## On the Parameterized Complexity of Layered Graph Drawing (2001)

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Venue: | PROC. 5TH ANNUAL EUROPEAN SYMP. ON ALGORITHMS (ESA '01 |

Citations: | 21 - 8 self |

### BibTeX

@INPROCEEDINGS{Dujmovic01onthe,

author = {V. Dujmovic and M. Fellows and M. Hallett and M. Kitching and G. Liotta and C. McCartin and N. Nishimura and P. Ragde and F. Rosamond and M. Suderman and S. Whitesides and D. R. Wood},

title = {On the Parameterized Complexity of Layered Graph Drawing},

booktitle = {PROC. 5TH ANNUAL EUROPEAN SYMP. ON ALGORITHMS (ESA '01},

year = {2001},

pages = {488--499},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight line-segments between vertices on adjacent layers. We prove that graphs admitting crossing-free h-layer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a linear-time algorithm to decide if a graph has a crossing-free h-layer drawing (for fixed h). This algorithm is extended to solve a large number of related problems, including allowing at most k crossings, or removing at most r edges to leave a crossing-free drawing (for fixed k or r). If the number of crossings or deleted edges is a non-fixed parameter then these problems are NP-complete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case the total span or the maximum span of edges can be minimized. In contrast to the so-called Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.

### Citations

906 | Parametrized Complexity
- Downey, Fellows
- 1999
(Show Context)
Citation Context ...allowed crossings is small, an algorithm whose running time is exponential in this parameter but polynomial in the size of the graph may be useful. The theory of parameterized complexity (surveyed in =-=[7]-=-) addresses complexity issues of this nature, in which a problem is specified in terms of one or more parameters. A parameterized problem with input size n and parameter size k is fixed parameter trac... |

434 |
Combinatorial Algorithms for Integrated Circuit Layout
- Lengauer
- 1990
(Show Context)
Citation Context ... of the vertices to layers. 1 Introduction Layered graph drawing [28,5,26] is a popular paradigm for drawing graphs, and has applications in visualization [6], in DNA mapping [29], and in VLSI layout =-=[21]-=-. In a layered drawing of a graph, vertices are arranged in horizontal layers, and edges are routed as polygonal lines between distinct layers. For acyclic digraphs, it may be required that edges poin... |

359 |
Methods for Visual Understanding of Hierarchical Systems
- Sugiyama, Tagawa, et al.
(Show Context)
Citation Context ... can be minimized. In contrast to the so-called Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers. 1 Introduction Layered graph drawing =-=[28,5,26]-=- is a popular paradigm for drawing graphs, and has applications in visualization [6], in DNA mapping [29], and in VLSI layout [21]. In a layered drawing of a graph, vertices are arranged in horizontal... |

345 |
Graph Drawing: Algorithms for the Visualization of Graphs
- Battista, Eades, et al.
- 1999
(Show Context)
Citation Context ... our algorithms do not assume a preassignment of the vertices to layers. 1 Introduction Layered graph drawing [28,5,26] is a popular paradigm for drawing graphs, and has applications in visualization =-=[6]-=-, in DNA mapping [29], and in VLSI layout [21]. In a layered drawing of a graph, vertices are arranged in horizontal layers, and edges are routed as polygonal lines between distinct layers. For acycli... |

312 |
Approximation algorithms for NP-complete problems on planar graphs
- BAKER
- 1983
(Show Context)
Citation Context ...s are k-outerplanar, where k = ⌈ 1 2 (h +1)⌉. One could use an algorithm to find such an embedding in O(k3n2 ) time [3], and then apply Baker’s approach of dynamic programming on k-outerplanar graphs =-=[1]-=-. However, this approach depends heavily on planarity, and so does not appear to be amenable to allowing crossings or edge deletions. If we relax the requirement of using h layers, recent work gives a... |

214 |
Crossing number is np-complete
- Garey, Johnson
- 1983
(Show Context)
Citation Context ...es; and the maximum number of vertices in one layer. Unfortunately, the question of whether a graph G can be drawn in two layers with at most k crossings, where k is part of the input, is NP-complete =-=[11,12]-=-, as is the question of whether r or fewer edges can be removed from G so that the remaining graph has a crossing-free drawing on two layers [27,10]. Both problems remain NP-complete when the permutat... |

199 |
A linear-time algorithm for finding tree-decompositions of small treewidth
- Bodlaender
- 1996
(Show Context)
Citation Context ...Each Pi is called a bag of P . It is easily seen that the set of vertices in a bag is a separator of the graph G. For fixed w, path decompositions of graphs of pathwidth w can be found in linear time =-=[4]-=-. A path decomposition P = P1,...,Pp, of a graph G of pathwidth w is a normalized path decomposition if (1) |Pi| = w + 1 for i odd; (2) |Pi| = w for i even; and (3) Pi−1 ∩ Pi+1 = Pi for even i. Given ... |

