Minimal disconnected cuts in planar graphs

The problem of finding a disconnected cut in a graph is NP-hard in general but polynomial-time solvable on planar graphs. The problem of finding a minimal disconnected cut is also NP-hard but its computational complexity is not known for planar graphs. We show that it is polynomial-time solvable on 3-connected planar graphs but NP-hard for 2-connected planar graphs. Our technique for the first result is based on a structural characterization of minimal disconnected cuts in 3-connected K 3,3 -free-minor graphs and on solving a topological minor problem in the dual. We show that the latter problem can be solved in polynomial-time even on general graphs. In addition we show that the problem of finding a minimal connected cut of size at least 3 is NP-hard for 2-connected apex graphs

Graph Minors and Parameterized Algorithm Design

Abstract. The Graph Minors Theory, developed by Robertson and Sey-mour, has been one of the most influential mathematical theories in pa-rameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct meta-algorithmic consequences, we present the algorithmic appli-cations of core theorems such as the grid-exclusion theorem, and we give a brief description of the irrelevant vertex technique

Algorithms and obstructions for linear-width and related search parameters

The linear-width of a graph G is defined to be the smallest integer k such that the edges of G can be arranged in a linear ordering (e 1 ; : : : ; e r ) in such a way that for every i = 1; : : : ; r \Gamma 1, there are at most k vertices incident to edges that belong both to fe 1 ; : : : ; e i g and to fe i+1 ; : : : ; e r g. In this paper, we give a set of 57 graphs and prove that it is the set of the minimal forbidden minors for the class of graphs with linear-width at most two. Our proof also gives a linear time algorithm that either reports that a given graph has linear-width more than two or outputs an edge ordering of minimum linear-width. We further prove a structural connection between linear-width and the mixed search number which enables us to determine, for any k 1, the set acyclic forbidden minors for the class of graphs with linear-width k. Moreover, due to this connection, our algorithm can be transfered to two linear time algorithms that check whether a graph has mixe..

Fast sub-exponential algorithms and compactness in planar graphs

We provide a new theory, alternative to bidimensionality, of sub-exponential parameterized algorithms on planar graphs, which is based on the notion of compactness. Roughly speaking, a parameterized problem is (r,q)-compact when all the faces and vertices of its YES-instances are "r-radially dominated" by some vertex set whose size is at most q times the parameter. We prove that if a parameterized problem can be solved in steps and is (r,q)-compact, then it can be solved by a cr.2.122.√q.kno(1) step algorithm (where k is the parameter). Our framework is general enough to unify the analysis of almost all known sub-exponential parameterized algorithms on planar graphs and improves or matches their running times. Our results are based on an improved combinatorial bound on the branchwidth of planar graphs that bypasses the grid-minor exclusion theorem. That way, our approach encompasses new problems where bidimensionality theory do not directly provide sub-exponential parameterized algorithms. © 2011 Springer-Verlag Berlin Heidelberg

Fixed-Parameter Algorithms for (k, r)-Center in Planar Graphs and Map Graphs

The (k, r)-center problem asks whether an input graph G has ≤ k vertices (called centers) such that every vertex of G is within distance ≤ r from some center. In this article, we prove that the (k, r)-center problem, parameterized by k and r, is fixed-parameter tractable (FPT) on planar graphs, i.e., it admits an algorithm of complexity f (k, r)nO(1) where the function f is independent of n. In particular, we show that f (k, r) = 2O(r log r) √ k, where the exponent of the exponential term grows sublinearly in the number of centers. Moreover, we prove that the same type of FPT algorithms can be designed for the more general class of map graphs introduced by Chen, Grigni, and Papadimitriou. Our results combine dynamic-programming algorithms for graphs of small branchwidth and a graphtheoretic result bounding this parameter in terms of k and r. Finally, a byproduct of our algorithm is the existence of a PTAS for the r-domination problem in both planar graphs and map graphs. Our approach builds on the seminal results of Robertson and Seymour on Graph Minors, and as a result is much more powerful than the previous machinery of Alber et al. for exponential speedup on planar graphs. To demonstrate the versatility of our results, we show how our algorithms can be extended to general parameters that are “large” on grids. In addition, our use of branchwidth instead of the usual treewidth allows us to obtain much faster algorithms, and requires more complicated dynamic programming than the standard leaf/introduce/forget/join structure of nice tree decompositions. Our results are also unique in that they apply to classes of graphs that are not minor-closed, namely, constant powers of planar graphs and map graphs. © 2005, ACM. All rights reserved

