Independent Trees in 4-Connected Graphs. Sean P. Curran

Transcription

1 Independent Trees in 4-Connected Graphs A Thesis Presented to The Academic Faculty by Sean P. Curran In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy School of Mathematics Georgia Institute of Technology June 24, 2003

2 Independent Trees in 4-Connected Graphs Approved by: Dr. Xingxing Yu, Committee Chair Dr. William T. Trotter Dr. Richard Duke Dr. R. Gary Parker School of Industrial Systems Engineering Dr. Margaret Symington Date Approved

3 For my wife Kyle, who, after all was said and done, had not killed me. iii

4 ACKNOWLEDGEMENTS I must first thank my wife for her support throughout the lean graduate years. This work would not exist without her. I also owe great thanks to my mother Jean and father Frank, whose unquestioning, oft-irrational faith in me is partially responsible for this work. I owe great thanks to my advisor Dr. Xingxing Yu, who has been patient and encouraging with this slow-witted student. I also thank my committee members: Dr. Richard Duke, Dr. Gary Parker, Dr. Margaret Symington, and Dr. Tom Trotter. All have been patient and supportive as I have struggled through this final leg of my Ph. D. journey. Thanks also to Dr. Presad Tetali and Dr. Robin Thomas, both of whom dutifully served on my oral exam committee. Additional thanks to Dr. Thomas, who allowed me opportunities to present this work in seminar, and who has been encouraging and supportive throughout my studies. Much thanks goes to Dr. Orlando Lee, whose expertise with algorithms and L A TEX was invaluable. Thanks to Dr. Evans Harrell and Dr. William Green, current and former graduate advisors, who guided me through numerous administrative hurdles throughout my studies. I thank Dr. Chris Heil, who gave selfless guidance and support in my early graduate career. I am convinced I would not have passed my written comprehensive exam without his aid. Finally, numerous friends have provided invaluable support, both technical and personal. Without these exceptionally supportive people, I ultimately would have been found locked in a padded room. While this list is far from complete, a few bear special mentioning. Thanks to Kyle Cantrell, Sarah Day, Andy and Chrissy Harris, Greg Lewis, Kevin Maguire, Russ Martin, Todd Moeller, Richard and Natalie Powell, Bryan Rasmussen, the Rueffert family, Marcus Sammer, and Laura Sheppardson. iv

5 TABLE OF CONTENTS DEDICATION ACKNOWLEDGEMENTS iii iv LIST OF FIGURES vii SUMMARY ix I INTRODUCTION Basic definitions and notation Overview II NONSEPARATING PATHS Planar graphs Disjoint paths in (k, S)-connected subgraphs Nonseparating paths III NONSEPARATING AND CYCLIC CHAINS Planar chains Nonseparating chains Planar cyclic chains IV THE CHAIN DECOMPOSITION Types of chains Internal chains Chain decomposition V USEFUL NUMBERINGS Motivation Ordering in planar graphs Numberings VI INDEPENDENT TREES Algorithm Trees Correctness and complexity VII OTHER RESULTS v

6 REFERENCES vi

7 LIST OF FIGURES 1 Component K in the proof of (2.1.5) Graphs G and G in the proof of (2.3.2) G and G as in the proof of Lemma (2.3.7) Example of a chain A planar chain H := v 0 B 1 v 1 B 2 v 2 B 3 v 3 B 4 v 4 B 5 v 5 in a graph G. The dashed edges may or may not exist, but they are not part of H G, a, b, P, B P, X P in (3.2.1) A nice bridge B and the graph G B (in boldface) as defined in the proof of Lemma (3.2.6) An example of an auxiliary graph K cuts determined by a component A j of K Example of a cyclic chain A planar cyclic chain H := v 0 B 1 v 1... v 5 B 6 v 6 rooted at r in a graph G. The dashed edges may or may not exist in G, but they are not part of H (a) An up F -chain, (b) a down F -chain, (c) an elementary F -chain, and (d) a triangle F -chain. The dashed edges may or may not exist, but they are not part of the chains Illustration for (4.2.1), (4.2.2), and (4.2.5): one example with r {a, a } and the other with r {a, a } Graph H in the proof of Lemma (4.2.7) Example for Notation (4.2.9) with X P = {r 1, r 2, r 3, r 4 }. Note that the edges r 1 a and r 4 a are not contained in any H i. H 1 is adjacent to F, but neither H 2 nor H 4 are adjacent to F Four independent trees in a plane graph forming an independent spanning system u 1,..., u p is a (T a, T c ) ordering of N G (b) {a, c} In each one of the examples above u v and u w. The bold lines denote the paths in T a, and the dashed lines denote the paths in T c Two disjoint paths, one in T a and the other in T d Three disjoint paths, one contained in each of T a, T c and T d In each example, v 2, v 3, u 3, and w 3 are the terminals of B Extending the numbering g when H i is an up G i 1 -chain vii

8 23 Extending the numbering g when H i is a down G i 1 -chain viii

9 SUMMARY The main result proves that a 4-connected graph has 4 independent spanning trees rooted at an arbitrary vertex. We provide an algorithm which constructs these trees in O( V (G) 3 ) time. Our method follows the method of Cheriyan and Maheshwari [3], who have a similar result for 3-connected graphs. We first construct a chain decomposition for 4-connected graphs, then use the decomposition to produce special numberings on V (G). The trees are finally produced in a natural way using the chain decomposition, a theorem of Huck [12], and the derived numberings. Chapter 2 provides numerous lemmas used throughout the work. Chapter 3 defines the basic tool in our decomposition, the planar chain, and provides an important result for later use. Chapter 4 constructs the decomposition, and Chapter 5 extracts the numberings. In Chapter 6, we construct the independent trees, using the results of the previous chapters. Some corollaries of these results are related to a classical connectivity conjecture. We discuss these related results in Chapter 7. ix

