the set E g of green edges into E g 1, Eg 2,..., Eg p as follows. Let E g 1 be the set of the edges in the bondary, and G 1 = G E g 1. Analogosly we can traerse the bondary of each component of G 1 following the edge directions. We repeatedly define E g i as the set of edges in the bondaries of the components of G (E g 1 E g i 1 ) ntil G (E g 1 Eg i ) has no edge for some i = p. Each set Eg i gies a collection of Elerian trail with no self-intersection. The set of all these trails forms a rooted tree strctre in which the trail in E g 1 is the root and a trail in Eg i is a child of a trail E g i 1 if these trails share a ertex f. We can combine all these trails into a single Elerian trail λ br with no self-intersection by traersing the trails in a DFS manner in the tree strctre. The aboe procedre can be implemented to rn in linear time by maintaining trails as dobly-linked lists so that two trails with a common ertex can be merged in constant time. This proes the if part. It is not difficlt to see that Steps 1-5 can be implemented to rn in linear time to 8

9 (a) Γ1 (b) Γ (c) Γ2 (d) Γ2 Figre 5: Two embeddings of a planar graph G = (V, E 1 E 2 ); (a) a plane embedding Γ 1 of G which has no separating cre λ; (b) A disjnctie and splitter-free plane embedding Γ 2 of G, where the circles represent the ertices in G, the sqares represent the new ertices f for the faces f F br, and the dashed arrows represent the green edges defined in the proof of Theorem 4; (c) A closed cre λ br isiting all br-ertices of G; (d) A closed cre λ isiting all ertices of G. obtain a separating cre λ, from which a 2PB-embedding of G can be obtained in linear time. This proes the second statement in the theorem. Notice that any plane embedding of a canonical instance is disjnctie, since red and ble edges meet at a br-ertex of degree 2. 5 Redction to Planarity Test Let G = (V, E 1 E 2 ) be a canonical instance. In a plane embedding Γ of a canonical instance G, a red cycle Q of G is a splitter if and only if Q is not a facial cycle of Γ. The only-if part is immediate, since no facial cycle can separate two ertices/edges. The if part follows from that each of the two regions R 1 and R 2 by a non-facial red cycle Q mst contain a ble edge, becase otherwise R i contains a red path joining two ertices on Q, contradicting that (V, E 1 ) is a cacts. Symmetrically a ble cycle becomes a splitter in Γ wheneer it is not a facial cycle of Γ. We call a plane embedding Γ of G proper if any red/ble cycle of G appears as a facial cycle of Γ. 9

10 Detecting a proper embedding Γ of G can redced to the standard planarity testing of an agmented graph as follows. For each cycle C in the cacts (V, E i ), i = 1, 2, we sbdiide each edge e in C with a new ertex w e, create a new ertex C, and add new edges ( C, w e ), e E(C). Fig. 6(a) shows the graph G obtained from the canonical instance in Fig. 3(c). Let G = (Ṽ, Ẽ1 Ẽ2) denote the reslting graph, where Ẽi is the set of edges obtained by sbdiiding an edge in E i or introdced to agment a cycle in the cacts (V, E i ). Lemma 5 A canonical instance G = (V, E 1 E 2 ) admits a proper embedding if and only if G is planar. A proper embedding of G can be obtained from a plane embedding of G in linear time. Proof. Only-if part: Let Γ G be a proper embedding of G. Then each cycle block C of the cacts (V, E i ) srronds no edge in E j (j i). Hence, we can draw the newly added ertices C, and w e, e E(C) and edges between them inside the empty region of C withot creating edge crossings. Ths the reslting embedding is a plane embedding of G. If part: Since G is biconnected, G is also biconnected. Let Γ G be a plane embedding of G, where withot loss of