Planar feedback vertex set and face cover: Combinatorial bounds and subexponential algorithms

The Planar Feedback Vertex Set problem asks, whether an n-vertex planar graph contains at most k vertices meeting all its cycles. The Face Cover problem asks, whether all vertices of a plane graph G lie on the boundary of at most k faces of G. Standard techniques from parameterized algorithm design indicate, that both problems can be solved by sub-exponential parameterized algorithms (where k is the parameter). In this paper, we improve the algorithmic analysis of both problems by proving a series of combinatorial results, relating the branchwidth of planar graphs with their face cover. Combining this fact with duality properties of branchwidth, allows us to derive analogous results on feedback vertex set. As a consequence, it follows that Planar Feedback Vertex Set and Face Cover can be solved in and steps, respectively. © 2008 Springer Berlin Heidelberg

Parameterized complexity of finding regular induced subgraphs

The r-Regular Induced Subgraph problem asks, given a graph G and a non-negative integer k, whether G contains an r-regular induced subgraph of size at least k, that is, an induced subgraph in which every vertex has degree exactly r. In this paper we examine its parameterization k-Size r-Regular Induced Subgraph with k as parameter and prove that it is W [1]-hard. We also examine the parameterized complexity of the dual parameterized problem, namely, the k-Almost r-Regular Graph problem, which asks for a given graph G and a non-negative integer k whether G can be made r-regular by deleting at most k vertices. For this problem, we prove the existence of a problem kernel of size O (k r (r + k)2). © 2008 Elsevier Inc. All rights reserved

Graph searching in a crime wave

We define helicopter cops and robber games with multiple robbers, extending previous research, which considered only the pursuit of a single robber. Our model is defined for robbers that are visible (their position in the graph is known to the cops) and active (they can move at any point in the game) but is easily adapted to other variants of the single-robber game that have been considered in the literature.We show that the game with many robbers is nonmonotone: that is, fewer cops are needed if the robbers are allowed to reoccupy positions that were previously unavailable to them. As the moves of the cops depend on the position of the visible robbers, strategies for such games should be interactive, but the game becomes, in a sense, less interactive as the initial number of robbers increases. We prove that the main parameter emerging from the game, which we denote mvams(G, r), captures a hierarchy of parameters between proper pathwidth and proper treewidth, and we completely characterize it for trees, extending analogous existing characterizations of the pathwidth of trees. Moreover, we prove an upper bound for mvams(G, r) on general graphs and show that this bound is reached by an infinite class of graphs. On the other hand, if we consider the robbers to be invisible and lazy, the resulting parameters collapse in all cases to either proper pathwidth or proper treewidth, giving a further case where the classical equivalence between visible, active robbers and invisible, lazy robbers does not hold. © 2008 Society for Industrial and Applied Mathematics

On self duality of pathwidth in polyhedral graph embeddings

Let G be a 3-connected planar graph and G* be its dual. We show that the pathwidth of G* is at most 6 times the pathwidth of G. We prove this result by relating the pathwidth of a graph with the cut-width of its medial graph and we extend it to bounded genus embeddings. We also show that there exist 3-connected planar graphs such that the pathwidth of such a graph is at least 1.5 times the pathwidth of its dual. © 2007 Wiley Periodicals, Inc

Paths of bounded length and their cuts: Parameterized complexity and algorithms

We study the parameterized complexity of two families of problems: the bounded length disjoint paths problem and the bounded length cut problem. From Menger's theorem both problems are equivalent (and computationally easy) in the unbounded case for single source, single target paths. However, in the bounded case, they are combinatorially distinct and are both NP-hard, even to approximate. Our results indicate that a more refined landscape appears when we study these problems with respect to their parameterized complexity. For this, we consider several parameterizations (with respect to the maximum length l of paths, the number k of paths or the size of a cut, and the treewidth of the input graph) of all variants of both problems (edge/vertex-disjoint paths or cuts, directed/undirected). We provide FPT-algorithms (for all variants) when parameterized by both k and l and hardness results when the parameter is only one of k and l. Our results indicate that the bounded length disjoint-path variants are structurally harder than their bounded length cut counterparts. Also, it appears that the edge variants are harder than their vertex-disjoint counterparts when parameterized by the treewidth of the input graph. © 2009 Springer-Verlag