10 CHAPTER I INTRODUCTION In this short chapter, we present basic notions of graph theory and provide an overview of the remainder of the thesis. 1.1 Basic definitions and notation In the interest of self containment, we first present some basic graph theoretic definitions and notation. We present here only the definitions necessary to state the main result. Other definitions will be made as necessary. The entirety of this section will be familiar to practitioners of graph theory, so the advanced reader may wish to skip it. We assume only that the reader knows what a graph is. Given a graph G, we let V (G) denote the vertex set, and we let E(G) denote the edge set. Throughout this work, we assume that all graphs are finite undirected simple graphs (no multiple edges or loops). We use the notation A := B to define A as B, or to rename B as A. If H is a graph such that V (H) V (G) and E(H) E(G), then we say H is a subgraph of G. We use the subset notation, H G, to represent this relationship, and we say, H is in G, for brevity. A subgraph H of G is said to be a spanning subgraph if V (H) = V (G). Let T G. We obtain a natural graph G[T ] from T by letting V (G[T ]) := V (T ), and taking E(G[T ]) to be the set of all edges of G haiving both ends in V (T ). G[T ] is said to be the subgraph of G induced by T. A subgraph H of G is said to be an induced subgraph if H = G[V (H)]. A graph C is a cycle if its vertices can be ordered v 1 v 2... v n such that E(C) = {v 1 v 2, v 2 v 3,... v n 1 v n, v n v 1 }. A graph P is a path if its vertices can be ordered v 1 v 2... v n such that E(P ) = {v 1 v 2, v 2 v 3,... v n 1 v n }. We say P is a v 1 v n path, and we call v 1 and v n the ends of P. Given vertices x and y of a path P, the subpath of P connecting x and 1

11 y is denoted P [x, y]. Note that a single vertex x may be thought of as an x x path, or trivial path. Given set T V (G) E(G), we let G T be the graph obtained from G by letting V (G T ) := V (G) T, and removing from E(G) any edge in T, as well as any edge in G incident to any vertex in T. If T is a single vertex or edge, say x, we will sometimes abuse notation and write G x instead of G {x}. Conversely, given a subgraph H of G and S V (G) E(G), we let H + S denote the graph with vertex set V (H) (S V (G)) and edge set E(H) (S E(G)), removing any duplicate edges which might arise. If no superset is indentified, G is assumed to be the complete graph. As above, if S = {x}, we will write H + x. A graph G is connected if for any distinct x, y V (G), there exists an x y path in G. For an integer k 1, we say graph G is k-connected if V (G) k + 1, and for any subset T V (G) with T < k, G T is connected. A graph T is called a forest if it contains no cycles. If T also is connected, we say it is a tree. It is well known that for any two vertices x and y of tree T, there exists a unique x y path in T. We naturally denote that path T [x, y]. It is also well known that any connected graph has a spanning tree. It is often important to specify a particular vertex r of a tree as being somehow distinct. Such a special vertex is commonly called a root of the tree. 1.2 Overview In 1984, Itai and Rodeh [14] proposed a multi-tree approach to reliability in distributed networks. Let G be a graph and r V (G). We may view G as a distributed network with a root r, with the vertices of G as processors. A fault-tolerant communication scheme can be designed for this network if we can find spanning trees of G which are independent [8, 14]. Let T and T be trees contained in a graph rooted at r. We say that T and T are independent if, for each vertex x V (T ) V (T ), the paths T [r, x] and T [r, x] have no vertex in common except for r and x. 2

12 Itai and Rodeh [14] developed a linear time algorithm that, given any vertex r in a 2- connected graph G, finds two independent spanning trees of G rooted at r. Later, Cheriyan and Maheshwari [3] proved that for any vertex r in a 3-connected graph G, there exist three independent spanning trees of G rooted at r; furthermore, they gave an O( V (G) 2 ) algorithm for finding these trees. Itai and Zehavi [15] proved independently that every 3-connected graph contains three independent spanning trees (rooted at any vertex), and they conjectured the following. (1.2.1) Conjecture. Let G be a k-connected graph and let r V (G). Then there exist k independent spanning trees of G rooted at r. A contractible edge in a k-connected graph is an edge whose contraction results in a new k-connected graph. Itai and Zehavi s proof for the 3-connected case relies on the existence of a contractible edge. It is well-known, however, that for every k 4 there exist infinitely many k-connected graphs with no contractible edges. In view of this fact, it would be interesting to know if (1.2.1) holds for k = 4. The 4-connected case of (1.2.1) is also important in terms of applications, since four independent spanning trees ensure at a reasonable cost a higher degree of reliability in distributed networks. Huck [12] proved (1.2.1) for planar 4-connected graphs. Miura, Nakano, Nishizeki and Takahashi [18] gave a linear algorithm for finding four independent rooted spanning trees in a planar 4-connected graph. The main result of this thesis is the following. (1.2.2) Theorem. Let G be a 4-connected graph and let r V (G). Then there exist four independent spanning trees of G rooted at r. Moreover, such trees can be found in O( V (G) 3 ) time. Itai and Rodeh s algorithm [14] for constructing two independent spanning trees relies on an ear decomposition of graphs. Cheriyan and Maheshwari [3] used the concept of a nonseparating ear decomposition to construct three independent spanning trees in 3

13 3-connected graphs. We extend this basic idea to the 4-connected case, extracting a nonseparating chain decomposition of 4-connected graphs. In Chapter 3, we define a basic structure in our decomposition: the nonseparating planar chain. Moreover, we provide an algorithm to extract these chains from 4-connected graphs and establish the existence of the first chain in our decomposition: the cyclic planar chain. In Chapter 4, we use the results of Chapter 3 to extract an algorithm which, given a partial nonseparating chain decomposition of a 4-connected graph, constructs the next chain in the decomposition. Chapter 6 introduces another algorithm which, given a nonseparating chain decomposition of a 4-connected graph, constructs the desired independent trees. Finally, Chapter 7 shows some applications of planar chains to other problems in graph theory. 4

14 CHAPTER II NONSEPARATING PATHS This chapter establishes basic tools which will be used throughout the remaining chapters. Many of these results are highly technical. 2.1 Planar graphs A graph G is a planar graph if it can be drawn in the plane with no crossing edges; such an embedding is a plane graph. The connected regions bounded by edges of a plane embedding of G are called faces of G. A particular type of planar graph appears frequently in our decomposition. Let G be a graph with distinct vertices a, b, c and d. (2.1.1) Definition. We say that the quintuple (G, a, b, c, d) is planar if G can be drawn in a closed disc in the plane with no pair of edges crossing, and such that a, b, c, d occur on the boundary of the disc in this cyclic order. Some of our results and algorithms require that we find an embedding of a planar quintuple (G, a, b, c, d) in a closed disc such that a, b, c, d occur on the boundary of the disc in that cyclic order. This can be done in linear time using an algorithm of Hopcroft and Tarjan [10] (or a more recent algorithm by Hsu and Shi [11]). For convenience, we state this result (without proof) as the following lemma. (2.1.2) Lemma. Let (G, a, b, c, d) be a planar graph. Then one can find in O( V (G) ) time an embedding of G in a closed disc such that a, b, c, d occur on the boundary of the disc in that cyclic order. Before we continue, we remind the reader of a few basic graph theoretic notions. Let G be a graph, let T V (G), and let (T ) be the set of edges in G which have exactly one end in T. We let N G (T ) be the set {x V (G) T xy (T )}. If T = {v} we write 5

