Monotone Grid Drawings of Planar Graphs

A monotone drawing of a planar graph $G$ is a planar straight-line drawing of $G$ where a monotone path exists between every pair of vertices of $G$ in some direction. Recently monotone drawings of planar graphs have been proposed as a new standard for visualizing graphs. A monotone drawing of a planar graph is a monotone grid drawing if every vertex in the drawing is drawn on a grid point. In this paper we study monotone grid drawings of planar graphs in a variable embedding setting. We show that every connected planar graph of $n$ vertices has a monotone grid drawing on a grid of size $O(n)\times O(n^2)$, and such a drawing can be found in O(n) time

Splitting Clusters To Get C-Planarity

In this paper we introduce a generalization of the c-planarity testing problem for clustered graphs. Namely, given a clustered graph, the goal of the S PLIT-C-P LANARITY problem is to split as few clusters as possible in order to make the graph c-planar. Determining whether zero splits are enough coincides with testing c-planarity. We show that S PLIT-C-P LANARITY is NP-complete for c-connected clustered triangulations and for non-c-connected clustered paths and cycles. On the other hand, we present a polynomial-time algorithm for flat c-connected clustered graphs whose underlying graph is a biconnected seriesparallel graph, both in the fixed and in the variable embedding setting, when the splits are assumed to maintain the c-connectivity of the clusters

The Power of Cut-Based Parameters for Computing Edge Disjoint Paths

This paper revisits the classical Edge Disjoint Paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our aim is to identify structural properties (parameters) of graphs which allow the efficient solution of EDP without restricting the placement of terminals in P in any way. In this setting, EDP is known to remain NP-hard even on extremely restricted graph classes, such as graphs with a vertex cover of size 3. We present three results which use edge-separator based parameters to chart new islands of tractability in the complexity landscape of EDP. Our first and main result utilizes the fairly recent structural parameter treecut width (a parameter with fundamental ties to graph immersions and graph cuts): we obtain a polynomial-time algorithm for EDP on every graph class of bounded treecut width. Our second result shows that EDP parameterized by treecut width is unlikely to be fixed-parameter tractable. Our final, third result is a polynomial kernel for EDP parameterized by the size of a minimum feedback edge set in the graph

Improved Approximation Algorithms for Box Contact Representations ⋆

Abstract. We study the following geometric representation problem: Given a graph whose vertices correspond to axis-aligned rectangles with fixed dimensions, arrange the rectangles without overlaps in the plane such that two rectangles touch if the graph contains an edge between them. This problem is called CONTACT REPRESENTATION OF WORD NETWORKS (CROWN) since it formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. CROWN is known to be NP-hard, and there are approximation algorithms for certain graph classes for the optimization version, MAX-CROWN, in which realizing each desired adjacency yields a certain profit. We present the first O(1)-approximation algorithm for the general case, when the input is a complete weighted graph, and for the bipartite case. Since the subgraph of realized adjacencies is necessarily planar, we also consider several planar graph classes (namely stars, trees, outerplanar, and planar graphs), improving upon the known results. For some graph classes, we also describe improvements in the unweighted case, where each adjacency yields the same profit. Finally, we show that the problem is APX-hard on bipartite graphs of bounded maximum degree.

Cover Contact Graphs.

We study problems that arise in the context of covering certain geometric objects (so-called seeds, e.g., points or disks) by a set of other geometric objects (a so-called cover, e.g., a set of disks or homothetic triangles). We insist that the interiors of the seeds and the cover elements are pair wise disjoint, but they can touch. We call the contact graph of a cover a cover contact graph (CCG). We are interested in two types of tasks: (a) deciding whether a given seed set has a connected CCG, and (b) deciding whether a given graph has a realization as a CCG on a given seed set. Concerning task (a) we give efficient algorithms for the case that seeds are points and covers are disks or triangles. We show that the problem becomes NP-hard if seeds and covers are disks. Concerning task (b) we show that it is even NP-hard for point seeds and disk covers (given a fixed correspondence between vertices and seeds)

The Complexity of Drawing a Graph in a Polygonal Region

We prove that the following problem is complete for the existential theory of the reals: Given a planar graph and a polygonal region, with some vertices of the graph assigned to points on the boundary of the region, place the remaining vertices to create a planar straight-line drawing of the graph inside the region. This strengthens an NP-hardness result by Patrignani on extending partial planar graph drawings. Our result is one of the first showing that a problem of drawing planar graphs with straight-line edges is hard for the existential theory of the reals. The complexity of the problem is open in the case of a simply connected region. We also show that, even for integer input coordinates, it is possible that drawing a graph in a polygonal region requires some vertices to be placed at irrational coordinates. By contrast, the coordinates are known to be bounded in the special case of a convex region, or for drawing a path in any polygonal region.Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

On Structural Parameterizations of the Edge Disjoint Paths Problem

In this paper we revisit the classical edge disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or FPT) algorithms. As our first result, we answer an open question stated in Fleszar et al. (Proceedings of the ESA, 2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain NP-hard even for treewidth two, a result by Zhou et al. (Algorithmica 26(1):3--30, 2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an FPT-algorithm has remained open since then. We show that this is highly unlikely by establishing the W[1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an FPT-algorithm for EDP by exploiting a novel structural parameter of the augmented graph

Planar graphs: theory and algorithms

Collected in this volume are most of the important theorems and algorithms currently known for planar graphs, together with constructive proofs for the theorems. Many of the algorithms are written in Pidgin PASCAL, and are the best-known ones; the complexities are linear or 0(nlogn). The first two chapters provide the foundations of graph theoretic notions and algorithmic techniques. The remaining chapters discuss the topics of planarity testing, embedding, drawing, vertex- or edge-coloring, maximum independence set, subgraph listing, planar separator theorem, Hamiltonian cycles, and single- or multicommodity flows. Suitable for a course on algorithms, graph theory, or planar graphs, the volume will also be useful for computer scientists and graph theorists at the research level. An extensive reference section is included