Erdős-Pósa Property and Its Algorithmic Applications Parity Constraints, Subset Feedback Set, and Subset Packing

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1 Erdős-Pósa Property and Its Algorithmic Applications Parity Constraints, Subset Feedback Set, and Subset Packing Naonori Kakimura Ken-ichi Kawarabayashi Yusuke Kobayashi Abstract The well-known Erdős-Pósa theorem says that for any integer k and any graph G, either G contains k vertexdisjoint cycles or a vertex set X of order at most c k log k (for some constant c) such that G X is a forest. Thomassen [39] extended this result to the even cycles, but on the other hand, it is well-known that this theorem is no longer true for the odd cycles. However, Reed [31] proved that this theorem still holds if we relax k vertex-disjoint odd cycles to k odd cycles with each vertex in at most two of them. These theorems initiate many researches in both graph theory and theoretical computer science. In the graph theory side, our problem setting is that we are given a graph and a vertex set S, and we want to extend all the above results to cycles that are required to go through a subset of S, i.e., each cycle contains at least one vertex in S (such a cycle is called an S-cycle). It was shown in [20] that the above Erdős-Pósa theorem still holds for this subset version. In this paper, we extend both Thomassen s result and Reed s result in this way. In the theoretical computer science side, we investigate generalizations of the following well-known problems in the framework of parameterized complexity: the feedback set problem and the cycle packing problem. Our purpose here is to consider the following problems: the feedback set problem with respect to the S-cycles, and the S-cycle packing problem. We give the first fixed parameter algorithms for the two problems. Namely; 1. For fixed k, we can either find a vertex set X of size k such that G X has no S-cycle, or conclude that such a vertex set does not exist in O(n 2 m) time (independently obtained in [7]). 2. For fixed k, we can either find k vertex-disjoint S-cycles, or conclude that such k disjoint cycles do not exist in O(n 2 m) time. We also extend the above results to those with the parity constraints as follows; 1. For a parameter k, there exists a fixed parameter algorithm that either finds a vertex set X of size k such that G X has no even S-cycle, or concludes that such a vertex set does not exist. 2. For a parameter k, there exists a fixed parameter algorithm that either finds a vertex set X of size k such University of Tokyo, Tokyo , Japan. Partly supported by Grant-in-Aid for Scientific Research and by Global COE Program The research and training center for new development in mathematics, MEXT, Japan. {kakimura, kobayashi}@mist.i.u-tokyo.ac.jp National Institute of Informatics, , Japan. Partly supported by JSPS, Grant-in-Aid for Scientific Research, by C & C Foundation, and by Inoue Research Award for Young Scientists. k keniti@nii.ac.jp that G X has no odd S-cycle, or concludes that such a vertex set does not exist. 3. For a parameter k, there exists a fixed parameter algorithm that either finds k vertex-disjoint even S- cycles, or concludes that such k disjoint cycles do not exist. 4. For a parameter k, there exists a fixed parameter algorithm that either finds k odd S-cycles with each vertex in at most two of them, or concludes that such k cycles do not exist. 1 Introduction Packing and covering are one of the central areas in both graph theory and theoretical computer science. The starting point of this research area goes back to the following well-known theorem due to Erdős and Pósa [10] in early 1960 s. Theorem 1.1. (Erdős and Pósa [10]) For any integer k and any graph G, either G contains k vertexdisjoint cycles or a vertex set X of order at most c k log k (for some constant c) such that G X is a forest. Theorem 1.1 is concerned with both packing, i.e., finding k disjoint cycles, and covering, i.e., finding at most ck log k vertices that hit all the cycles in a graph. Starting with this theorem, there are many results in this direction in both graph theory and theoretical computer science. In fact, Theorem 1.1 gives rise to the Erdős-Pósa property for a family of graphs. A family F of graphs is said to have the Erdős-Pósa property if for every integer k there is an integer f(k, F) such that every graph G contains either k vertex-disjoint subgraphs each isomorphic to a graph in F or a set X of at most f(k, F) vertices such that G X has no subgraph isomorphic to a graph in F. The term Erdős- Pósa property arose because of Theorem 1.1 which proves that the family of cycles has this property. In this paper, we study the Erdős-Pósa property for the family of cycles under various constraints and its algorithmic aspects. 1.1 Erdős-Pósa Property with Parity Constraints We first discuss a generalization of Theorem 1726 Copyright SIAM.

