## The Thickness of Graphs: A Survey (1998)

Venue: | Graphs Combin |

Citations: | 20 - 0 self |

### BibTeX

@ARTICLE{Mutzel98thethickness,

author = {Petra Mutzel and Thomas Odenthal and Mark Scharbrodt},

title = {The Thickness of Graphs: A Survey},

journal = {Graphs Combin},

year = {1998},

volume = {14},

pages = {59--73}

}

### Years of Citing Articles

### OpenURL

### Abstract

We give a state-of-the-art survey of the thickness of a graph from both a theoretical and a practical point of view. After summarizing the relevant results concerning this topological invariant of a graph, we deal with practical computation of the thickness. We present some modifications of a basic heuristic and investigate their usefulness for evaluating the thickness and determining a decomposition of a graph in planar subgraphs. Key words: Thickness, maximum planar subgraph, branch and cut 1 Introduction In VLSI circuit design, a chip is represented as a hypergraph consisting of nodes corresponding to macrocells and of hyperedges corresponding to the nets connecting the cells. A chip-designer has to place the macrocells on a printed circuit board (which usually consists of superimposed layers), according to several designing rules. One of these requirements is to avoid crossings, since crossings lead to undesirable signals. It is therefore desirable to find ways to handle wi...

### Citations

1259 |
Graph Theory
- Harary
- 1969
(Show Context)
Citation Context ...to the general pattern. 2 Planarity and Maximal Planarization We assume familiarity with standard graph-theoretic terminology. For a survey, we refer the reader to Beineke and Wilson [BW78] or Harary =-=[Har69]-=-. As planarity is a basic concept for the thickness of a graph and maximal planarization algorithms are used in section 5, we briefly list some results. Among all graphs, planar graphs are of special ... |

485 |
Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-tree Algorithms
- Booth, Lueker
- 1976
(Show Context)
Citation Context ...lved efficiently for planar graphs even if they are intractable in the general case. After detecting nonplanarity of a graph, using a standard planarity testing algorithm (see, e.g., Booth and Lueker =-=[BL76]-=- or Hopcroft and Tarjan [HT74]), it is often favorable to first extract a possible large planar subgraph and treat the remaining edges independently. If no edge can be added to this subgraph without d... |

395 | How to draw a graph
- Tutte
- 1963
(Show Context)
Citation Context ...checked all planar subgraphs with nine nodes!) of Selfridge's conjecture. In other words, K 9 is not biplanar, i.e., not the union of two planar subgraphs. Generalizing the term of biplanarity, Tutte =-=[Tut63b]-=- defined the thickness of a graph. For notational convenience, we refer to the planar subgraphs in the following as layers. Evidently, `(G) = 1 if and only if G is planar. A further observation is tha... |

238 | Efficient planarity testing
- Hopcroft, Tarjan
- 1974
(Show Context)
Citation Context ...aphs even if they are intractable in the general case. After detecting nonplanarity of a graph, using a standard planarity testing algorithm (see, e.g., Booth and Lueker [BL76] or Hopcroft and Tarjan =-=[HT74]-=-), it is often favorable to first extract a possible large planar subgraph and treat the remaining edges independently. If no edge can be added to this subgraph without destroying planarity, the subgr... |

58 |
Edge-Disjoint Spanning Trees of Finite Graphs
- Nash-Williams
- 1961
(Show Context)
Citation Context ...silon(G) of a graph G is the minimal number of forests whose union is G. In contrast to other topological invariants of a graph, the arboricity can be exactly determined using Nash-Williams' formula (=-=[NW61]-=-) \Upsilon(G) = maxH`G mH nH \Gamma1 for an induced subgraph H of G with mH edges and nH nodes. Clearly, `(G)s\Upsilon(G) and \Upsilon(G)s3 \Delta `(G), since a maximal planar subgraph is at most thre... |

32 |
Selected Topics in graph Theory
- Beineke, Wilson
- 1978
(Show Context)
Citation Context ...ich do not fit into the general pattern. 2 Planarity and Maximal Planarization We assume familiarity with standard graph-theoretic terminology. For a survey, we refer the reader to Beineke and Wilson =-=[BW78]-=- or Harary [Har69]. As planarity is a basic concept for the thickness of a graph and maximal planarization algorithms are used in section 5, we briefly list some results. Among all graphs, planar grap... |

30 | Maximum planar subgraphs and nice embeddings: Practical layout tools
- Junger, Mutzel
- 1996
(Show Context)
Citation Context ... the Thick PQ heuristic is by Jayakumar, Thulasiraman and Swamy in an implementation of Winter [Win93]. Finally, the Thick JM heuristic is founded on the branch and cut algorithm of Junger and Mutzel =-=[JM93b]-=-. First, the three heuristics are compared on the complete and complete bipartite graphs, since the exact values of the thickness of these graphs are known. Table 1 and Table 2 show that the ThickHT a... |

29 |
On representations of some thicknesstwo graphs
- Hutchinson, Shermer, et al.
- 1999
(Show Context)
Citation Context ...thout K 5 -minors, then `(G)s2. Moreover, graphs with thickness two have drawn some attention in the field of graph-drawing, where there are used in the study of so-called rectangle-visibility graphs =-=[HSV95]-=-. To our knowledge, the thickness of no other graph class has been settled yet. Moreover, we cannot expect to find a nice formula describing the thickness of an arbitrary graph, since the thickness pr... |