15 N G (v), and for any x N G (v), we say x is a neighbor of v in G. If it is clear which graph is referred to, we will abbreviate this notation to N(T ). If G is connected, and T V (G) such that G T is not connected (and nonempty), then we call T a vertex cut set of G. If k := T, we say T is a k-cut. If T = {v}, we say v is a cut vertex. We can similarly define edge cut sets and cut edges. If G is not connected, then G is a union of maximally connected subgraphs. Each such subgraph is called a component of G. (2.1.3) Definition. Let F be a subgraph of a graph K. An F -bridge of K is a subgraph of K which is induced by either (1) an edge in E(K) E(F ) with both ends on F, or (2) edges of a component D of K V (F ) together with the edges of K from D to F. For an F -bridge B of K, the set V (B) V (F ) is the set of attachments of B on F. In the case that V (F ) = {v}, we call B a v-bridge. A common subproblem we will encounter is how to reroute certain paths through subgraphs which are almost 3-connected or 4-connected. By almost k-connected, we mean the following. (2.1.4) Definition. Let G be a graph, let S V (G), and let k be a positive integer. We say that G is (k, S)-connected if V (G) S + 1, G is connected, and for any T V (G) with T k 1, every component of G T contains an element of S. This definition is partially motivated by the following observation. Let G be a k- connected graph, let S V (G), and let K be a component of G S. Then G[V (K) S] is (k, S)-connected. Now, let (G, a, b, a, b ) be a planar graph. Then any a-a path in G {b, b } separates b from b in G. The next lemma shows that one can find efficiently an a-a path P in G {b, b } such that G V (P ) has exactly two components. (2.1.5) Lemma. Let (G, a, b, a, b ) be a planar graph and suppose G is (4, {a, a, b, b })- connected. Then there exists an induced a-a path P in G such that G V (P ) has exactly 6

16 two components K and K with b V (K) and b V (K ). Moreover, such a path can be found in O( V (G) + E(G) ) time. Proof. Take an embedding of G in a closed disc such that a, b, a, b occur on the boundary of the disc in the cyclic order listed. By Lemma (2.1.2), this can be done in O( V (G) + E(G) ) time. Let G := (G b ) + {ab, a b}. We claim that G is 2-connected. Suppose for a contradiction that G is not 2-connected. Let x be a cut vertex of G. Since G is (4, {a, a, b, b })-connected, G {b, b } contains an a-a path, and hence, {a, a, b} is contained in a cycle in G. Therefore, {a, a, b} is contained in an x-bridge of G, and G has another x-bridge B such that (V (B) {x}) {a, a, b} =. Hence, B x is a component of G T, where T := {x, b } and (V (B) {x}) {a, a, b, b } =, which contradicts the assumption that G is (4, {a, a, b, b })-connected. Thus, we can assume that ab, a b are in the cycle bounding the infinite face of G. Let P be the a-a subpath of this cycle which avoids b. Note that N G (b ) V (P ) and P can be computed in O( V (G) + E(G) ) time. We claim that G V (P ) is connected. Suppose for a contradiction that G V (P ) is not connected. Let K be the set of components of G V (P ) which do not contain b. For any K K, let u K, u K V (P ) such that N G (K) V (P ) V (P [u K, u K ]) and P [u K, u K ] is minimal with respect to this property. If K 2, choose K K such that for any K K, if E(P [u K, u K ]) E(P [u K, u K ]), then P [u K, u K ] P [u K, u K ]; such a component must exist because of planarity. If K = 1, let K = {K}. In either case, N G (P (u K, u K )) V (K) {u K, u K, b }. Thus, K P (u K, u K ) is contained in a component of G {u K, u K, b } that does not contain any vertex in {a, a, b, b }, which contradicts the assumption that G is (4, {a, a, b, b })-connected. So G V (P ) = G (V (P ) {b }) is connected. Hence, G V (P ) has exactly two components K and K with b V (K) and b V (K ). We now show that P is an induced path in G. Suppose on the contrary that P is not induced. Let e = xy E(G) E(P ) with x, y V (P ). Then V (P (x, y)). Moreover, by planarity N G (P (x, y)) {x, y, b }. Then P (x, y) is contained in a component of G {x, y, b } that does not contain any vertex in {a, a, b, b }, which contradicts again 7

17 a PSfrag replacements b K b a Figure 1: Component K in the proof of (2.1.5). the assumption that G is (4, {a, a, b, b })-connected. Thus, P is a path as required. Moreover, it is easy to see that such a path can be found in O( V (G) + E(G) ) time. We conclude this section with another lemma which concerns nonseparating induced paths in planar graphs. (2.1.6) Lemma. Let (G, a, a, b, b ) be a planar graph, and suppose G is (4, {a, a, b, b })- connected and G = K1,4. Then there exists a nonseparating induced a-a path P in G such that V (P ) {b, b } =. Moreover, such a path can be found in O( V (G) + E(G) ) time. Proof. For convenience, let S := {a, a, b, b }. Take an embedding of G in a closed disc such that a, a, b, b occur on the boundary of the disc in the cyclic order listed. By Lemma (2.1.2), this can be done in O( V (G) + E(G) ) time. Let G := G + {ab, a b}. We claim that G is 2-connected. Suppose for a contradiction that G is not 2-connected. Let x be a cut vertex of G. Since G (and hence, G ) is (4, S)-connected, it follows that any component of G x either contains only vertices in S, or contains at least one vertex in V (G) S and at least three vertices in S. Since a b, ab E(G ), G x cannot have both kinds of components. Therefore, every component of G x contains only vertices in S. Moreover, since V (G) 5, x S. But then, it is easy to see that (G, a, a, b, b ) must be isomorphic to K 1,4 with x as the vertex of degree four, which contradicts the hypothesis. Hence, G is 2-connected. 8