generality that the oter facial cycle f o contains a br-ertex z of G. Note that z is not in any cycle of a cacts (V, E i ). Let ΓẼi denote the plane embedding indced from Γ G by the edges in ΓẼi. We first show that for each block C in (V, E) agmented with ertex C, the ertex C is srronded by the cycle C (i.e., C separates C from f o ) in ΓẼi. If for some C, both C and f o are otside C, then by the way of agmentation, only one ertex V (C) can be adjacent to ertices otside C (see Fig. 6(b)), and wold be a ct-ertex separating C and z in G, contradicting the biconnectiity of G. Now C of each cycle is located inside the sbdiided cycle C in ΓẼi. Next we show that no other ertex than C is located inside the sbdiided cycle C in Γ G. Assme that for some C, a ertex y ( C ) is inside the sbdiided C in Γ G (see Fig. 6(c)). Since G is connected, y is connected to a ertex in the the sbdiided C. Let be the one closest to y among sch. Then in G, V (C), and is adjacent to two edges e, e E(C). Now in the plane embedding Γ G, the cycle (, w e, C, w e ) srrond the set Y of all ertices reachable from y withot passing throgh. This, howeer, implies that is a ct-ertex separating y and z in G, a contradiction. Hence, each sbdiided C encloses no other ertex than C in Γ G. Let Γ be the plane embedding of G indced from Γ G by G; i.e., remoe the agmented ertices C and ignore the introdced ertices w e for all cycles C in cacti (V, E 1 ) and (V, E 2 ). Clearly, each C of a cacts (V, E i ) encloses no edges/ertices in Γ. Since each red (resp., ble) cycle Q in G is a (facial) cycle of (V, E 1 ) (resp., (V, E 2 )), the reslting embedding Γ is proper. Since testing planarity and constrcting a plane embedding if any can be done in linear time, we can find a 2PB-embedding of a gien instance if any in linear time. When Γ G is not planar, it contains a sbdiision of K 5 or K 3,3. Howeer, sch a sbgraph is not a direct eidence of a gien infeasible instance de to the agmentation. In the next section, we gie an algorithm that detects a forbidden sbgraph of a gien instance. 6 Forbidden Sbgraphs to Two-page Book Embeddings In this paper, we also characterize the instances G that do not admit 2PB-embeddings. 10

11 z z C f o f o C we C C y we we C ~ (a) G (b) (c) Figre 6: Illstration for a plane embedding of the agmented graph G; (a) Graph G obtained from the graph in Fig. 3(c) by agmenting each cycle in red and ble cacti with stars; (b) f o and C are otside the sbdiided cycle C; (c) other ertex y than C is inside the sbdiided cycle C. A graph H is called psedo-triconnected if a sbdiision of a triconnected graph G. Obsere that a graph H is psedo-triconnected if and only if H has at least three ertices of degree 3 and eery two ertices and of degree 3 in H has three internally disjoint paths. We call a psedo-triconnected sbgraph S of G a forbidden sbgraph if a cycle Q with E i in a plane embedding of S separates two edges e 1, e 2 E i (i {1, 2} {j}). By the niqeness of plane embedding of S, sch a sbgraph S (and hence G) cannot admit a 2PB-embedding. In this paper, we show that the conerse is tre establishing the following reslt. Theorem 6 Let G = (V, E 1 E 2 ) be a planar graph with a partition E 1 and E 2 of E(G) sch that each E i indces a simple and oterplanar sbgraph (V, E i ). Then G admits a 2PB-embedding if and only if there is no forbidden sbgraph in G. Frthermore, either a 2PB-embedding or a forbidden sbgraph of G can be fond in linear time. A completer characterization of forbidden sbgraphs is sefl since it can answer the soltion to restricted graph classes. For example, when each of E 1 and E 2 indces a forest, G cannot hae a forbidden sbgraph. Hence the theorem implies