Complexity and Approximability of the k -Way Vertex Cut* André Berger, Alexander Grigoriev, and Ruben van der Zwaan Operations Research Group, Department of Quantitative Economics, School of Business and Economics, Maastricht University, The Netherlands In this article, we consider k -way vertex cut: the prob- lem of finding a graph separator of a given size that decomposes the graph into the maximum number of components. Our main contribution is the derivation of an efficient polynomial-time approximation scheme for the problem on planar graphs. Also, we show that k -way vertex cut is polynomially solvable on graphs of bounded treewidth and fixed–parameter tractable on planar graphs with the size of the separator as the parameter. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol. 63(2), 170–178 2014 Keywords: planar graph; separator; connectivity; computational complexity; approximation scheme 1. INTRODUCTION In this article, we address the following graph separation problem: given an n-vertex graph G = (V , E) and integers k ≤ n and s ≤ n, find a cut-set S ⊆ V of size at most s such that the subgraph of G induced by V \S has at least k components. We refer to this problem as k -way vertex cut. 1.1. Related Problems Surprisingly enough, despite its very basic and quite generic setting, k -way vertex cut did not receive much atten- tion in the literature. We are aware of only one paper by Marx [29] investigating the fixed-parameter (in)tractability of k -way vertex cut. He showed that k -way vertex cut is W[1]- hard with parameter k . This is in contrast to the amount of research concerning the following closely related problems. • The k-way edge cut, also referred as k-cut or minimum k- cut. This problem can be regarded as the “edge-version” of k-way vertex cut: given a graph G = (V , E) and integers k and s, remove at most s edges so as to split the graph Received June 2012; accepted August 2013 Correspondence to: A. Grigoriev; e-mail: a.grigoriev@maastrichtuniversity .nl *A preliminary version of this article has been presented at the 8th Annual Conference on Theory and Applications of Models of Computation [3] DOI 10.1002/net.21534 Published online 15 October 2013 in Wiley Online Library (wileyonlinelibrary.com). © 2013 Wiley Periodicals, Inc. into at least k components. Goldschmidt and Hochbaum [18] have shown that k-way edge cut is NP-hard and admits an n O(k 2 ) -time algorithm. Later, Karger and Stein [25] developed a randomized n O(k) -time algorithm for this problem. Kami- dori, et al. [24] presented a deterministic O(n (4+o(1))k )-time algorithm which was further improved by Thorup [32]. Saran and Vazirani [31] gave a polynomial time approximation algo- rithm that finds at most 2s edges which deletion splits the graph into at least k components. For dense graphs with (n 2 ) edges, Arora et al. [1] developed a polynomial time approximation scheme for k-way edge cut. Recently, Kawarabayashi and Thorup [26] have shown that k-way edge cut is fixed parameter tractable with respect to parameter s, whereas parameterized by k the problem is W[1]-hard [14]. Therefore, this problem, when considered with paramter s, is one of the few problems known where the vertex version is W[1]-hard, but the edge ver- sion is FTP. There are several papers that studied k-way edge cut on planar graphs. For k = 3, Hochbaum and Shmoys [22] constructed a quadratic time algorithm for k-way edge cut on planar graphs. He [21] gave an improved algorithm for this special case with running time O(n log n). Kawarabayashi and Thorup [26] presented an 2 O(s 2 g 2 ) · n–time algorithm for graphs of genus g. Their results were recently improved by Chtinis et al. [8]. • Bounded fragmentation [20]: a graph G = (V , E) has bounded fragmentation if after removing any S ⊆ V of size at most s the number of components in the obtained graph is bounded by a function of s. • Multiway cut [7, 11–13, 17, 29, 33]: given a graph G, a set of k terminals and an integer s, find a set of at most s edges (or vertices) whose removal separates the terminals from each other. • Multicut [6, 9, 10, 16, 19, 30]: given a graph G, a set of k terminal-pairs and an integer s, find a set of at most s edges (or vertices) whose deletion separates the vertices in each terminal-pair from each other. 1.2. Our Results In this article we study the computational complexity and approximability of k -way vertex cut. We consider the prob- lem parametrized with parameter s, the size of the cut-set. The objective is to maximize the number of additional com- ponents that can be obtained by removing at most s vertices. Clearly, any α–approximation for this problem also yields an α–approximation for the problem of maximizing the total NETWORKS—2014—DOI 10.1002/net