2 1.1 to impose parity constraints on the length of cycles. Thomassen [39] showed that the Erdős-Pósa property holds for the family of cycles of length divisible by a fixed p. This immediately provides the Erdős-Pósa property for the family of even cycles, i.e., the cycles of even length. Wollan [40] discusses the Erdős-Pósa property for the family of cycles under another kind of modularity constraints. However, the family of odd cycles is known not to have the Erdős-Pósa property. In fact, Lovász and Schrijver (see also Dejter and Neumann-Lara [8]) give infinitely many pairs l and m such that the Erdős-Pósa property does not hold for the family of cycles of length l mod m. On the other hand, Reed [31] showed that a half-integral packing of odd cycles has the Erdős-Pósa property. Here a half-integral packing of k odd cycles is a set of k odd cycles such that each vertex is in at most two of these odd cycles. Reed [31] also found the only obstruction preventing the Erdős-Pósa property for the odd cycles from holding. An Escher wall is a wall of height h together with h twisted paths from the top to the bottom of the wall that are internally disjoint from the wall (see e.g., [31] for the precise definition). It is shown in [31] that for a positive integer k, there exists a constant f(k) such that if a graph G does not contain an Escher wall then G has either k vertex-disjoint odd cycles or a vertex set X V with size f(k) such that G X has no odd cycle. 1.2 Erdős-Pósa Property through Prescribed Vertex Set Another direction to generalize the cycle packing problem is packing cycles that are required to go through a given set of vertices. For a graph G = (V, E) with S V, an S-cycle is a cycle which has a vertex in S. The problem called S-cycle packing is that we are given a graph G and a subset S of its vertices, and the goal is to find as many vertex/edge-disjoint cycles that intersect S as possible. For this problem, approximation algorithms have been studied in theoretical computer science (e.g., [29]), while it seems that the Erdős-Pósa type result has not been explored yet. This is one of our motivations of this paper. It was proved in [20] that the Erdős-Pósa type result holds for the S-cycle packing problem. Namely: Theorem 1.2. Let k be a positive integer. Then there exists a constant f(k) such that any graph G = (V, E) with S V has either k vertex-disjoint S-cycles or a vertex set X of order at most f(k) such that G X has no S-cycles. Note that the case where S coincides with the whole vertex set V, any cycle in a graph is an S-cycle. Thus this is a generalization of Theorem 1.1 to the subset version. Recently, Pontecorvi and Wollan [30] showed f(k) = Θ(k log k), which is the same bound as Theorem Our Contribution for the Erdős-Pósa Property In this paper, we first present a common generalization of Thomassen s classical result [39] and Theorem 1.2. That is, we show the Erdős-Pósa property for the family of S-cycles of length divisible by a fixed p. Theorem 1.3. Let k and p be positive integers. Then there exists a constant f(k, p) such that any graph G = (V, E) with S V has either k vertex-disjoint S-cycles, each of which has length divisible by p, or a vertex set X of order at most f(k, p) such that G X has no such S-cycles. As a corollary, we know that the family of even S-cycles, i.e., when p = 2, has the Erdős-Pósa property. Corollary 1.1. Let k be a positive integer. Then there exists a constant f(k) such that any graph G = (V, E) with S V has either k vertex-disjoint even S- cycles or a vertex set X of order at most f(k) such that G X has no even S-cycles. We also extend the Reed s result [31] mentioned above to packing odd S-cycles in a half-integral way. Theorem 1.4. Let k be a positive integer. Then there exists a constant f(k) such that any graph G = (V, E) with S V has either a half-integral packing of k odd S-cycles or a vertex set X of order at most f(k) such that G X has no odd S-cycles. One might expect to extend Reed s result for the obstruction to packing odd S-cycles, that is, if a graph G does not contain an Escher wall, then the Erdős- Pósa property for odd S-cycles holds. However, this extension does not hold. In fact, consider the graph depicted as in Figure 1, i.e., the graph G defined to be an elementary wall of height h = Ω( V ) with h edges connecting the bottom vertices and h paths of length two between the top vertices that are internally disjoint from the wall. Define a vertex set S to be the set of vertices not in the wall. This graph contains no Escher wall, but it is easy to check that G has no two vertexdisjoint odd S-cycles, while we have to delete at least h = Ω( V ) vertices to make G have no odd S-cycles. 1.4 Subset Feedback Set and S-cycle Packing in view of Fixed Parameter Tractability The Erdős- Pósa property also initiates some areas in theoretical computer science. This property is clearly related to 1727 Copyright SIAM.