the next. Corollary 7 Let G = (V, E 1 E 2 ) be a graph with a partition E 1 and E 2 of E(G) sch that each E i indces a forest (V, E i ). Then G admits a 2PB-embedding if and only if G is planar. Frthermore, a 2PB-embedding of G if any can be fond in linear time. To proe Theorem 6, we first design a linear-time algorithm for detecting a forbidden sbgraph of a gien canonical instance. It is not difficlt see that een if a canonical instance G is obtained from the original instance G by Definition 1, a forbidden sbgraph S of G gies a forbidden sbgraph of G. We then proe that a canonical instance with no forbidden sbgraphs admits a proper embedding (and hence a 2PB-embedding) by designing a linear-time algorithm for constrcting a proper embedding for sch an instance. 11

12 6.1 SPQR tree representations To consider the all possible plane combinatorial embeddings of a biconnected graph, we se the SPQR tree by di Battista and Tamassia [9, 10] (see Appendix 1 for detail). The SPQR tree T of a biconnected graph G represents the adjacency of the triconnected components of G. Each node ν in T corresponds to a triconnected component of G, and is associated with a graph σ(ν) called the skeleton of ν, sch that a sbdiision of appears as a sbgraph of G (hence V (σ(ν)) V (G)). For each edge e = (, ) E ν, G has an indced sbgraph G e which shares {, } as a ct-pair with the complementary part G (V (G e ) {, }). Hence the skeleton σ(ν) proides an abstract strctre of the entire graph G; there is a sbgraph S of G that is a sbdiision of σ(ν), which is obtained by replacing each irtal edge (, ) with a, -path of G. For each R-node ν, its skeleton σ(ν) is triconnected and its plane embedding is niqe p to choices of oter faces. For each edge of the skeleton of a node ν of the SPQR treet of a canonical instance G = (V, E 1 E 2 ), the graph G e is called an r-graph (resp., b-graph) if it has a red (resp., ble), -path. Note that G e cannot hae both a red, -path and a ble, -path, since and are br-ertices of degree 2 and cannot be adjacent to other ertices in V V (G e ). A irtal edge e is called an r-edge (resp., b-edge) if G e is an r-graph (resp., b-graph). Again no irtal edge can be an r-edge and b-edge at the same time. We also treat a red (resp., ble) real edge as an r-edge (resp., b-edge). Fig. 7(b) shows the skeleton σ(ν) of the R-node ν with V (σ(ν)) = { 1, 3, 5, 7, z 1, z 2, z 3 } of the graph in Fig. 7(a). For a sbgraph H of the skeleton σ(ν) of a node ν, let E r (H) (resp., E b (H)) denote the set of r-edges (resp., b-edges) in H. Note that each of E r (H) and E b (H) indces a cacts, since each (V, E i ) is a cacts. Hence each P-node ν of T satisfies 6.2 Splitters and forbidden sbgraphs E r (σ(ν)) + E b (σ(ν)) 2. (1) Now we examine the strctre of red/ble splitters in canonical instances. A cycle Q in the skeleton σ(ν) of a node ν is called an r-cycle (resp., b-cycle) if E(Q ) E r (σ(ν)) (resp., E(Q ) E b (σ(ν))). For example, Q = ( 1, 2, 3, 4, 5, 6, 7, 8 ) in Fig. 7(b) is a non-facial r-cycle in the skeleton σ(ν) of the R-node ν. In fact, splitters and forbidden sbgraphs are eqialent in the following sense. Lemma 8 Let G = (V, E 1 E 2 ) be a canonical instance. (i) Let Q be a non-facial r-cycle in the skeleton σ(ν) of an R-node ν. Then the sbgraph S of G obtained from σ(ν) by replacing each irtal edge e = (, ) E(Q ) (resp., e = (, ) E(σ(ν)) (Q )) with a, -path of Q (resp., G) is a forbidden sbgraph of G. (ii) Let S be a forbidden sbgraph sch that a red cycle Q in S separates two ble edges e 1, e 2 of G. Then for the R-node ν sch that V (ν) contains all the eritces of degree 3 of S, the set of r-edges e E b (σ(ν)) with E(G e ) Q is a non-facial r-cycle Q in the skeleton σ(ν). Proof. (i) It sffices to show that the r-cycle Q in σ(ν) corresponding to Q separates two ble edges of G. Since Q is a non-facial r-cycle in σ(ν), Q separates two non-r-edges e 1, e 2 12