88 |
Edge crossings in drawings of bipartite graphs
- Eades, Wormald
- 1994
(Show Context)
Citation Context ...es; and the maximum number of vertices in one layer. Unfortunately, the question of whether a graph G can be drawn in two layers with at most k crossings, where k is part of the input, is NP-complete =-=[11,12]-=-, as is the question of whether r or fewer edges can be removed from G so that the remaining graph has a crossing-free drawing on two layers [27,10]. Both problems remain NP-complete when the permutat... |

69 | 2-layer straightline crossing minimization: Performance of exact and heuristic algorithms
- Jünger, Mutzel
- 1997
(Show Context)
Citation Context ...such a drawing with straight line-segments [16], even for edges which span multiple layers [9]. Integer linear programming formulations have been developed for crossing minimization in layered graphs =-=[15,18,19]-=-, and for 2-layer planarization [22,24]. The special case of two layers is important for the layer-by-layer sweep approach. Junger and Mutzel [19] summarize the many heuristics for 2layer crossing min... |

60 | Straight-line drawing algorithms for hierarchical graphs and clustered graphs
- Eades, Feng, et al.
- 1996
(Show Context)
Citation Context ...ial time algorithms to test if a layered graph admits a crossing-free drawing [15,17], and if so, to produce such a drawing with straight line-segments [16], even for edges which span multiple layers =-=[9]-=-. Integer linear programming formulations have been developed for crossing minimization in layered graphs [15,18,19], and for 2-layer planarization [22,24]. The special case of two layers is important... |

50 |
Crossing theory and hierarchy mapping
- Warfield
- 1977
(Show Context)
Citation Context ... can be minimized. In contrast to the so-called Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers. 1 Introduction Layered graph drawing =-=[28,5,26]-=- is a popular paradigm for drawing graphs, and has applications in visualization [6], in DNA mapping [29], and in VLSI layout [21]. In a layered drawing of a graph, vertices are arranged in horizontal... |

47 |
Automatic display of hierarchized graphs for computer aided decision analysis
- Carpano
(Show Context)
Citation Context ... can be minimized. In contrast to the so-called Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers. 1 Introduction Layered graph drawing =-=[28,5,26]-=- is a popular paradigm for drawing graphs, and has applications in visualization [6], in DNA mapping [29], and in VLSI layout [21]. In a layered drawing of a graph, vertices are arranged in horizontal... |

32 |
Drawing graphs in two layers
- Eades, Whitesides
- 1994
(Show Context)
Citation Context ...sings, where k is part of the input, is NP-complete [11,12], as is the question of whether r or fewer edges can be removed from G so that the remaining graph has a crossing-free drawing on two layers =-=[27,10]-=-. Both problems remain NP-complete when the permutation of vertices in one of the layers is given [11,10]. When, say, the maximum number of allowed crossings is small, an algorithm whose running time ... |

29 | Edge crossings in drawings of bipartite graphs” Algorithmica - Eades, Wormald - 1994 |

28 | Computing Crossing Numbers in Quadratic Time
- Grohe
(Show Context)
Citation Context ... crossings or edge deletions. If we relax the requirement of using h layers, recent work gives an f(k) · n2 algorithm for recognizing graphs that can be embedded in the plane with at most k crossings =-=[14]-=-. A very similar approach would work for deleting r edges to leave a graph planar. Unfortunately, the approach relies on deep structure theorems from the Robertson-Seymour graph minors project, and so... |

28 | An alternative method to crossing minimization on hierarchial graphs
- Mutzel
- 1997
(Show Context)
Citation Context ... [16], even for edges which span multiple layers [9]. Integer linear programming formulations have been developed for crossing minimization in layered graphs [15,18,19], and for 2-layer planarization =-=[22,24]-=-. The special case of two layers is important for the layer-by-layer sweep approach. Junger and Mutzel [19] summarize the many heuristics for 2layer crossing minimization. Our companion paper [8] addr... |

27 |
Interval graphs and maps of DNA
- Waterman, Griggs
- 1986
(Show Context)
Citation Context ...ot assume a preassignment of the vertices to layers. 1 Introduction Layered graph drawing [28,5,26] is a popular paradigm for drawing graphs, and has applications in visualization [6], in DNA mapping =-=[29]-=-, and in VLSI layout [21]. In a layered drawing of a graph, vertices are arranged in horizontal layers, and edges are routed as polygonal lines between distinct layers. For acyclic digraphs, it may be... |

21 | Level planarity testing in linear time
- Jünger, Leipert, et al.
- 1998
(Show Context)
Citation Context ...he second phase of the Sugiyama algorithm has received much attention in the graph drawing literature. Notable are polynomial time algorithms to test if a layered graph admits a crossing-free drawing =-=[15,17]-=-, and if so, to produce such a drawing with straight line-segments [16], even for edges which span multiple layers [9]. Integer linear programming formulations have been developed for crossing minimiz... |

20 | Level Planar Embedding in Linear Time
- Jünger, Liepert
- 1999
(Show Context)
Citation Context ...he graph drawing literature. Notable are polynomial time algorithms to test if a layered graph admits a crossing-free drawing [15,17], and if so, to produce such a drawing with straight line-segments =-=[16]-=-, even for edges which span multiple layers [9]. Integer linear programming formulations have been developed for crossing minimization in layered graphs [15,18,19], and for 2-layer planarization [22,2... |