18 Thus, we can assume that ab, a b are in the cycle bounding the infinite face of G. Let P be the a-a subpath of this cycle which avoids b and b. Note that P is an a-a path in G, and such a path can be found in O( V (G) + E(G) ) time. We claim that P is nonseparating in G. Suppose for a contradiction that G V (P ) is not connected. Note that b and b are contained in a component of G V (P ). Let K be the set of components of G V (P ) which contain neither b nor b. For any K K, let u K, u K V (P ) such that N G (K) V (P ) V (P [u K, u K ]) and P [u K, u K ] is minimal with respect to this property. If K 2, choose K K such that for any K K, if E(P [u K, u K ]) E(P [u K, u K ]), then P [u K, u K ] P [u K, u K ]; such a component must exist because of planarity. If K = 1, let K = {K}. In either case, N G (P (u K, u K )) V (K) {u K, u K }. Thus, K P (u K, u K ) is contained in a component of G {u K, u K } that does not contain any vertex in S, which contradicts the assumption that G is (4, S)- connected. Thus, G V (P ) is connected. Next we show that P is an induced path in G. Suppose by contradiction that P is not induced. Let e = xy E(G) E(P ) such that x, y V (P ). Then V (P (x, y)). Moreover, by planarity N G (P (x, y)) {x, y}. Then P (x, y) is contained in a component of G {x, y} that does not contain any vertex in S, which contradicts again the assumption that G is (4, S)-connected. Thus, P is a nonseparating induced a-a path in G such that V (P ) {b, b } = as required. 2.2 Disjoint paths in (k, S)-connected subgraphs The disjoint paths problem can be defined as follows: given a graph G and distinct vertices a, b, c, d of G, find disjoint paths from a to b, and from c to d, respectively, or certify that they do not exist. This problem was solved independently in [1], [20], [21], and [23]. We state Seymour s version ([20], Theorem 4.1). (2.2.1) Theorem. Let a, b, c, d be distinct vertices of a graph G. Then exactly one of the following holds: 9

19 (1) G contains disjoint paths from a to b, and from c to d, respectively, or (2) for some integer k 0, there exist pairwise disjoint sets A 1,..., A k V (G) {a, b, c, d} such that for 1 i j k, N G (A i ) A j =, for 1 i k, N G (A i ) 3, and if G is the graph obtained from G by, for each i, deleting A i and adding new edges joining every pair of distinct vertices in N G (A i ), and also, adding the edges ab and cd, then G can be drawn in the plane with no pair of edges crossing except ab and cd, which cross once. Let G be a graph and S := {a, b, c, d} V (G). Shiloach [21] gave an O( V (G) E(G) ) algorithm for the disjoint paths problem. We need to solve a special case of the disjoint paths problem: when G is (4, S)-connected. We show that Shiloach s algorithm can solve the disjoint paths problem in O( V (G) + E(G) ) time for (4, S)-connected graphs. (2.2.2) Lemma. Let G be a graph and let S := {a, b, c, d} V (G). Suppose that G is (4, S)-connected. Then exactly one of the following holds: (1) there exist disjoint paths from a to b and from c to d, respectively, or (2) (G, a, c, b, d) is planar. Moreover, one can in O( V (G) + E(G) ) time find paths as in (1) or certify that (2) holds. Proof. First, we prove that exactly one of (1) and (2) holds. Clearly, (1) and (2) are mutually exclusive because of planarity. We know that either (1) or (2) of (2.2.1) holds. If (1) of (2.2.1) holds, then (1) of Lemma (2.2.2) holds. So assume (2) of (2.2.1) holds. Let A 1,..., A k be as described in (2) of (2.2.1). Then S A i = for 1 i k. Hence, G[A i ] consists of those components of G N G (A i ) containing no element of S, contradicting our assumption that G is (4, S)-connected because N G (A i ) 3. So no A i can exist. Let G be described as in (2) of Theorem (2.2.1). Observe that (G {ab, cd}, a, c, b, d) is planar. Since G {ab, cd} = G, (2) of Lemma (2.2.2) holds. Therefore, either (1) or (2) holds. 10

20 Let us prove the algorithmic part of Lemma (2.2.2). First, we give a sketch of Shiloach s algorithm. It has as input a graph G and vertices a, b, c, d of G (with no connectivity hypothesis on G). The algorithm consists of reductions R1,...,R6 which reduces the general problem to a restricted one. R1: The algorithm initially reduces the problem to 3-connected graphs, in O( V (G) + E(G) ) time. R2: If the graph is planar, then a specialized O( V (G) + E(G) ) algorithm for 3- connected planar graphs [19] is used to solve the problem. R3: Assume that G is nonplanar. This is the most time-consuming step of the algorithm. It reduces the problem using network flow techniques to graphs satisfying some connectivity constraints involving S := {a, b, c, d}. Namely, the resulting graph G satisfies the following property: for any subset S of vertices of G with S 4, there exist four disjoint paths connecting S to S (these paths can share ends in S though). In fact, this step is not executed at once, but it is interspersed with reductions R4, R5 and R6 in the algorithm. Whenever the algorithm finds a set S which is not connected to S by four disjoint paths, a reduction is performed. The total time spent with these reductions during the whole algorithm is O( V (G) E(G) ). For simplicity, suppose that no such set S exists. By Menger s theorem, this is equivalent to saying that G is (4, S)-connected. Note that the graph G in the statement of Lemma (2.2.2) is (4, S)-connected. Thus, so far G is 3-connected, nonplanar, and (4, S)-connected. The algorithm then finds a subdivision of a Kuratowski graph (K 5 or K 3,3 ). Shiloach gave an O( V (G) 2 ) algorithm to find such a subdivision, but this can be improved as we show below using an algorithm of Hsu and Shih [11]. R4: If a subdivision of K 5 is found, Shiloach claims that the required two disjoint paths can be found in O( V (G) + E(G) ) time, using a result of Watkins [26]. R5 and R6: If a subdivision of K 3,3 is found, then Shiloach s algorithm finds the required two disjoint paths in O( V (G) + E(G) ) time. Let us show how to improve the running time of the algorithm for (4, S)-connected 11