13 1 1 z 1 z z 5 z 4 3 z 3 7 f 2 3 f 3 z 3 f 5 f 6 4 z 2 f 1 f 4 z (a) G : r-edges : b-edges 5 (b) σ(ν) : r-edges : b-edges : non r-, non b-edges Figre 7: (a) A graph G which has a non-facial r-cycle Q = ( 1, 2, 3, 4, 5, 6, 7, 8 ); (b) The skeleton σ(ν) of the R-node ν with V (σ(ν)) = { 1, 3, 5, 7, z 1, z 2, z 3 } of G in (a). in the two regions R 1 and R 2 in a plane embedding γ ν of σ(ν). This means that R i contains at least one ble edge of G. (ii) Since S is a psedo-triconnected sbgraph of G, there is an R-node ν in which all ertices of degree 3 of S appear. Then σ(ν) has the r-cycle Q corresponding to Q. By the niqeness of embedding of S and σ(ν), Q still separates the ble edges e 1 and e 2, and this means that Q is not a facial cycle in a plane embedding γ ν of σ(ν). This shows that Q is a non-facial r-cycle in the skeleton σ(ν) of the R-node ν. Lemma 9 Gien E r (σ(ν)) (resp., E b (σ(ν))) for the skeleton σ(ν) of an R-node ν, testing if there is a non-facial r-cycle (b-cycle) in a plane embedding γ ν of σ(ν) can be done in O( E(σ(ν)) ) time. Proof. We consider the case of r-cycles (the case of b-cycles can be treated symmetrically). Since the red graph (V, E 1 ) is a cacts and no two ertices in V are joined by three ertex-disjoint paths. This means that the r-edges in the skeleton σ(ν) indce a cacts (V (σ(ν)), E r (σ(ν))). Since no two cycles share an edge in a cacts, the total size of all r-cycles in the cacts (V (σ(ν)), E r (σ(ν))) is O( E(σ(ν)) ). Hence een if we test whether each r-cycle in σ(ν) is a facial cycle of γ ν or not, the total time complexity is O( E(σ(ν)) ). Lemma 9 implies that once we know E r (σ(ν)) (resp., E b (σ(ν))) for all nodes ν in the SPQR tree, a non-facial r-/b-cycle in the skeleton of an R-node can be fond in linear time. In the rest of the section, we show how to compte the r-edges and b-edges in the skeletons of all nodes in the SPQR tree. 6.3 Rooted SPQR trees When a node in the SPQR tree of G is designated as the root, the SPQR tree is treated as a rooted tree, which defines a parent-child relationship among the nodes of the tree. For a node ν, a node µ adjacent to ν in SPQR tree is called the parent of ν if µ is closer to the 13

14 root than ν is, and is called a child of ν otherwise. Let Ch(ν) denote the set of all children of ν. We denote the graph formed from σ(ν) by deleting its parent irtal edge pe(ν) as σ (ν), if ν is not the root of T. Let G (ν) denote the sbgraph indced from G by the set of all ertices in the graphs σ (µ) for all descendants µ of ν, inclding ν itself. For each irtal edge e E(σ(ν)), let µ e Ch(ν) denote the corresponding child node. For each non-root node ν, we denote by G(ν) the graph G (V (G (µ)) {, }). See Fig. 8 for illstrations of the skeleton of a node n and its children. σ(ν) e 3 e2 e 1 σ(µe 2 ) σ(ν) : parent irtal edges σ(µe 1 ) σ(µe 3 ) (a) The skeleton σ(ν) of a P-node ν and the skeletons of its children µe i 1 e 2 2 e 3 e 1 σ(µe 1 ) σ(µe 2 ) σ(µe 3 ) (b) The skeleton σ(ν) of an S-node ν and the skeletons of its children µe i e 4 e 5 e σ(µe 5 ) 1 e 3 e 2 2 σ(ν) σ(µe 4 ) σ(µe 3 ) σ(µe 1 ) 1 1 σ(µe 2 ) (c) The skeleton σ(ν) of an R-node ν and the skeletons of its children µe i Figre 8: The skeleton σ(ν) of a ν and the skeletons σ(µ ei ) of its children µ ei where ν is a P-node in (a), an S-node in (b) and an R-node (c). Ch(ν), 6.4 Compting the color of edges in the skeletons of all nodes along rooted SPQR trees We first gie an oeriew of the algorithm