3 S S S Figure 1: An example not satisfying the Erdős-Pósa property for odd S-cycles. the cycle packing problem, which asks to find maximum number of vertex-disjoint(or edge-disjoint) cycles in an input graph G. For the edge-disjoint variant of this problem, Caprara et al. [5] designed an O(log n)-approximation algorithm, where n is the number of vertices of the input graph, and Krivelevich et al. [29] proved that this algorithm yields O( log n)- approximation factor. For the vertex-disjoint one, the proof of Theorem 1.1 due to Simonovits [37] leads to an O(log n)-approximation algorithm. For the S-cycle packing problem, Pontecorvi and Wollan [30] showed that their proof of Theorem 1.2 derives an O(log n)- approximation algorithm for both vertex/edge-disjoint cases. Also, covering leads to the well-known concept feedback set in theoretical computer science. The problem of finding a minimum feedback vertex set (FVS) in a graph, i.e., the smallest set of vertices whose deletion makes the graph acyclic, has many applications and its history can be traced back to the early 60 s (see the survey of Festa et al. [12]). It is also one of the classical NP-complete problems from Karp s list [21]. Thus not surprisingly, for several decades, many different algorithmic approaches were tried on this problem including approximation algorithms [2, 3], linear programming [6], polyhedral combinatorics [4, 14], exact algorithms [13] and parameterized complexity [16]. The problem called subset feedback set is a natural generalization of the feedback set problem, which corresponds to the linear programming dual of the S-cycle packing problem. The subset feedback set problem is that we are given a graph G and a subset S of its vertices, and the goal is to find a vertex set X of minimum order such that G X has no S-cycle. For this problem, Even et al. [11] gives an 8-approximation algorithm. In this paper, we are interested in the framework of parameterized complexity developed by Downey and Fellows [9] for both the S-cycle packing and subset feedback set problems. The standard goal of parameterized analysis is to take some parameter, say k, out of the exponent in the running time. A problem is called fixedparameter tractable (FPT) if it can be solved in time O(f(k)n c ), where n is the number of vertices of the input graph, c is a constant not depending on k, and f is an arbitrary function. An algorithm with such a running time is also called FPT. We can trivially determine whether or not G has a vertex set X of order at most k such that G X has no S-cycle in O(n k+2 ) time by enumerating all k-tuples of vertices in G. Although this is polynomial time for each fixed k, it is practically too slow for large inputs, even if k is relatively small. Our first main result here is an FPT algorithm for the subset feedback set problem. For a vertex set S V, we say that a vertex set X V is an S-cycle feedback vertex set (S-FVS for short) if G X contains no S-cycles. Theorem 1.5. For a graph G = (V, E), a vertex set S V, and a fixed integer k, we can either find an S- cycle feedback vertex set X of size k, or conclude that such a vertex set does not exist in O(n 2 m) time, where n is the number of vertices and m is the number of edges. Note that, in 2010, an FPT algorithm for the subset feedback set problem was also given in [7] independently. But our algorithm has a few more appealing points. Our proof is somehow shorter, and in addition, the same framework can be applied to solve the S-cycle packing problem below and these parity-constrained problems as in Section 1.5. Our second result is the first FPT algorithm for the S-cycle packing problem. Theorem 1.6. For a graph G = (V, E), a vertex set S V, and a fixed integer k, we can either find k vertexdisjoint S-cycles, or conclude that such k vertex-disjoint cycles do not exist in O(n 2 m) time. Let us observe that if S = V (G), then we can find k vertex-disjoint cycles in linear time for fixed k, if they exist. Indeed, if a given graph has large tree-width, then we can do this from the existence of a large grid minor, and otherwise we can use the dynamic programming to find vertex-disjoint cycles. On the other hand, this would not work for the S- cycle packing problem. In fact, the problem setting is closer to the well-known disjoint paths problem for a fixed number of terminals [33] as pointed out in [29]. Using the result in [33], we can determine whether or not G has k vertex-disjoint S-cycles in O(n 2k+3 ) time as follows: we enumerate all k pairs of vertices such that each pair contains at least one vertex in S, and 1728 Copyright SIAM.

4 for such pairs we apply Robertson-Seymour s O(n 3 ) time algorithm for finding k vertex-disjoint paths [33]. We emphasize here that even obtaining an n O(k) time algorithm is non-trivial without using the graph minor theory, which implies that the S-cycle packing problem is harder than the subset feedback set problem. Thus we shall use some tools from the graph minor theory to obtain our results. 1.5 Parity-Constrained Subset FVS and S- cycle Packing Since we have shown the Erdős-Pósa property for the family of S-cycles with parity conditions, we also investigate parameterized complexity for the parity-constrained version of subset FVS and S- cycle packing. That is, we further generalize the subset feedback set problem and the S-cycle packing problem to those with parity constraints on the length of cycles. Let us remark that our problem setting includes packing and covering parity-constrained cycles (when S = V ), especially packing odd cycles [27, 35] and odd cycle transversal [26, 32]. Our contributions are summarized in Table 1. Concerning the S-cycle feedback set problems with parity constraints, we show the following two results. For a vertex subset S V, we say that a vertex set X V is an even/odd S-cycle feedback vertex set (even/odd S-FVS for short, respectively) if G X contains no even/odd S-cycles. Theorem 1.7. For a graph G = (V, E), a vertex set S V, and a fixed integer k, we can either find an even S-cycle feedback vertex set X of size k, or conclude that such a vertex set does not exist in O(n 3 m) time, where n is the number of vertices and m is the number of edges. Theorem 1.8. For a graph G = (V, E), a vertex set S V, and a fixed integer k, we can either find an odd S-cycle feedback vertex set X of size k, or conclude that such a vertex set does not exist in O(n 3 m) time, where n is the number of vertices and m is the number of edges. Note that Theorem 1.7 generalizes an FPT algorithm for the problem without parity constraint (see Theorem 1.5) because if we subdivide each