21 graphs. Let G be a graph, let S := {a, b, c, d} V (G) and suppose that G is (4, S)- connected. Let G + := G + {ac, cb, bd, da}. Since G is (4, S)-connected, G + is (4, S)- connected. Because each of a, b, c, d has degree at least three in G +, it follows that G + is 3-connected. Moreover, if there exist disjoint paths P and Q in G + from a to b, and from c to d, respectively, then P and Q are both paths in G, and vice-versa. We describe now how to solve the two disjoint paths problem for G + in O( V (G) + E(G) ) time. Hsu and Shih [11] developed a O( V (H) + E(H) ) algorithm that given a graph H, either finds an embedding of H, or finds a subdivision of a Kuratowski graph in H. Applying this algorithm to G +, either we find an embedding of G +, or, a subdivision of K 5 or K 3,3. If the former occurs, then G + is planar, and we can use step R2 to solve the two disjoint paths problem in O( V (G) + E(G) ) time. Otherwise, there exists a subdivision of K 5 (or K 3,3 ), and we can use steps R4 (or, R5 and R6, respectively) of Shiloach s algorithm to find the required two disjoint paths. Thus, we can find the two disjoint paths P and Q, if they exist, in O( V (G) + E(G) ) time. 2.3 Nonseparating paths The rest of this chapter deals with nonseparating induced paths in graphs with certain connectivity properties. We also show how to find these paths efficiently. We will use the following result. (2.3.1) Theorem. Let G be a 3-connected graph, let e E(G), and let u V (G) be nonincident to e. Then G has a nonseparating induced cycle through e and avoiding u. Moreover, such a cycle can be found in O( V (G) + E(G) ) time. The existence of this cycle was proven by Tutte [25], and the algorithmic part was proven by Cheriyan and Maheshwari [3]. (2.3.2) Lemma. Let G be a connected graph, S V (G), {a, a } S, and let P be an a-a path in G. Suppose (i) G is (3, S)-connected, and 12

22 PSfrag replacements U u a a a a G G Figure 2: Graphs G and G in the proof of (2.3.2). (ii) S {a, a } is contained in a component U of G V (P ). Then there exists a nonseparating induced a-a path P in G such that V (P ) V (U) =. Moreover, such a path can be found in O( V (G) + E(G) ) time. Proof. We may assume that P is induced, otherwise, we can find in O( V (G) + E(G) ) time an induced a-a path in G satisfying (ii). If P is nonseparating, then P := P is the required path. If V (P ) = 2 then by (i) every component of G V (P ) contains a vertex of S, and so by (ii) G V (P ) = U, which implies that P is nonseparating. So we may assume that V (P ) 3 and G V (P ) is not connected. Let G be the graph obtained from G by contracting U to a single vertex u, adding the edges aa, ua and ua, and removing multiple edges. See Figure 2. Note that a, a belong to the cycle P + aa. We claim that H := G u is 2-connected. Suppose for a contradiction that there exists v V (H) such that H v is disconnected. Since a, a are vertices of P +aa which is a cycle in H, there exists a component K in H v which does not contain any vertex of P. But then K is a component of G v which does not contain any vertex in S, contradicting (i). Thus, G u is 2-connected. We now see that G must be 3-connected. Suppose for a contradiction that G is not 3- connected. Then there is a vertex cut set T in G with T 2. Since G u is 2-connected, u T. Moreover, since {u, a, a } induces a triangle in G, there exists a component K of G T which does not contain any of u, a, a. But then K is also a component of G T which does not contain any element of S, contradicting (i). Hence, G is 3-connected. 13

23 By Theorem (2.3.1) (with G, aa, u as G, e, u respectively), there exists a nonseparating induced cycle C in G containing aa and avoiding u. Moreover, such a cycle can be found in O( V (G ) + E(G ) ) time (and hence, in O( V (G) + E(G) ) time). Thus, P := C aa is a nonseparating induced path in G such that V (P ) V (U) =. As an application of (2.3.2), we derive the following strengthening of Lemma (2.2.2). (2.3.3) Lemma. Let G be a graph and S := {a, a, b, b } V (G). Suppose that G is (4, S)-connected. Then exactly one of the following holds: (1) there exists a nonseparating induced a-a path P in G such that V (P ) {b, b } =, or (2) (G, a, b, a, b ) is planar. Moreover, one can in O( V (G) + E(G) ) time find a path as in (1) or certify that (2) holds. Proof. By Lemma (2.2.2), either (a) there exist disjoint paths P and Q in G from a to a and from b to b, respectively, or (b) (G, a, b, a, b ) is planar. Moreover, one can in O( V (G) + E(G) ) time find paths as in (a) or certify that (b) holds. If (b) holds, then (2) holds. So assume (a) holds. Let U be the component of G V (P ) containing S {a, a } = {b, b }. Since G is (4, S)-connected (and hence, (3, S)-connected), G, P, S, U and {a, a } satisfy the hypothesis of (2.3.2). Thus, by (2.3.2) there exists a nonseparating a-a path P in G such that V (P ) V (U) =, and such a path can be found in O( V (G) + E(G) ) time. Hence, V (P ) {b, b } =, and P satisfies (1). The next lemma is a variation of (2.3.2) (and (2.3.3) as well) in which we prove the existence of a specific nonseparating induced path. However, here it is not possible to specify the ends of the desired path. Moreover, in the hypotheses of (2.3.4) there are some technical conditions which arise when we try to produce an internal chain (see Section 4.2. Note that conditions (iii), (iv), and (v) are automatically satisfied if G is (4, S {b, b })- connected. Actually, this is the case in all applications of (2.3.4) with the exception of the proof of (4.2.15), where the more complicated conditions are required. (2.3.4) Lemma. Let G be a graph, let S V (G), and let {b, b } V (G) S. Suppose 14

24 (i) G S is 2-connected, (ii) every element of S has a neighbor in V (G) (S {b, b }), (iii) G is (3, S {b, b })-connected, (iv) if S = 2, then G is (4, S {b, b })-connected, and (v) if S 3, then there exists some component of G (S {b, b }) which has at least two neighbors in S. Then exactly one of the following holds: (1) there exist a, a S and an induced a-a path P in G such that V (P ) {b, b } =, V (P ) S = {a, a }, and G (V (P ) S) is connected; (2) S = 2, and the elements of S can be labelled as a, a such that (G, a, b, a, b ) is planar. Moreover, one can in O( V (G) + E(G) ) time find a path as in (1) or certify that (2) holds. Proof. First, suppose that S = 2. Let a, a denote the vertices in S. By (iv) G is (4, {a, a, b, b })-connected. Thus, by Lemma (2.3.3) exactly one of the following holds: (a) there exists a nonseparating induced a-a path P such that V (P ) {b, b } =, or (b) (G, a, b, a, b ) is planar. Moreover, one can in O( V (G) + E(G) ) time find a path as in (a) or certify that (b) holds. If (a) holds, then P is the required path in (1). If (b) holds, then (2) holds. Thus, we may assume that S 3. First, we prove the following Claim. There exist a, a S and an a-a path Q in G (S {a, a }) such that b and b are contained in a component of G V (Q). Moreover, such a path can be found in O( V (G) + E(G) ) time. Proof of Claim. We consider two cases. Case 1: G (S {b, b }) is not connected. 15