for compting the r-edges in the skeletons of all nodes in the SPQR tree of G (compting b-edges can be done symmetrically): 1. First choose a node as the root of the SPQR tree of G, which introdces a parent-child relationship among the nodes of the tree. 2. By traersing the rooted SPQR tree in a bottom-p manner, compte the r-edges in the skeleton σ (ν) of each node ν (except for the parent irtal edge pe(ν)), based on the comptation of r-edges in the skeletons of children of ν. 3. By traersing the rooted SPQR tree in a top-down manner, identify the children µ Ch(ν) of each node ν sch that the parent irtal edge pe(µ) is an r-edge. Since G is not a simple cycle, the SPQR tree of G has an R- or P-node. We choose an R- or P-node ν as the root of the SPQR tree T. For a leaf node ν in the rooted SPQR tree T, we know that E r (σ (ν)) is the set of the red edges in the sbgraph G (ν). The next lemma says that the r-edges in the skeletons σ (ν) of all other nodes in the SPQR tree T can be compted in linear time. Lemma 10 Gien E r (σ (µ)) for a node ν in T, the set E r (σ (ν)) of all r-edges in the skeleton σ (ν) except pe(ν) can be compted in O( E(σ(ν)) + µ Ch(ν) E(σ(µ)) ) time. 14

15 Proof. An edge e = (, ) in σ (ν) is an r-edge if and only if G (µ e ) of the corresponding child µ e Ch(ν) has a red, -path, which is eqialent to that skeleton σ (µ e ) has a, - path P that consists of r-edges. Finding sch a path P in σ (µ e ) takes O( E(σ(µ e )) ) time. A real edge e in σ (ν) is an r-edge if and only if it is a red edge. Therefore, the set E r (σ (ν)) of all r-edges can be obtained in the claimed time bond. For the root ν, we know E r (σ(ν ). The next lemma says that the parent irtal r-edges of the skeletons σ(ν) of all other nodes in the SPQR tree T can be compted in linear time. Lemma 11 Gien E r (σ (ν)) for a node ν in T, the set of all children µ Ch(ν) sch that pe(µ) is an r-edge can be compted in O( E(σ(ν)) ) time. Proof. Obsere that for each edge e = (, ) E(σ (ν)), its parent irtal edge pe(µ e ) is an r-edge if and only if σ(ν) {e} contains a path that consists of r-edges. We compte the connected components C 1,..., C k of the graph (V (σ(ν)), E r (σ(ν))) indced from σ(ν) by the r-edges, and identify all bridges in the components C i, i = 1,..., k (where C i may consists of a single ertex). This can be done in linear time by the depth-first search algorithm. Then for each non-r-edge e = (, ) E(σ (ν)), pe(µ e ) is an r-edge if and only if and belong to the same component C i. On the other hand, for each r-edge e = (, ) E(σ (ν)), pe(µ e ) is an r-edge if and only if e is not a bridge of the same component C i to which and belong. Based on these characterizations, we can find all children µ Ch(ν) sch that pe(µ) is an r-edge in O( E(σ(ν)) ) time. By Lemma 8, any forbidden sbgraph can be fond as a non-facial r- or b-cycle in the skeleton of an R-node. After we compte the r-edges in the skeletons of all nodes in the SPQR tree in linear time by Lemmas 10 and 11, we test if each R-node has a non-facial r-cycle in its skeleton in time linear of the size of the skeleton by Lemma 9, which takes O( V + E ) time in total oer all R-nodes. Symmetrically we can find a non-facial b-cycle in the skeleton of an R-node in linear time. Hence finding a forbidden sbgraph of a canonical instance, if any can be done in linear time. To proe the theorem 6, the remaining task is to design a linear-time algorithm for constrcting a proper embedding for a canonical instance with no forbidden sbgraphs. 