edge once, then the resulting graph is bipartite, and an even S-FVS given by Theorem 1.7 in the bipartite graph clearly gives rise to an S-cycle feedback set in the original graph. Concerning the even/odd S-cycle packing problems, we show the following two results. Theorem 1.9. For a graph G = (V, E), a vertex set S V, and a parameter k, there exists an FPT algorithm that either finds k vertex-disjoint even S- cycles, or concludes that such k vertex-disjoint cycles do not exist. Theorem For a graph G = (V, E), a vertex set S V, and a parameter k, there exists an FPT algorithm that either finds a half-integral packing of k odd S-cycles, or concludes that such k cycles do not exist. Again, Theorem 1.9 generalizes an FPT algorithm for the problem without parity constraint (see Theorem 1.6) because if we subdivide each edge once, then the resulting graph is bipartite, and k vertex-disjoint even S-cycles given by Theorem 1.9 in the bipartite graph clearly gives rise to k vertex-disjoint S-cycles in the original graph. 1.6 Organization In this paper, we mainly discuss parameterized complexity for packing S-cycles, i.e., the proof of Theorem 1.6. The details of other results can be found in the full version [17, 18, 19, 24]. Since our FPT algorithm follows the framework of Robertson- Seymour s algorithm for the disjoint paths problem [33], we provide some useful tools in graph minor theory and overview their algorithm in Section 2. In Section 3, we describe the proof of Theorem 1.6, and finally we present proof sketches of other theorems. 2 Preliminaries 2.1 Odd Paths, S-paths, and Odd Clique Models We first recall the results of Geelen et al. [15] for packing odd paths. Theorem 2.1. (Geelen et al. [15]) Let G = (V, E) be a graph with T V. Then, in O(kn 2 ) time, we can find either k vertex-disjoint paths, each of which has an odd number of edges and its end points in T, or a vertex set Z V with Z 2k 2 that intersects every such path. For S, T V, an S-path with respect to T is a path with end vertices in T such that it has at least one vertex of S. The following theorem is derived in [20] from Theorem 2.1. Theorem 2.2. Let G = (V, E) be a graph, and S, T V. Then, in O(kn 2 ) time, we can find either k vertex-disjoint S-paths with respect to T, or a vertex set Z V with Z 2k 2 that intersects every S-path with respect to T Copyright SIAM.

5 Table 1: Summary of Our Contribution No parity Even constraint Odd constraint Erdős-Pósa property [20] this paper this paper (half integral) FPT for subset FVS [7] and this paper this paper this paper FPT for S-cycle packing this paper this paper this paper (half integral) For an integer p, K p is the complete graph with p vertices. A graph G contains a K p -model if there exists a function σ with domain V (K p ) E(K p ) such that 1. for each vertex v V (K p ), σ(v) is a tree of G, and the trees σ(v) (v V (K p )) are pairwise vertexdisjoint, and 2. for each edge e = uv E(K p ), σ(e) is an edge f E(G), such that f is incident in G with a vertex in σ(u) and with a vertex in σ(v). We also say that G contains a K p -minor if and only if G contains a K p -model. We call the tree σ(v) (v V (K p )) the node of the K p -model. The image of σ, which is a subgraph of G, is called the K p -model. A K p -model K is even if K is bipartite. It is known that every graph containing a large clique model has an even clique model. More precisely, it was shown in [15] that there exists a constant c such that every graph containing a K t -model with t c p log p also contains an even K p -model. For a bipartite subgraph H, a path in G is parity-breaking with respect to H if its end points are in H and the parity is different from the paths in H between them. A block of a graph means a maximal subgraph with the property that it is either 2-connected or a 1- or 2-vertex complete graph. We say that a K p -model K is odd if for each cycle C of K the number of edges of C that belong to clique nodes is even. Let σ be a K p -model, and v V (K p ). A center for σ(v) is a vertex t V (σ(v)) such that for each component H of σ(v) t, the number of edges e E(K p ) such that σ(e) is incident in G with a vertex of H is at most half the number of edges in K p incident with v. It is not hard to see that every node σ(v) has a center (perhaps more than one). Thus we assume that for each node, one of its centers has been selected, and we often speak of the center of a node without further explanation. Proposition 2.1. (Geelen et al. [15]) Let K be an even K 12p -model in G. (a) If there are 4p vertex-disjoint parity-breaking paths with respect to K such that the 8p endpoints of these paths are the centers of distinct nodes of K, then one can find in O(kn 2 ) time an odd K p -model in K from the union of K and such 4p disjoint paths. (b) Otherwise, i.e., if G has no such 4p disjoint paritybreaking paths, then one can find in O(kn 2 ) time Z V (G) with Z 8p 2 such that the (unique) block of G Z that intersects all the clique nodes disjoint from Z is bipartite. Note that we can decide whether (a) or (b) holds in O(kn 2 ) time by Theorem 2.1. Proposition 2.1 implies the following lemma. Lemma 2.1. Let p be a positive integer, and K be a K 12p -model in a graph G. (a) If there are 4p vertex-disjoint S-paths with respect to V (K) such that the 8p endpoints of these paths are the centers of distinct nodes of K, then one can find in O(kn 2 ) time a K p -model such that each cycle in it has a vertex of S. (b) Otherwise, i.e., if G has no such 4p disjoint S- paths, one can find in O(kn 2 ) time Z V (G) with Z 8p 2 such that the (unique) block of G Z that intersects all the clique nodes disjoint from Z has no vertices of S. 2.2 Flat and Dividing Walls Let h be a positive integer. A wall W of height h is defined to be a graph which is isomorphic to a subdivision of the graph W h with vertex set V (W h ) = {(i, j) 0 i h, 0 j 2h} in which two vertices (i, j) and (i, j ) are adjacent if and only if either (1) i = i and j {j 1, j + 1} or (2) j = j and i = i + ( 1) i+j. The graph W h is called an elementary wall. An example of a wall can be found in Figure 2. A cycle of length six in W h is called a brick of W h. A brick of a wall is defined similarly. The nails of a wall are the vertices of degree three within it. Any wall has a unique planar embedding. The perimeter of a wall W, denoted per(w ) is the unique face in this embedding which contains more than six nails. The set of the nails on the perimeter, called perimeter-nails, is denoted by pn(w ). For any wall W in a graph G, there is a unique component U of G per(w ) containing W per(w ). We call U 1730 Copyright SIAM.

