Pim van ’t Hof

Obtaining a bipartite graph by contracting few edges

We initiate the study of the Bipartite Contraction problem from the perspective of parameterized complexity. In this problem we are given a graph G on n vertices and an integer k, and the task is to determine whether we can obtain a bipartite graph from G by a sequence of at most k edge contractions. Our main result is an f(k) n^{O(1)} time algorithm for Bipartite Contraction. Despite a strong resemblance between Bipartite Contraction and the classical Odd Cycle Transversal (OCT) problem, the methods developed to tackle OCT do not seem to be directly applicable to Bipartite Contraction. To obtain our result, we combine several techniques and concepts that are central in parameterized complexity: iterative compression, irrelevant vertex, and important separators. To the best of our knowledge, this is the first time the irrelevant vertex technique and the concept of important separators are applied in unison. Furthermore, our algorithm may serve as a comprehensible example of the usage of the irrelevant vertex technique.

Minimal dominating sets in graph classes: Combinatorial bounds and enumeration

Abstract The number of minimal dominating sets that a graph on n vertices can have is known to be at most 1.715 9 n . This upper bound might not be tight, since no examples of graphs with 1.570 5 n or more minimal dominating sets are known. For several classes of graphs, we substantially improve the upper bound on the number of minimal dominating sets. At the same time, we give algorithms for enumerating all minimal dominating sets, where the running time of each algorithm is within a polynomial factor of the proved upper bound for the graph class in question. In several cases, we provide examples of graphs containing the maximum possible number of minimal dominating sets for graphs in that class, thereby showing the corresponding upper bounds to be tight.

A new characterization of P6-free graphs

We study P 6 -free graphs, i.e., graphs that do not contain an induced path on six vertices. Our main result is a new characterization of this graph class: a graph Gis P 6 -free if and only if each connected induced subgraph of Gon more than one vertex contains a dominating induced cycle on six vertices or a dominating (not necessarily induced) complete bipartite subgraph. This characterization is minimal in the sense that there exists an infinite family of P 6 -free graphs for which a smallest connected dominating subgraph is a (not induced) complete bipartite graph. Our characterization of P 6 -free graphs strengthens results of Liu and Zhou, and of Liu, Peng and Zhao. Our proof has the extra advantage of being constructive: we present an algorithm that finds such a dominating subgraph of a connected P 6 -free graph in polynomial time. This enables us to solve the Hypergraph 2-Colorability problem in polynomial time for the class of hypergraphs with P 6 -free incidence graphs.

Obtaining planarity by contracting few edges

The Planar Contraction problem is to test whether a given graph can be made planar by using at most k edge contractions. This problem is known to be NP-complete. We show that it is fixed-parameter tractable when parameterized by k.

Partitioning Graphs into Connected Parts

The 2-DISJOINT CONNECTED SUBGRAPHS problem asks if a given graph has two vertex-disjoint connected subgraphs containing pre-specified sets of vertices. We show that this problem is NP-complete even if one of the sets has cardinality 2. The LONGEST PATH CONTRACTIBILITY problem asks for the largest integer l for which an input graph can be contracted to the path P l on l vertices. We show that the computational complexity of the LONGEST PATH CONTRACTIBILITY problem restricted to P l-free graphs jumps from being polynomially solvable to being NP-hard at l=?6, while this jump occurs at l=?5 for the 2-DISJOINT CONNECTED SUBGRAPHS problem. We also present an exact algorithm that solves the 2-DISJOINT CONNECTED SUBGRAPHS problem faster than O*(2n) for any n-vertex P l-free graph. For l=?6, its running time is O*(1.5790n). We modify this algorithm to solve the LONGEST PATH CONTRACTIBILITY problem for P 6-free graphs in O*(1.5790n) time.

Contracting graphs to paths and trees

Vertex deletion and edge deletion problems play a central role in Parameterized Complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. The study of analogous edge contraction problems has so far been left largely unexplored from a parameterized perspective. We consider two basic problems of this type: Tree Contraction and Path Contraction. These two problems take as input an undirected graph G on n vertices and an integer k, and the task is to determine whether we can obtain an acyclic graph or a path, respectively, by a sequence of at most k edge contractions in G. We present an algorithm with running time 4.98knO(1) for Tree Contraction, based on a variant of the color coding technique of Alon, Yuster and Zwick, and an algorithm with running time 2k+o(k)+nO(1) for Path Contraction. Furthermore, we show that Path Contraction has a kernel with at most 5k+3 vertices, while Tree Contraction does not have a polynomial kernel unless NP ⊆ coNP/poly. We find the latter result surprising, because of the connection between Tree Contraction and Feedback Vertex Set, which is known to have a kernel with 4k2 vertices.

Contracting chordal graphs and bipartite graphs to paths and trees

We study the following two graph modification problems: given a graph G and an integer k, decide whether G can be transformed into a tree or into a path, respectively, using at most k edge contractions. These problems, which we call Tree Contraction and Path Contraction, respectively, are known to be NP-complete in general. We show that on chordal graphs these problems can be solved in O(n+m) and O(nm) time, respectively. As a contrast, both problems remain NP-complete when restricted to bipartite input graphs.

Parameterized complexity of vertex deletion into perfect graph classes

Vertex deletion problems are at the heart of parameterized complexity. For a graph class F, the F-Deletion problem takes as input a graph G and an integer k. The question is whether it is possible to delete at most k vertices from G such that the resulting graph belongs to F. Whether Perfect Deletion is fixed-parameter tractable, and whether Chordal Deletion admits a polynomial kernel, when parameterized by k, have been stated as open questions in previous work. We show that Perfect Deletion (k) and Weakly Chordal Deletion (k) are W[2]-hard. In search of positive results, we study restricted variants such that the deleted vertices must be taken from a specified set X, which we parameterize by |X|. We show that for Perfect Deletion and Weakly Chordal Deletion, although this restriction immediately ensures fixed parameter tractability, it is not enough to yield polynomial kernels, unless NP ⊆ coNP/poly. On the positive side, for Chordal Deletion, the restriction enables us to obtain a kernel with O(|X|4) vertices.

Obtaining a Bipartite Graph by Contracting Few Edges

The Bipartite Contraction problem takes as input an

Detecting Fixed Patterns in Chordal Graphs in Polynomial Time

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