7 Constrcting Proper Embeddings In this section, we assme that a gien canonical instance G = (V, E 1 E 2 ) has no forbidden sbgraph, i.e., no non-facial r- or b-cycle in the skeleton of any R-node, and present a procedre for constrcting a proper plane embedding. In what follows, for a graph H = σ (ν) or H = G (ν) of each non-root node ν in T, an embedding ψ of H means a plane embedding of the graph sch that the both end ertices and of the parent irtal edge pe(ν) = (, ) of ν appear in the bondary of the plane embedding. When we traerse the bondary of ψ, we denote the path along the bondary from to (resp., from to ) by B, (ψ) (resp., B, (ψ)). The path B, (ψ) is called r-rimmed B, (ψ) is the niqe r-, -path. We define b-rimmed bondaries symmetrically. Fig. 9(a) and (b) shows an embedding γ ν of the skeleton σ(ν) and an embedding Γ ν of the graph G (ν) for an R-node ν, where B, (γ ν ) and B, (Γ ν ) are r-rimmed while B, (γ ν ) and B, (Γ ν ) are both no r-rimmed. 15

16 An embedding ψ of H is called proper if (i) eery r-cycle/b-cycle in ψ is a facial cycle; and (ii) B, (ψ) or B, (ψ) is r-rimmed (resp., b-rimmed) when pe(ν) is an r-edge (resp., b-edge). A plane embedding ψ of H = G or H = σ(ν) for the root node ν is called proper if eery r-cycle/b-cycle in ψ is a facial cycle. Assming that each edge e E(σ (ν)) of a node ν admits a proper embedding Γ e of G e, we show that a proper embedding Γ ν of G (ν) can be obtained from these embeddings Γ e. For this, we first obsere that the skeleton of each node admits a proper embedding e1 e 6 e 2 e 3 e 7 r-cycle 2 e e 4 4 e 8 Γe Γe 2 Γe 3 Γe 4 Γe parent irtal edge : r-edges : b-edges : non r-, non b-edges (a) γν r-rimmed r-rimmed r-rimmed r-rimmed 4 Γe 6 Γe 7 Γe 8 (b) Γ ei r-cycle (c) Γν 4 r-rimmed Figre 9: (a) A proper embedding γ ν of the skeleton σ(ν) of an R-node ν, where e 1,..., e 8 are the irtal edges of σ(ν); (b) A proper embedding Γ ei of sbgraph G e of each irtal edge e i, i = 1, 2,..., 8; and (c) A proper embedding Γ ν of sbgraph G (ν) obtained from γ ν by replacing each irtal edge e i with Γ ei. Lemma 12 Let G be a canonical graph with no forbidden sbgraph, and ν be a node in the SPQR tree. Then: (i) When ν is an S- or R-node or a P-node with E r (σ(ν)) + E b (σ(ν)) 1, any plane embedding γ ν of the skeleton σ(ν) of the root ν or of σ (ν) of a non-root ν is proper. (ii) When ν is a P-node with E r (σ(ν)) = 2 (resp., E b (σ(ν)) = 2), a plane embedding γ ν of the skeleton σ(ν) of the root ν or of σ (ν) of a non-root ν is proper if one of the two edges in E r (σ(ν)) (resp., E b (σ(ν))) appear consectiely (possibly one appears as B, (γ ν ) and the other as the parent irtal edge pe(ν)). Proof. (i) The lemma is immediate for a P-node ν with E r (σ(ν)) + E b (σ(ν)) 1 in (1) or an S-node. Let ν be a non-root R-node (the case of root R-node can be treated analogosly). Let γ ν be a plane embedding of σ (ν) Since G has no forbidden sbgraph, any r-/b-cycle of σ(ν) is a facial cycle of any plane embedding of σ(ν). Assme that the parent irtal edge pe(ν) of ν is an r-edge (the case where the parent irtal edge of ν is a b-edge can be shown symmetrically). Hence the parent irtal edge pe(ν)) = (, ) is an r-edge in σ(ν). If σ (ν) has an r-, -path P, then pe(ν) and P form an r-cycle Q, which again mst be 16

17 a facial cycle in a plane embedding of σ(ν); i.e., P is B, (γ ν ) or B, (γ ν ), and σ (ν) has no r-, -path other than P. This proes that γ ν is proper. (ii) For a P-node ν with E r (σ(ν)) = 2 or E b (σ(ν)) = 2, the embedding stated in the lemma is proper, since σ(ν) has exactly one r-cycle or b-cycle. Based on proper embeddings of all nodes in the SPQR tree, we show how to constrct a proper embedding for H = G traersing the rooted SPQR tree T in a bottom-p manner. We first constrct proper embeddings for leaf nodes of T. Then we constrct a proper embedding of a