6 Figure 2: A wall of height 3 It is known that the folio can be solved in polynomial time if the tree-width is bounded. Theorem 2.3. (See [1, 33]) For integers w and k, there exists a (k + w) O(k+w) O(n 2 ) time algorithm for computing the folio relative to a set of k vertices in graphs of tree-width w. Furthermore, if w and k are fixed, there exists an O(n) time algorithm. the interior of W, denoted by int(w ). The compass of W, denoted comp(w ), is a subgraph induced by V (int(w )) V (per(w )). A subwall of a wall W is a wall which is a subgraph of W. A subwall of W of height h is proper if it consists of h consecutive bricks from each of h consecutive rows of W. A wall is flat if its compass does not contain two vertex-disjoint paths connecting the diagonally opposite corners. Note that if the compass of W has a planar embedding whose infinite face is bounded by the perimeter of W then W is clearly flat. It is shown in [36, 38] that a wall W is flat if and (C1) X is a vertex set with X ( p 2), only if there are pairwise disjoint sets A 1,..., A l (C2) W is a flat wall of height h in G X, V (comp(w )) containing no corners of W such that (1) for 1 i, j l with i j, N(A i ) A j =, where N(A i ) is the set of neighbors of A i, (2) for 1 i l, N(A i ) 3, and (3) if W is the graph obtained from comp(w ) by deleting A i and adding new edges joining every pair of distinct vertices in N(A i ) for each i, then W may be drawn in a plane so that all corners of W are on the outer face boundary. If such A 1,..., A l exist, we say that comp(w ) can be embedded into a plane up to 3-separations, and an embedding as in (3) is called a flat embedding. 2.3 Folio As we mentioned in Section 1, we use some tools from graph minor theory. In this subsection, we state some results of Robertson and Seymour. In [33], Robertson and Seymour gave a polynomialtime algorithm for the disjoint paths problem for a fixed number of terminals. Actually, they solved a generalized problem called folio. For a vertex set X, a partition X = {X 1,..., X q } of X is realizable if there are disjoint trees T 1,..., T q in G such that X i V (T i ) for i = 1,..., q. We say that a vertex v V \ X is irrelevant with respect to X when a partition of X is realizable in G v if and only if it is also realizable in G. The list of realizable partitions of X is called the folio relative to X, and the problem of computing it is also called the folio. When tree-width is large, in Robertson-Seymour s algorithm for the disjoint paths problem or the folio, they first find a large clique minor or a large almost flat wall by the following theorem. Theorem 2.4. ([33, Theorem (9.8)]) For any p and any h there are computable constants g 1 (p, h) and g 2 (p, h) such that, if a given graph G has tree-width at least g 1 (p, h), then there is an O(nm) time algorithm to find either a K p -minor or a pair (X, W ) satisfying the following conditions: (C3) all the components A 1,..., A l (as in the definition up to 3-separations ) have tree-width at most g 2 (p, h). Note that there is now an O(n) time algorithm to obtain either a K p -minor or a wall satisfying (C1) (C3) [25]. Then they find an irrelevant vertex if the graph contains a large clique minor or a large flat wall. The following theorem plays a crucial role in their algorithm when the graph has a large clique minor. Actually, this theorem is used to find an irrelevant vertex in a clique model in O(m) time. Theorem 2.5. ([33, Theorem (5.3)]) Let Z be a vertex set with Z = 2k in a given graph G. Suppose that there is a clique model K of order at least 3k in G, and there is no separation (A, B) of order at most 2k 1 in G such that A contains Z and B A contains at least one node of the clique model. Then, we can find mutually disjoint connected subgraphs H 1,..., H 2k of G such that V (H i ) Z = 1 for every i and there is an edge between H i and H j which is contained in E(K) for every i j, in O(m) time. Moreover, if the order of K is 3k +1, we can take H 1,..., H 2k not intersecting with some node of K, and the vertices in the node of K are irrelevant to the folio relative to Z. If we have a large flat wall, we can find an irrelevant vertex by the following theorem Copyright SIAM.