25 In this case, there exist edge-disjoint subgraphs G 1, G 2 of G S such that G 1 G 2 = G S, V (G 1 ) V (G 2 ) = {b, b }, and V (G 1 ) 3 V (G 2 ). Note that such a partition can be found in O( V (G) + E(G) ) time. By (v) there exists some component K of G (S {b, b }) which has at least two neighbors in S. Note that such a component can be found in O( V (G) + E(G) ) time. We may assume that V (K) V (G 1 ). Let a, a N G (K) S and let Q be an a-a path in G[V (K) {a, a }]. Since G 2 is connected, b, b are contained in a component of G V (Q). Moreover, such a path can be found in O( V (G) + E(G) ) time. Case 2: G (S {b, b }) is connected. Since G S is 2-connected by (i), one can find in O( V (G) + E(G) ) time two internally disjoint b-b paths P 1, P 2 in G S. Let a 1, a 2, a 3 be distinct vertices in S. For i = 1, 2, 3, let v i N G (a i ) V (G) (S {b, b }) (they exist by (ii)). Since G (S {b, b }) is connected by assumption, for each i = 1, 2, 3, there exists a path Q i from v i to some vertex u i in (V (P 1 ) V (P 2 )) {b, b } internally disjoint from V (P 1 ) V (P 2 ). Moreover, such paths can be found in O( V (G) + E(G) ) time. Note that at least two (not necessarily distinct) vertices in u 1, u 2, u 3 lie on the same path P 1 {b, b } or P 2 {b, b }. By symmetry, we may assume that u 1, u 2 V (P 1 ) {b, b }. Then there exist disjoint paths in G (S {a 1, a 2 }) from a 1 to a 2 (the path contained in Q 1 Q 2 (P 1 {b, b })) and from b to b (the path P 2 ), respectively. Thus, the result follows by taking a = a 1 and a = a 2. Moreover, it is not hard to see that such paths can be found in O( V (G) + E(G) ) time. Now given a, a and Q, we will describe how to find a S and an induced a-a path P such that V (P ) {b, b } =, V (P ) S = {a, a }, and G (V (P ) S) is connected. Let G be the graph obtained from G by identifying the vertices in S {a} to a single vertex a and removing the resulting multiple edges. Let S := {a, a, b, b }. We claim that G is (3, S )-connected. Suppose for a contradiction that there exists T V (G ) such that T 2 and G T has a component K with V (K) S =. Clearly a T, because G is (3, S {b, b })-connected (by (iii)); then either a T or T {a } is a vertex cut of G S, which is a contradiction since G S is 2-connected (by (i)). Thus, G is (3, S )-connected. 16

26 Note that the a-a path Q in G corresponds to an a-a path P in G, and S {a, a } = {b, b } is contained in a component U of G V (P ). Thus, the hypotheses of (2.3.2) are satisfied with G, S, P, a, a, U as G, S, P, a, a, U respectively. Hence, there exists a nonseparating induced a-a path P in G such that V (P ) V (U) =. Moreover, such a path P can be found in O( V (G ) + E(G ) ) time (hence, in O( V (G) + E(G) ) time). The path P corresponds to an induced a-a path P in G for some a S {a} such that V (P ) {b, b } = and V (P ) S = {a, a }. Since P is nonseparating in G, G (V (P ) S) is connected. Therefore, a, a and P satisfy (1), and they can be found in O( V (G) + E(G) ) time. The following lemma is a variation of (2.3.4) (by letting b = b ), and its proof is essentially the same. For the sake of completeness, we include it. (2.3.5) Lemma. Let G be a graph, let S V (G), and let b V (G) S. Suppose (i) G S is 2-connected, (ii) every element of S has a neighbor in V (G) (S {b}), and (iii) G is (3, S {b})-connected. Then there exist a, a S and an induced a-a path P in G such that V (P ) {b} =, V (P ) S = {a, a }, and G (V (P ) S) is connected. Moreover, such a path can be found in O( V (G) + E(G) ) time. Proof. Since G is (3, S {b})-connected (by (iii)), S 2, so let a, a S. Since G S is 2-connected (by (i)), G (S {b}) is connected. Since a and a have a neighbor in V (G) (S {b}) (by (ii)), there exists an a-a path Q in G ((S {a, a }) {b}). Clearly, such a path can be found in O( V (G) + E(G) ) time. Let G be the graph obtained from G by identifying the vertices in S {a} to a single vertex a and removing the resulting multiple edges. Let S := {a, a, b}. We claim that G is (3, S )-connected. Suppose for a contradiction that there exists T V (G ) such that T 2 and G T has a component K with V (K) S =. Clearly, 17

27 a T because G is (3, S {b})-connected (by (iii)). But then either a T or T {a } is a vertex cut of G S, which is a contradiction since G S is 2-connected (by (i)). Thus, G is (3, S )-connected. Note that the a-a path Q in G corresponds to an a-a path P in G, and S {a, a } = {b} is contained in a component U of G V (P ). Thus, by (2.3.2) (with G, S, P, a, a, U as G, S, P, a, a, U respectively), there exists a nonseparating induced a-a path P in G such that V (P ) V (U) =. Moreover, such a path P can be found in O( V (G ) + E(G ) ) time (and hence, in O( V (G) + E(G) ) time). The path P corresponds to an induced a-a path P in G for some a S {a} such that V (P ) {b} = and V (P ) S = {a, a }. Since P is nonseparating in G, G (V (P ) S) is connected. So a, a and P are as required, and they can be found in O( V (G) + E(G) ) time. The following result of Cheriyan and Maheshwari ([3], p. 516) is in fact the core of the linear algorithm in [3] for finding a nonseparating induced cycle as described in Theorem (2.3.1). (2.3.6) Theorem. Let G be a 3-connected graph, let aa E(G), and let C be a nonseparating induced cycle in G containing aa. Then there exists another nonseparating induced cycle C in G such that V (C ) V (C) = {a, a } and E(C ) E(C) = {aa }. Moreover, such a cycle can be found in O( V (G) + E(G) ) time. Our next result is in the same spirit as of (2.3.6), but we relax the 3-connectivity condition. Therefore, it is more convenient to use. (2.3.7) Lemma. Let G be a connected graph, let a, a be distinct vertices of G with degree at least two, and let P be a nonseparating induced a-a path in G. Suppose that G is (3, V (P ))-connected. Then there exists another nonseparating induced a-a path P in G such that V (P ) V (P ) = {a, a } and E(P ) E(P ) =. Moreover, such a path can be found in O( V (G) + E(G) ) time. Proof. For convenience, let H := G V (P ). Since P is a nonseparating path in G, H is connected. Moreover, both a and a have a neighbor in V (H) because both have degree at 18

