Related papers: Counting and Enumerating Crossing-free Geometric G…
In the subgraph counting problem, we are given a input graph $G(V, E)$ and a target graph $H$; the goal is to estimate the number of occurrences of $H$ in $G$. Our focus here is on designing sublinear-time algorithms for approximately…
We show that for every fixed non-negative integer k there is a quadratic time algorithm that decides whether a given graph has crossing number at most k and, if this is the case, computes a drawing of the graph in the plane with at most k…
We explore various techniques for counting the number of straight-edge crossing-free graphs that can be embedded on a planar point set. In particular, we derive a lower bound on the ratio of the number of such graphs with $m+1$ edges to the…
We present an algorithm that computes the girth of the intersection graph of $n$ given line segments in the plane in $O(n^{1.483})$ expected time. This is the first such algorithm with $O(n^{3/2-\varepsilon})$ running time for a positive…
We consider the NP-hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamic-programming algorithms for general and…
The crossing number is the smallest number of pairwise edge-crossings when drawing a graph into the plane. There are only very few graph classes for which the exact crossing number is known or for which there at least exist constant…
The Max-Cut problem is known to be NP-hard on general graphs, while it can be solved in polynomial time on planar graphs. In this paper, we present a fixed-parameter tractable algorithm for the problem on `almost' planar graphs: Given an…
Given an edge-weighted graph $G$ on $n$ nodes, the NP-hard Max-Cut problem asks for a node bipartition such that the sum of edge weights joining the different partitions is maximized. We propose a fixed-parameter tractable algorithm…
We give a simple polynomial-time algorithm to exactly count the number of Euler Tours (ETs) of any Eulerian generalized series-parallel graph, and show how to adapt this algorithm to exactly sample a random ET of the given generalized…
An algorithm observes the trajectories of random walks over an unknown graph $G$, starting from the same vertex $x$, as well as the degrees along the trajectories. For all finite connected graphs, one can estimate the number of edges $m$ up…
Drawing a graph in the plane with as few crossings as possible is one of the central problems in graph drawing and computational geometry. Another option is to remove the smallest number of vertices or edges such that the remaining graph…
Subgraph enumeration problems ask to output all subgraphs of an input graph that belongs to the specified graph class or satisfy the given constraint. These problems have been widely studied in theoretical computer science. As far, many…
The crossing number of a graph is the minimum number of edge crossings that a graph can have when drawn in the plane. Determining this number, known as the Crossing Number problem, is a celebrated problem in combinatorial optimization. It…
A transversal of a hypergraph is a set of vertices intersecting each hyperedge. We design and analyze new exponential-time algorithms to enumerate all inclusion-minimal transversals of a hypergraph. For each fixed k>2, our algorithms for…
I present an algorithm that, given a number $n \geq 1$, computes a compact representation of the set of all noncrossing acyclic digraphs with $n$ nodes. This compact representation can be used as the basis for a wide range of dynamic…
The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve $c$ represented by a closed walk of length at…
A planar orthogonal drawing of a planar 4-graph G (i.e., a planar graph with vertex-degree at most four) is a crossing-free drawing that maps each vertex of G to a distinct point of the plane and each edge of $G$ to a sequence of horizontal…
We study the $P_3$-convexity, the path convexity generated by all three-vertex paths, and focus on the problem of counting the $P_3$-convex vertex sets of a graph $G$, denoted by $\noc(G)$. First, we settle the associated extremal question:…
We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane to so-called \emph{pseudo-simultaneously flippable edges}. Such edges are related to the notion of convex…
In this paper, we present a new exact algorithm for counting perfect matchings, which relies on neither inclusion-exclusion principle nor tree-decompositions. For any bipartite graph of $2n$ nodes and $\Delta n$ edges such that $\Delta \geq…