7 Theorem 2.6. ([34], also [33, Theorem (10.2)]) For fixed integers k, p, there is a computable constant h 1 (k, p) satisfying the following: if there is a subset X V (G) of order at most p such that there is a flat wall W of height h 1 (k, p) in G X, then there is a vertex v in W such that v is irrelevant to the folio relative to a set of k vertices. Furthermore, if all the components G[A 1 ],..., G[A l ] have tree-width bounded by a fixed constant, where A 1,..., A l are as in the flat embedding of comp(w ), we can find in O(m) time the irrelevant vertex v. Note that Robertson and Seymour actually showed that the middle vertices of a large flat wall are irrelevant. With these theorems, Robertson and Seymour [33] gave an O(n 3 ) time algorithm for the folio. Note that the running time of their algorithm is improved to O(n 2 ) time in [25]. 3 FPT Algorithm for Packing S-cycles In this section, we prove Theorem 1.6, that is, we give an O(n 2 m) time algorithm for the following problem for fixed k. S-cycle Packing. Input. A graph G = (V, E), a vertex set S V, and a fixed integer k (parameter). Problem. Find k vertex-disjoint S-cycles in G, or conclude that such cycles do not exist. Our FPT framework follows Robertson-Seymour s algorithm for the disjoint paths problem described in Section 2.3. The first step is to examine whether or not the tree-width is large. If it is bounded by a fixed constant, then one can apply dynamic programming to a tree-decomposition of bounded tree-width similarly to Theorem 2.3. Otherwise, we apply Theorem 2.4 to G (p and h will be given later) and obtain either a K p - minor K or a pair (X, W ) satisfying (C1) (C3). For both cases, we shall find an irrelevant vertex v, i.e., a vertex v such that G has a solution if and only if so does G v. We remove v and go back to determine whether or not the tree-width is bounded. By repeating this at most V times, we find k vertex-disjoint S-cycles in G, or conclude that such cycles do not exist. Therefore, the remaining task in this algorithm is to find an irrelevant vertex in the K p -minor K or the almost flat wall W efficiently. For that purpose, we first try to find many (depending only on k) vertexdisjoint S-paths with respect to K and W, respectively. If such paths exist, we can construct k vertex-disjoint S-cycles explicitly. Otherwise, it follows from Theorem 2.2 that there exists a vertex subset Z of bounded size such that deleting Z makes G have no such paths. That is, each s S is separated with at most one cut vertex from the clique model K in G Z (the same is true for the flat wall case). The union of Z and the set U of such cut vertices for all s S separates K and all the vertices of S. The family of disjoint S-cycles leads to the family of disjoint paths with end vertices in Z U on the clique side, and we do not need to take care of S on this side. Hence it suffices to find an irrelevant vertex as for the disjoint paths problems relative to Z U on the clique side. Reducing the size of U carefully described in Sections 3.1 and 3.2, we can find an irrelevant vertex in O(nm) time. Thus this algorithm runs in O(n 2 m) time. 3.1 Large Clique Minor Set p = p + 3k when we apply Theorem 2.4, and suppose that G has a K p -model K. We may assume that K is minimal, i.e., for every vertex v and every edge e in K, neither K v nor K e has a K p -model. If there exists a node of the K p -model that contains a vertex in S, then we can find an S- cycle through three nodes of the K p -model. That is, we obtain an S-cycle and a K p 3 -model that are mutually disjoint. By finding a node that contains a vertex in S repeatedly, we can obtain either k vertex-disjoint S- cycles or a K p -model K containing no vertices of S. Therefore, in what follows in this subsection, we assume that we have such a K p -model K. Let T be the set of centers of K, and suppose that p 36k. Then if there are 12k vertex-disjoint S-paths P 1,..., P 12k with respect to T, Lemma 2.1 implies that we can construct a K 3k -model such that each cycle in it has a vertex of S, which follows in a similar way to [15]. Hence this model contains k vertex-disjoint S- cycles such that each of the S-cycles contains exactly three nodes of this model. Therefore, by Lemma 2.1, we can find in O(kn 2 ) time either k vertex-disjoint S- cycles, or Z V with Z 24k 2 such that G Z contains no S-paths with respect to T. Let p be a sufficiently large integer (the definition will be given later). If we find k vertex-disjoint S- cycles, then we are done. Thus, in what follows, we consider the case when we have a subset Z V with Z 24k 2 such that G Z contains no S-paths with respect to the set T. Let S S be the subset of S contained in a connected component of G Z intersecting with T. By Menger s theorem, for any vertex s S, there exists a vertex τ(s) V such that the connected component of G Z {τ(s)} containing s, say G s, does not intersect with any vertex in T. We take such a vertex τ(s) V Z so that G s is maximal. We denote s S {τ(s)} by U = {u 1,..., u q }, and let V i be the vertex set defined by V i = {V (G s ) s S, τ(s) = u i } 1732 Copyright SIAM.