17 |
On the complexity of embedding planar graphs to minimize certain distance measures
- Bienstock, Monma
- 1990
(Show Context)
Citation Context ...to laying out proper h-layer planar graphs is to use the observation that such graphs are k-outerplanar, where k = ⌈ 1 2 (h +1)⌉. One could use an algorithm to find such an embedding in O(k3n2 ) time =-=[3]-=-, and then apply Baker’s approach of dynamic programming on k-outerplanar graphs [1]. However, this approach depends heavily on planarity, and so does not appear to be amenable to allowing crossings o... |

17 | A polyhedral approach to the multi-layer crossing minimization problem
- Jünger, Lee, et al.
- 1997
(Show Context)
Citation Context ...such a drawing with straight line-segments [16], even for edges which span multiple layers [9]. Integer linear programming formulations have been developed for crossing minimization in layered graphs =-=[15,18,19]-=-, and for 2-layer planarization [22,24]. The special case of two layers is important for the layer-by-layer sweep approach. Junger and Mutzel [19] summarize the many heuristics for 2layer crossing min... |

15 | A linear-time algorithm for tree-decompositions of small treewidth - Bodlaender - 1996 |

14 |
On planarization algorithms of 2-level graphs
- Tomii, Kambayashi, et al.
- 1977
(Show Context)
Citation Context ...sings, where k is part of the input, is NP-complete [11,12], as is the question of whether r or fewer edges can be removed from G so that the remaining graph has a crossing-free drawing on two layers =-=[27,10]-=-. Both problems remain NP-complete when the permutation of vertices in one of the layers is given [11,10]. When, say, the maximum number of allowed crossings is small, an algorithm whose running time ... |

11 | Fast layout algorithm for k-level graphs
- Buchheim, Jünger, et al.
- 2000
(Show Context)
Citation Context ...cessive pairs of neighbouring layers, the permutation of one layer is fixed, and a good permutation of the other layer is determined. The third phase of the method assigns coordinates to the vertices =-=[2]-=-. A disadvantage of the Sugiyama approach is that after the vertices have been assigned to layers in the first phase, these layer assignments are not changeds490 V. Dujmović etal. during the crossing ... |

11 | A fixed-parameter approach to twolayer planarization
- Dujmovic, Fellows, et al.
- 2002
(Show Context)
Citation Context ... [22,24]. The special case of two layers is important for the layer-by-layer sweep approach. Junger and Mutzel [19] summarize the many heuristics for 2layer crossing minimization. Our companion paper =-=[8]-=- addresses the 2-layer case. The remainder of this paper is organized as follows. Section 2 gives definitions and discusses pathwidth, a key concept for our algorithms. The overall framework for our a... |

6 | Two-layer planarization in graph drawing
- Mutzel, Weiskircher
- 1998
(Show Context)
Citation Context ... [16], even for edges which span multiple layers [9]. Integer linear programming formulations have been developed for crossing minimization in layered graphs [15,18,19], and for 2-layer planarization =-=[22,24]-=-. The special case of two layers is important for the layer-by-layer sweep approach. Junger and Mutzel [19] summarize the many heuristics for 2layer crossing minimization. Our companion paper [8] addr... |

5 | P.: Embeddings of k-connected graphs of pathwidth k
- Gupta, Nishimura, et al.
- 2005
(Show Context)
Citation Context ... + 1 for i odd; (2) |Pi| = w for i even; and (3) Pi−1 ∩ Pi+1 = Pi for even i. Given a path decomposition, a normalized path decomposition of the same width (and Θ(n) bags) can be found in linear time =-=[13]-=-.sOn the Parameterized Complexity of Layered Graph Drawing 491 A proper h-layer drawing of a (directed or undirected) graph G consists of a partition of the vertices V (G) intoh layers L1,L2,...Lh suc... |

5 | Crossing theory and hierarchy mapping - War - 1977 |

4 | The vertex-exchange graph and its use in multi-level graph layout
- Healy, Kuusik
- 1999
(Show Context)
Citation Context ...he second phase of the Sugiyama algorithm has received much attention in the graph drawing literature. Notable are polynomial time algorithms to test if a layered graph admits a crossing-free drawing =-=[15,17]-=-, and if so, to produce such a drawing with straight line-segments [16], even for edges which span multiple layers [9]. Integer linear programming formulations have been developed for crossing minimiz... |

4 |
Optimization in leveled graphs
- Mutzel
(Show Context)
Citation Context ...d r, can G be drawn so that the deletion of at most r edges removes all crossings?). The exact solution of the r-planarization problem for h ≥ 3 layers is stated as an open problem in a recent survey =-=[23]-=-, even with vertices preassigned to layers. Our algorithm can be modified to handle acyclic directed graphs whose edges must be drawn pointing downward. We also consider drawings whose edges are allow... |