non-leaf node ν by assembling the proper embeddings of all children of ν. Leaf nodes: For each leaf S- or R-node ν of T, any plane embedding of G (ν) = σ (ν) is proper by Lemma 12. Note that there is no leaf P-node since G is simple. Internal nodes: Let ν be an internal node of T, pe(ν) = (, ) be the parent irtal edge of ν, and γ ν be a proper embedding of σ (ν) in Lemma 12. For each irtal edge e E(σ (ν)), let Γ e denote a proper embedding of G e. (1) S-node: Let ν be an S-node. In this case, γ ν is a single path joining and. Let Γ ν be an embedding of G (ν) obtained from γ ν by replacing each irtal edge e E(σ (ν)) with Γ e. If G (ν) is not an r-graph or a b-graph, then the reslting embedding Γ ν of G (ν) is already proper. Assme that G (ν) is an r-graph (the case of a b-graph can be treated symmetrically). Then, (α) for each edge e in B, (ξ ν ), we flip each Γ e if necessary so that the r-rimmed bondary of ψ e appears along the oter face of γ ν. The reslting embedding Γ ν of G (ν) is now proper. (2) R-node: Let ν be an R-node, and γ ν be a proper embedding of σ (ν), where we assme withot loss of generality that B, (γ ν ) is the niqe r-, -path of σ (ν). See Fig. 9, where (a) shows a proper embedding γ ν of σ (ν) of the R-node ν, and (b) shows a proper embedding Γ ei of G ei of each irtal edge e i E(σ(ν)). Consider the case where G (ν) is an r-graph and the parent irtal edge of ν is an r-edge (the other case can be treated analogosly). We replace each irtal edge e in σ (ν) so that (α) for each edge e in B, (γ ν ), the r-rimmed bondary of Γ e of appears along the oter face of γ ν ; and (β) for each edge e in a facial r-cycle Q of γ ν, the r-rimmed bondary of Γ e of appears facing the interior of Q. Let Γ ν be the reslting embedding of G (ν). See Fig. 9(c) for the reslting embedding Γ ν obtained from the embeddings in Fig. 9(a) and (b). We see that Γ ν is proper, since γ ν and all ψ e are proper. (3) P-node: Let ν be a P-node. Let γ ν be a proper embedding of σ (ν). That is, two edges in E r (σ(ν)) (resp., E r (σ(ν))), if any, appear consectiely (possibly one as B, (γ ν ) and the other as the parent edge pe(ν)). We replace each irtal edge e in σ (ν) by Γ e according to the same rles (α) and (β). We easily see that the reslting embedding Γ ν of G (ν) is proper. Root nodes: For the root R- or P-node ν, we can obtain a proper embedding Γ of G by replacing each irtal edge e in a proper embedding γ ν of σ(ν) with a proper embedding Γ e according to the same rle (β). 17

18 This completes an indctie proof for the existence of proper embeddings in canonical instances when there is no forbidden sbgraph. It is not difficlt see that the aboe procedre for constrcting a proper embedding of G can be implemented to rn in linear time. Therefore, this completes the proof of Theorem 6. References [1] P. Angelini, M. D. Bartolomeo, and G. D. Battista, Implementing a partitioned 2PBembedding testing algorithm, In 20th International Symposim on Graph Drawing (GD 12), Springer-Verlag, Lectre Notes in Compter Science, [2] F. Bernhart and P. C. Kainen, The book thickness of a graph, J. Combin. Theory Ser. B 27(3), , [3] F. R. K. Chng, F. T. Leighton, and A. L. Rosenberg, Embedding graphs in books: A graph layot problem with applications to VLSI design, SIAM J. Algebraic Discrete Methods, 8, pp , [4] G. Di Battista and R. Tamassia, On-line planarity testing, SIAM J. on Compt. 