8 for i = 1,..., q. Then the collection of V i s is mutually disjoint by the definition of τ(s). Let V 0 = V Z U i V i. Then G[V 0 ] intersects with K because T V 0. Let U 0 U be the vertex set defined by U 0 = {u i U G[V i ] contains an S-cycle}, and define U 1 = U \U 0. Note that we can easily compute U 0. If U 0 k, then we can immediately find k vertexdisjoint S-cycles, since the collection of V i s is mutually disjoint. Suppose that U 0 < k. Since a path internally disjoint from V 0 with end vertices in V 0 must contain at least one vertex in Z, we observe the following: (1) If G has vertex-disjoint S-cycles, then they intersect with at most 2 Z sets of {V i u i U 1 }. For simplicity, first consider the case where U 1 is bounded by a fixed constant. Then we can find a vertex that is irrelevant to the existence of k vertexdisjoint S-cycles if K is large enough as follows. We find a separation (A, B) of G of minimum order such that A contains all vertices in S and B A contains at least one node of K. Then, we find a vertex that is irrelevant to the folio relative to V (A) V (B) in B using Theorem 2.5, which is a desired vertex. However, the number of elements in U 1 is not necessarily bounded. Our main idea is that we only need 2 Z elements in each equivalence class of U 1 by Observation (1). We now describe how to divide U 1 into the equivalence classes. We introduce a new concept weak folio, which is a weaker concept than the folio. Let G = (V, E) be a graph and X V be a set. We say that (s, t, γ), where s and t are distinct vertices in X and γ {0, 1}, is admissible if G has a path P from s to t such that P has a vertex of S if γ = 1. Let X = {(s, t, γ) (s, t, γ) is admissible, s, t X}, called the weak folio relative to X. Note that X is bounded by a function of X. For each u i U 1, we compute the weak folio X i relative to Z {u i } in G[V i Z {u i }]. This can be computed in O(nm) time. Let U1 1,..., U1 r be the partition of U 1 depending on the weak folios, that is, u i and u i are in the same set if and only if the weak folio relative to Z {u i } in G[V i Z {u i }] and that relative to Z {u i } in G[V i Z {u i }] are the same by exchanging u i and u i. Note that r is bounded by a function of Z + 1, say f 1 ( Z ). If U j 1 > 2 Z for some j, then we replace the vertex sets V i s corresponding to all vertices in U j 1 by 2 Z new vertices, and add all edges between these new vertices and the vertices in U j j 1. Let U 1 be the set of these 2 Z vertices, and we replace V 0 and U j 1 by V 0 U j j 1 and U 1, respectively. After executing this reduction for each j, we can find a vertex that is irrelevant to the existence of k vertex-disjoint S-cycles by Theorem 2.5 in O(m) time, since U 2 Z f 1 ( Z ). More precisely, we find a separation (A, B) of G[V 0 Z U] of minimum order such that A contains all vertices in Z U and B A contains at least one node of K. Then, we find a vertex that is irrelevant to the folio relative to V (A) V (B) in B using Theorem 2.5. Note that by the conditions of H i in Theorem 2.5, we do not need to consider paths (or trees) not intersecting with K when we find an irrelevant vertex in K. Thus, by Observation (1), we can see that this vertex is also irrelevant to the existence of k vertexdisjoint S-cycles. Note that, for the above arguments, we define p as an integer at least 3 (2 Z f 1 ( Z )+ Z )+1 3 U Z +1, and the total running time to find an irrelevant vertex is O(nm). 3.2 Large Wall We may assume that each connected component of G is 2-connected. Suppose we are given a graph G = (V, E), a vertex set S, and a pair (X, W ) in G T satisfying (C1) (C3) in Theorem 2.4, where h will be given later. Then W contains k disjoint flat walls of height h/k. If the compass of each wall contain an S-cycle then we are done. Otherwise, we can find a flat wall W of height h/k such that the unique block containing W in the compass has no vertices of S. We now use the following lemma. Lemma 3.1. ([20]) Let k be a positive integer. Assume G has a cycle C with no vertices of S. If G has 4k log 2 (k + 10) vertex-disjoint S-paths with respect to V (C), then there are k vertex-disjoint S-cycles. Moreover, such k vertex-disjoint S-cycles can be found in linear time. By applying Theorem 2.2 and Lemma 3.1 to the wall W of height h/k that contains no vertices of S, we can find either k vertex-disjoint S-cycles, or Z V with Z 8k log 2 (k + 10) 2 such that G Z contains no S-paths with respect to V (per(w )), in O(kn 2 ) time. If we have k vertex-disjoint S-cycles, we are done. Hence, we may assume that we obtain a subset Z V with Z 8k log 2 (k + 10) 2 such that G Z contains no S-paths with respect to T = V (per(w )). By replacing Z with Z X, we use the same argument as the previous subsection. Then, there exists a vertex set U such that removal of Z U X separates S and W. By Observation (1), the intersection of k vertexdisjoint S-cycles and G[V 0 Z X U] consists of at most 2 Z X paths whose end vertices are in Z X U, where V 0 V is defined in the same way as Section 3.1. Thus, it suffices to find a vertex in W that is irrelevant 1733 Copyright SIAM.