28 least two in G and P is induced. Let v 1 = a, v 2,..., v k = a be the neighbors of H on P PSfrag replacements in this order from a to a (see Figure 3 for an illustration). Note that k 3 because G is (3, V (P ))-connected. H H P C a = v 1 v a = v 6 a = v 1 a 2 v 3 v4 v 5 v 2 v 3 v 4 v 5 = v 6 G G Figure 3: G and G as in the proof of Lemma (2.3.7). Let G be the graph obtained from G by adding the edge aa, and by replacing, for each 1 i k 1, the path P [v i, v i+1 ] by an edge v i v i+1. Note that C := G V (H) is a cycle in G. See again Figure 3. We claim that G is 3-connected. Suppose for a contradiction that G is not 3-connected. Then there is a vertex cut T in G with T 2. Note that T V (C), since H is connected and every vertex of C has a neighbor in H. But then G T has a component K such that V (K) V (C) = V (K) V (P ) =. Hence, K is also a component of G T with V (K) V (P ) =, contradicting the assumption that G is (3, V (P ))-connected. Hence, G is 3-connected. By (2.3.6) (with G, C, a, a as G, C, a, a respectively), there exists a nonseparating cycle C in G such that V (C ) V (C) = {a, a } and E(C ) E(C) = {aa }. Moreover, such a cycle can be found in O( V (G ) + E(G ) ) time (and hence, in O( V (G) + E(G) ) time). Thus, P := C aa is a nonseparating induced a-a path in G such that V (P ) V (P ) = {a, a } and E(P ) E(P ) =. 19

29 CHAPTER III NONSEPARATING AND CYCLIC CHAINS In order to describe precisely our decomposition, we must define the chain and planar chain. Moreover, we must show that these chains exist and provide an algorithm to extract them from a 4-connected graph. These are the goals of this chapter. 3.1 Planar chains A block of a graph G is either a maximally 2-connected subgraph of G, or a subgraph of G induced by a cut edge. A block is nontrivial if it is 2-connected, and trivial if it is induced by a cut edge. (3.1.1) Definition. A connected graph H is a chain if its blocks can be labelled B 1,..., B k, where k 1 is an integer, and its cut vertices can be labelled v 1,..., v k 1 such that (i) V (B i ) V (B i+1 ) = {v i } for 1 i k 1, and (ii) V (B i ) V (B j ) = if i j 2 and 1 i, j k. PSfrag We replacements let H := B 1 v 1 B 2 v 2... v k 1 B k denote this arrangement. If k 2, v 0 V (B 1 ) {v 1 } and v k V (B k ) {v k 1 }, or if k = 1, {v 0, v k } V (B 1 ), and v 0 v k, then we say that H is a v 0 -v k chain; we denote this by H := v 0 B 1 v 1... v k 1 B k v k. We usually fix v 0 and v k, and we refer to them as the ends of H i. See Figure 4 for an example with k = 5. B 1 B 2 B 3 B 4 B 5 v 0 v 1 v 2 v 3 v 4 v 5 Figure 4: Example of a chain. 20

30 (3.1.2) Definition. Let G be a graph and let H := v 0 B 1 v 1... v k 1 B k v k be a chain. If H is an induced subgraph of G, then we say that H is a chain in G. We say that H is a planar chain in G if, for each 1 i k with V (B i ) 3 (or equivalently, B i is 2- connected), there exist distinct vertices x i, y i V (G) V (H) such that (G[V (B i ) {x i, y i }] x i y i, x i, v i 1, y i, v i ) is planar, and B i {v i 1, v i } is a component of G {x i, y i, v i 1, v i }. We also say that H is a planar v 0 -v k chain. See Figure 5 for two drawings of an example with PSfrag replacements k = 5. r G V (H) v 5 v 5 v 6 y 2 = x 4 y 5 B 5 x 5 y 5 y 4 B 4 B 5 B 5 x 5 v 4 v4 y 4 v 3 y 2 = x 4 v B 3 3 v B 3 2 B 2 v 2 B 2 B 2 v 1 v 1 B 4 B 4 B 6 x 2 x 2 G V (H) B 1 B1 v 0 v 0 Figure 5: A planar chain H := v 0 B 1 v 1 B 2 v 2 B 3 v 3 B 4 v 4 B 5 v 5 in a graph G. The dashed edges may or may not exist, but they are not part of H. Now we are ready to describe the main result of Chapter 3. It is stated in a form which can be used conveniently in following chapters. See Figure 6 for an illustration of the hypotheses of the theorem. (3.1.3) Theorem. Let G be a graph, let a, b be distinct vertices of G, let P be a nonseparating induced path in G between a and b, let B P be a nontrivial block of G V (P ), and let X P := N G (G V (B P )). Suppose G (V (B P ) X P ) is (4, X P {a, b})-connected. Then there exists a planar a-b chain H in G such that B P G V (H) and G V (H) is 21

31 2-connected. Moreover, such a chain can be found in O( V (G) E(G) ) time. b b X P X P PSfrag replacements P P r cut vertices of G V (P ) N G (P ) V (B P ) B P a G G (V (B P ) X P ) a Figure 6: G, a, b, P, B P, X P in (3.2.1). 3.2 Nonseparating chains The main goal of this section is to design an algorithm to solve the following problem: given G, a, b, P, B P as in (3.1.3), find a planar a-b chain H in G such that G V (H) is 2-connected and V (B P ) V (G) V (H). For convenience, we fix the following notation throughout this section. (3.2.1) Notation and assumptions. Let G be a graph, let a, b be distinct vertices of G, let P be a nonseparating induced a-b path in G, let B P be a nontrivial block of G V (P ), and let X P := N G (G V (B P )). Suppose that G (V (B P ) X P ) is (4, X P {a, b})-connected. See Figure 6. Let P P be the set of nonseparating induced a-b paths P in G with B P G V (P ). Note that P P P. For each P P P let B P denote the nontrivial block of G V (P ) containing B P. We say that P P P is a B P -augmenting path if V (B P ) < V (B P ). 22

