429 research outputs found
Strip Planarity Testing of Embedded Planar Graphs
In this paper we introduce and study the strip planarity testing problem,
which takes as an input a planar graph and a function and asks whether a planar drawing of exists
such that each edge is monotone in the -direction and, for any
with , it holds . The problem has strong
relationships with some of the most deeply studied variants of the planarity
testing problem, such as clustered planarity, upward planarity, and level
planarity. We show that the problem is polynomial-time solvable if has a
fixed planar embedding.Comment: 24 pages, 12 figures, extended version of 'Strip Planarity Testing'
(21st International Symposium on Graph Drawing, 2013
Planarization With Fixed Subgraph Embedding
The visualization of metabolic networks using techniques of graph drawing has recently become an important research area. In order to ease the analysis of these networks, readable layouts are required in which certain known network components are easily recognizable. In general, the topology of the drawings produced by traditional graph drawing algorithms does not reflect the biologists' expert knowledge on particular substructures of the underlying network. To deal with this problem we present a constrained planarization method---an algorithm which computes a graph layout in the plane preserving the predefined shape for the specified substructures while minimizing the overall number of edge-crossings
A Note on the Practicality of Maximal Planar Subgraph Algorithms
Given a graph , the NP-hard Maximum Planar Subgraph problem (MPS) asks for
a planar subgraph of with the maximum number of edges. There are several
heuristic, approximative, and exact algorithms to tackle the problem, but---to
the best of our knowledge---they have never been compared competitively in
practice. We report on an exploratory study on the relative merits of the
diverse approaches, focusing on practical runtime, solution quality, and
implementation complexity. Surprisingly, a seemingly only theoretically strong
approximation forms the building block of the strongest choice.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Planarization With Fixed Subgraph Embedding
The visualization of metabolic networks using techniques of graph drawing has recently become an important research area. In order to ease the analysis of these networks, readable layouts are required in which certain known network components are easily recognizable. In general, the topology of the drawings produced by traditional graph drawing algorithms does not reflect the biologists' expert knowledge on particular substructures of the underlying network. To deal with this problem we present a constrained planarization method---an algorithm which computes a graph layout in the plane preserving the predefined shape for the specified substructures while minimizing the overall number of edge-crossings
NodeTrix Planarity Testing with Small Clusters
We study the NodeTrix planarity testing problem for flat clustered graphs
when the maximum size of each cluster is bounded by a constant . We consider
both the case when the sides of the matrices to which the edges are incident
are fixed and the case when they can be chosen arbitrarily. We show that
NodeTrix planarity testing with fixed sides can be solved in
time for every flat clustered graph that can be
reduced to a partial 2-tree by collapsing its clusters into single vertices. In
the general case, NodeTrix planarity testing with fixed sides can be solved in
time for , but it is NP-complete for any . NodeTrix
planarity testing remains NP-complete also in the free sides model when .Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Recognizing and Drawing IC-planar Graphs
IC-planar graphs are those graphs that admit a drawing where no two crossed
edges share an end-vertex and each edge is crossed at most once. They are a
proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph
with vertices, we present an -time algorithm that computes a
straight-line drawing of in quadratic area, and an -time algorithm
that computes a straight-line drawing of with right-angle crossings in
exponential area. Both these area requirements are worst-case optimal. We also
show that it is NP-complete to test IC-planarity both in the general case and
in the case in which a rotation system is fixed for the input graph.
Furthermore, we describe a polynomial-time algorithm to test whether a set of
matching edges can be added to a triangulated planar graph such that the
resulting graph is IC-planar
Synchronized planarity with applications to constrained planarity problems
We introduce the problem Synchronized Planarity. Roughly speaking, its input is a loop-free multi-graph together with synchronization constraints that, e.g., match pairs of vertices of equal degree by providing a bijection between their edges. Synchronized Planarity then asks whether the graph admits a crossing-free embedding into the plane such that the orders of edges around synchronized vertices are consistent. We show, on the one hand, that Synchronized Planarity can be solved in quadratic time, and, on the other hand, that it serves as a powerful modeling language that lets us easily formulate several constrained planarity problems as instances of Synchronized Planarity. In particular, this lets us solve Clustered Planarity in quadratic time, where the most efficient previously known algorithm has an upper bound of O(n⁸)
Cubic Augmentation of Planar Graphs
In this paper we study the problem of augmenting a planar graph such that it
becomes 3-regular and remains planar. We show that it is NP-hard to decide
whether such an augmentation exists. On the other hand, we give an efficient
algorithm for the variant of the problem where the input graph has a fixed
planar (topological) embedding that has to be preserved by the augmentation. We
further generalize this algorithm to test efficiently whether a 3-regular
planar augmentation exists that additionally makes the input graph connected or
biconnected. If the input graph should become even triconnected, we show that
the existence of a 3-regular planar augmentation is again NP-hard to decide.Comment: accepted at ISAAC 201
- …