9 to the existence of such 2 Z X paths. By executing the same reductions as the previous subsection, we may assume that Z U X is bounded by a function of k, say f 2 (k). That is, it suffices to consider the case when there is a separation (A, B) of bounded order such that A contains all vertices in S and B contains W. Then it follows from Theorem 2.6 that if we have a wall of height h 1 (f 2 (k), X ), then removing some vertex of the wall does not affect the folio of B relative to V (A) V (B), where h 1 is defined as in Theorem 2.6. Therefore, if we set h k h 1 (f 2 (k), X ), then we can find an irrelevant vertex in O(m) time. Thus the total running time to find an irrelevant vertex is O(nm). 3.3 Algorithm Finally in this subsection, we describe our algorithm for the S-cycle packing. Assume that p and h are given as in Sections 3.1 and 3.2. Recall that g 1 is the function defined in Theorem 2.4. Algorithm for the S-cycle Packing. Input. A graph G = (V, E), a vertex set S V, and a fixed integer k. Output. Find k vertex-disjoint S-cycles in G, or conclude that such cycles do not exist. Step 1. Determine whether the tree-width of G is at most g 1 (p, h) or not. If it is at most g 1 (p, h), then solve the problem by Theorem 2.3. Otherwise, go to Step 2. Step 2. Apply Theorem 2.4 to G and obtain either a K p -minor or a pair (X, W ) satisfying (C1) (C3) in Theorem 2.4. If we have a K p -minor, then find an irrelevant vertex as in Section 3.1. Otherwise, we find an irrelevant vertex as in Section 3.2. Then, remove the irrelevant vertex and go to Step 1. Since we can find an irrelevant vertex in O(nm) time when k is fixed, this algorithm runs in O(n 2 m) time. This completes the proof of Theorem FPT Algorithms for Parity-Constrained Problems In Section 3, we have exploited graph minor theory to solve the S-cycle packing. For the parity-constrained problems, we additionally make use of recent results on odd clique minors [15] and the parity disjoint paths problem [28]. The parity disjoint path problem is that we are given a graph G = (V, E), a k pair of vertices (s 1, t 1 ), (s 2, t 2 ),..., (s k, t k ) in G, and a parity l i {0, 1} for each i with 1 i k, and we aim at finding k vertexdisjoint paths P 1,..., P k such that P i joins s i and t i and its parity is l i for each i. Recently, Kawarabayashi, et al. [28] showed that this problem can be solved in polynomial time by extending the Robertson Seymour s algorithm for the disjoint paths problem. Similarly to the non-parity S-cycle packing in Section 3, we may consider when the tree-width is large, i.e., when there exists a large clique minor or a large almost flat wall, and the main task is to find an irrelevant vertex in these structures. Our FPT framework in Section 3, however, is not enough to find an irrelevant vertex, since we have to control the parity of cycles. To tract the parity constraints, we further divide the cliqueminor case into the following two cases, depending on whether G has a large odd clique minor or not. That is, we consider (1) A large odd clique minor with a huge clique minor. (2) A huge clique minor, but no large odd clique minors. (3) An almost flat wall of large height. First assume that we have a large odd clique minor. Then it is known that we can find an irrelevant vertex for the parity disjoint paths problem. By using this fact instead of Theorem 2.5, we can find an irrelevant vertex in the odd clique minor in a similar way to Section 3.1. We next discuss the case (2) where we have no odd clique minors. In this case, we know that it has a large even clique minor, and we apply Theorem 2.1 to obtain a vertex set X such that the unique block containing most of the clique nodes in G X is bipartite. Then we follow the arguments in Section 3.1 for G X, and find a vertex set Z such that G X Z contains no S-paths with respect to the clique minor. The block, say L, having the clique minor in G X Z is bipartite, and L has no vertices of S. Hence we could use Theorem 2.5 for the disjoint paths problem with respect to X Z (instead of Z in Section 3.1) in L to find an irrelevant vertex. Thus we may assume that there is no huge clique minor, which corresponds to the case (3). Let us remind that in Section 3.2, we separate S and the flat wall with small vertex set Z, and then apply Theorem 2.6 to find an irrelevant vertex. Here we can also obtain such Z by Theorem 2.2, and find an irrelevant vertex by results for the parity disjoint paths problem [28] instead of Theorem 2.6. Note that for the parity-constrained FVS problems, we do not need results in [28], and can find an irrelevant vertex in a simpler way by Theorem 2.1. Indeed, we can find a small vertex set Z that separates S and the flat wall, or a small vertex set Y such that the block having the wall in G Y is bipartite, and, for both cases, we can find an irrelevant vertex by Theorem 2.6 with a careful analysis. See [19] for details. Finally, let us remark that these algorithms are inspired by the proofs of Theorems 1.3 and 1.4. In the proofs, assuming the induction hypothesis on k, we first observe that a counterexample graph G has large 1734 Copyright SIAM.

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