25(5), pp , [5] G. Di Battista and R. Tamassia, On-line maintenance of triconnected components with SPQR-trees, Algorithmica, 15, pp , [6] V. Djmoić and D. R. Wood, On linear layots of graphs Discrete Math. Theor. Compt. Sci, 6, pp , [7] V. Djmoić and D. R. Wood, Stacks, qees and tracks: Layots of graph sbdiisions, Discrete Math. Theor. Compt. Sci, 7, pp , [8] S. Hong, and H. Nagamochi, Two-page book embedding and clstered graph planarity, TR[ ], Dept. of Applied Mathematics and Physics, Uniersity of Kyoto, Japan, [9] J. E. Hopcroft and R. E. Tarjan, Diiding a graph into triconnected components, SIAM J. on Compt., 2, pp , [10] A. L. Rosenberg, The Diogenes approach to testable falt-tolerant arrays of processors, IEEE Trans. Compt. C-32, pp , [11] W. T. Ttte, Graph Theory, Encyclopedia of Mathematics and Its Applications, Vol. 21, Addison-Wesley, Reading, MA, [12] M. Yannakakis, Embedding planar graphs in for pages, J. Compt. System Sci. 38, pp , [13] A. Wigderson, The complexity of the Hamiltonian circit problem for maximal planar graphs, Technical Report 298, EECS Department, Princeton Uniersity,

19 Appendix 1: SPQR tree We reiew the definition of triconnected components [17] (or 3-blocks) and a ariation of the SPQR tree [9, 10] of a biconnected graph. First we reiew the definition of triconnected components [17]. If G is triconnected, then G itself is the niqe triconnected component of G. Otherwise, let, be a ct-pair of G. We split the edges of G into two disjoint sbsets E 1 and E 2, sch that E 1 > 1, E 2 > 1, and the sbgraphs G 1 and G 2 indced by E 1 and E 2 only hae ertices and in common. Form the graph G 1 from G 1 by adding an edge (called a irtal edge) between and that represents the existence of the other sbgraph G 2 ; similarly form G 2. We contine the splitting process recrsiely on G 1 and G 2. The process stops when each reslting graph reaches one of three forms: a triconnected simple graph, a set of three mltiple edges (a triple bond), or a cycle of length three (a triangle). The triconnected components of G are obtained from these reslting graphs: a triconnected simple graph; a bond, formed by merging the triple bonds into a maximal set of mltiple edges; a polygon, formed by merging the triangles into a maximal simple cycle. The triconnected components of G are niqe. See [17] for frther details. One can define a tree strctre, sometimes called as the 3-block tree, sing triconnected components as follows. The nodes of the 3-block tree are the triconnected components of G. The edges of the 3-block tree are defined by the irtal edges, that is, if two triconnected components hae a irtal edge in common, then the nodes that represent the two triconnected components in the 3-block tree are joined by an edge that represents the irtal edge. There are many ariants of the 3-block tree in the literatre; the first was defined by Ttte [20]. In this paper, we se the terminology of the SPQR tree, as defined by di Battista and Tamassia [9, 10]. We now briefly reiew this terminology. Each node ν in the SPQR tree is associated with a graph G = (V, E) called the skeleton of ν, denoted by σ(ν) = (V ν, E ν ) (V ν V ). For each edge e = (, ) E ν, G has an indced sbgraph G e which shares {, } as a ct-pair with the complementary part G (V (G e ) {, }). Hence the skeleton σ(ν) corresponds to a triconnected component, proiding an abstract strctre of the entire graph G. There are for types of nodes in the SPQR tree. The node types and their skeletons are: 1. Q-node: the skeleton consists of two ertices connected by two mltiple edges. Each Q-node corresponds to an edge of the original graph. 2. S-node: the skeleton is a simple cycle with at least three ertices (this corresponds to a polygon triconnected component). 3. P-node: the skeleton consists of two ertices connected by at least three edges (this corresponds to a bond triconnected component). 4. R-node: the skeleton is a triconnected graph with at least for ertices. 19