32 Note that X P consists of cut vertices of G V (P ) contained in V (B P ) and the neighbors of V (P ) contained in V (B P ). Our next result shows that the paths in P P are well-behaved. (3.2.2) Lemma. Let P P P. Let X P := N(G V (B P )). Then G (V (B P ) X P ) is (4, X P {a, b})-connected. Proof. For convenience, let G := G (V (B P ) X P ). Suppose for a contradiction that G is not (4, X P {a, b})-connected. Then there exists some T V (G ) with T 3 and there exists some component K of G T such that V (K) (X P {a, b}) =. Since V (B P ) V (B P ), for each x X P, either x V (G ) or x X P. Thus, V (K) X P =. But then, K is a component of (G (V (B P ) X P )) T which does not contain any vertex in X P {a, b}. This contradicts the assumption that G (V (B P ) X P ) is (4, X P {a, b})- connected. Therefore, G is (4, X P {a, b})-connected. Let us describe the basic idea of the algorithm we will design. At the beginning of each iteration we have a nonseparating a-b path P and a nontrivial block B P of G V (P ). The algorithm then tries to find a B P -augmenting path P P P. If the algorithm finds such a path P, then it starts a new iteration with P as P (note that, by Lemma (3.2.2), G (V (B P ) X P ) is (4, X P {a, b})-connected). If the algorithm does not find a B P - augmenting path, then it finds a planar a-b chain as required in (3.1.3). In order to describe this algorithm more precisely, we require the following definitions and lemmas. (3.2.3) Notation. Let B denote the set of (nontrivial) B P -bridges of G V (P ). For each B B, V (B P ) V (B) consists of exactly one vertex (which is contained in X P ), and we let r B denote this vertex. For any x, y V (P ), we denote x y if x V (P [a, y]). If x y and x y, then we write x < y. In this case, we say that x is lower than y, or y is higher than x. Since G is (4, X P {a, b})-connected, for each B B, B r B has at least three neighbors on P. Let l B and h B denote the lowest and the highest neighbor of B r B on P, respectively. See Figure 7 for an example. (3.2.4) Lemma. The following holds for any B B: 23

33 (1) V (P (l B, h B )) and N G (P (l B, h B )) (V (B) {r B }), and (2) N G (P (l B, h B )) V (B) V (P ). Proof. (1) holds because B r B has at least three neighbors on P, and (2) holds because P is an induced path in G and {r B, l B, h B } is not a 3-vertex cut of G. Next, we describe members of B which we can use to produce a B P -augmenting path. (3.2.5) Definition. For each vertex x of G V (P ), we define x as follows. If x V (B) for some B B, then let x := r B. If x V (B P ), then x := x. We say that a member B of B is a nice bridge if there exist x, y N G (P (l B, h B )) ((V (B) {r B }) V (P )) such that x y. See Figure 7 for an example. PSfrag replacements y y b r x h B r B = x B P B r B B P y z = z x x l B a Figure 7: A nice bridge B and the graph G B (in boldface) as defined in the proof of Lemma (3.2.6). The next lemma shows that any nice bridge can be used to find a B P -augmenting path. (3.2.6) Lemma. Let B B be a nice bridge. Then there exists an induced l B -h B path Q in G[V (B) {l B, h B }] such that P := (P V (P (l B, h B ))) Q is a B P -augmenting path in G. Moreover, such a path Q can be found in O( V (G) + E(G) ) time. 24

34 Proof. Since B is a nice bridge, there exist x, y N G (P (l B, h B )) ((V (B) {r B }) V (P )) such that x y. See Figure 7. Let G B be the subgraph of G induced by (V (B) {r B }) V (P [l B, h B ]). Since B is a B P -bridge, B r B is connected. Thus, P [l B, h B ] is a nonseparating induced path in G B. Furthermore, since G is (4, X P {a, b})-connected, for any T V (G B ) with T 2, every component of G B T contains a vertex of V (P [l B, h B ]) (otherwise, T {r B } is a 3-cut of G, and G (T {r B }) has a component not containing any element of X P {a, b}). Thus, G B is (3, V (P [l B, h B ]))-connected. By Lemma (2.3.7) (with G B, l B, h B, P [l B, h B ] as G, a, a, P respectively), there exists a nonseparating induced l B -h B path Q in G B disjoint from P (l B, h B ). Moreover, such a path Q can be found in O( V (G B ) + E(G B ) ) time. Since V (G B ) + E(G B ) = O( V (G) + E(G) ), such a path Q can be found in O( V (G) + E(G) ) time. Clearly, the path P = (P V (P (l B, h B ))) Q is an induced a-b path in G. Let us prove that P is nonseparating in G. It suffices to prove that for every v / V (B P ) V (P ), there exists a {v}-v (B P ) path in G V (P ). First, suppose v V (B ) for some B B with B B. Since V (B ) V (P ) =, there exists a v-r B path in B (and hence in G V (P )). So we may assume v (V (B) {r B }) V (P (l B, h B )). Since N G (P (l B, h B )) V (B) V (P ) (by (2) of (3.2.4)) and V (Q(l B, h B )) V (P (l B, h B )) =, and because Q is a nonseparating path in G B, there exists a v-v (B P ) path in G V (P ). Hence, P is nonseparating in G. Thus, P P P. It remains to show that V (B P ) < V (B P ). Note that G contains disjoint paths P x and P y from x to x and from y to y, respectively, and P x and P y are disjoint from P (B r B ) (B P {x, y }). Let x, y V (P (l B, h B )) such that xx, yy E(G). Then both B P and the path (P x P [x, y ] P y ) + {xx, yy } are contained in B P. Hence, V (B P ) < V (B P ), and so, P is a B P -augmenting path. In what follows we prove several lemmas which will help us find nice bridges (and hence, B P -augmenting paths by (3.2.6)). But first, we need the following. (3.2.7) Definition. We say that two B P -bridges B and B in B are overlapping if the paths P [l B, h B ] and P [l B, h B ] have an edge in common. Define an auxiliary graph K such 25