29 |
On the deletion of nonplanar edges of a graph
- Liu, Geldmacher
- 1977
(Show Context)
Citation Context ...ed worst-case ratio of 4 9 can be achieved using the algorithm of Calinescu et al. [CFFK96], which uses a socalledstriangle cactus approach. Although determining a maximum planar subgraph is NP-hard (=-=[LG77]-=-), the branch and cut algorithm of Junger and Mutzel ([JM93a, JM93b]) solves many problems to optimality (particular problems on sparse graphs) or achieves a good approximation. 3 Theoretical Results ... |

25 | O(n 2 ) algorithms for graph planarization - Jayakumar, Thulasiraman, et al. - 1989 |

24 |
Graphs with forbidden subgraphs
- Chartrand, Geller, et al.
- 1971
(Show Context)
Citation Context ...hat we cannot recommend this approach in general. 6 Miscellaneous Results This last section is devoted to some results which cannot be subsumed under a general topic. Chartrand, Geller and Hedetniemi =-=[CGH71] define a -=-line-partitionnumber 0 n (G) of a graph G, using a "Property P n ", which makes it possible to view the arboricity, the outerthickness and the thickness of a graph in a general framework, si... |

21 |
The thickness of the complete graph
- Beineke, Harary
(Show Context)
Citation Context ...4 e, if G has no triangles. 3 In the 1960's the cornerstone of the thickness-work on special graph classes was laid by Harary and Beineke, who published the first results on the thickness of complete =-=[BH65] and compl-=-ete bipartite graphs [BHM64]. But the determination of a "nice" formula describing the thickness of complete graphs has a long history and was completed by Alekseev and Goncakov [AG76]. By t... |

17 |
Thickness of arbitrary complete graphs
- Alekseev, Gončakov
(Show Context)
Citation Context ...plete [BH65] and complete bipartite graphs [BHM64]. But the determination of a "nice" formula describing the thickness of complete graphs has a long history and was completed by Alekseev and=-= Goncakov [AG76]-=-. By the way, the question of whether `(K 16 ) = 3 or `(K 16 ) = 4 gives rise to a little anecdote, since this question was the subject of a mathematical competition (exemplarily correcting the pictur... |

17 | An O(m log n)-time algorithm for the maximal planar subgraph problem
- Cai, Han, et al.
- 1993
(Show Context)
Citation Context ...d on the Hopcroft and Tarjan planarity testing algorithm and the other based on a PQtree approach. The fastest algorithm using the first method is the O(jEj log jV j) algorithm of Cai, Han and Tarjan =-=[CHT91]-=-. The history of maximal planarization 2 algorithms using the PQ-approach is quite interesting [OT81, JTS89, Kan92] and appears to be still incomplete [Lei94]. All these algorithms only guarantee that... |

17 | An O(n2) maximal planarization algorithm based on PQ-trees - Kant - 1992 |

16 | On the thickness of graphs of given degree
- Halton
- 1991
(Show Context)
Citation Context ...ry graphs. In the early 90's, two new results dealing with upper bounds were published. Dean, Hutchinson and Scheinerman [DHS91] correlate the thickness of a graph with the number of edges and Halton =-=[Hal91]-=- uses the maximal degree of a graph to compute an upper bound of the thickness of a graph. In the following theorem we summarize these three approaches. Theorem 3.8 If G = (V; E) is a graph with jV j ... |

14 |
On the thickness of the complete bipartite graph
- Beineke, Harary, et al.
- 1964
(Show Context)
Citation Context ...1960's the cornerstone of the thickness-work on special graph classes was laid by Harary and Beineke, who published the first results on the thickness of complete [BH65] and complete bipartite graphs =-=[BHM64]. But the -=-determination of a "nice" formula describing the thickness of complete graphs has a long history and was completed by Alekseev and Goncakov [AG76]. By the way, the question of whether `(K 16... |

14 |
On the thickness and arboricity of a graph
- Dean, Hutchinson, et al.
- 1991
(Show Context)
Citation Context ...mula of the thickness of complete graphs operates as an upper bound for arbitrary graphs. In the early 90's, two new results dealing with upper bounds were published. Dean, Hutchinson and Scheinerman =-=[DHS91]-=- correlate the thickness of a graph with the number of edges and Halton [Hal91] uses the maximal degree of a graph to compute an upper bound of the thickness of a graph. In the following theorem we su... |

12 |
Multilayer grid embeddings for VLSI, Algorithmica 6
- Aggarwal, Klawe, et al.
- 1991
(Show Context)
Citation Context ... Kyoto University, Kyoto 606, Japan contact cuts are used, the manufacturing cost measure of this method is the number of layers. An application of this approach was given by Aggarwal, Klawe and Shor =-=[AKS91]-=-. They proposed a layout-algorithm with a provably good layoutarea. Since the algorithm needs a priori a decomposition of the graph in planar subgraphs, a graph-theoretic treatment seems helpful. Inde... |

11 | Relations among embedding parameters for graphs
- Dean, Hutchinson
- 1991
(Show Context)
Citation Context ...for the genus is not that easy. However, Asano [Asa87] proved that `(G)sfl(G) + 1 holds, if G contains no triangle. Furthermore, he showed that a graph of genus 1 has thickness 2. Dean and Hutchinson =-=[DH88]-=- strengthened Asano's result in proving that `(G)s6 + q 2 \Delta fl(G) \Gamma 2. In the subsequent lines we deal with restrictions on the shape of the planar subgraphs of the decomposition. The arbori... |

11 | A Graph-planarization Algorithm and its Applications to Random Graphs," Graph Theory and Algorithms, Lecture Notes in Computer Science 108(1981),95-107 [14J F.P. & R. Tamassia, "Fully Dynamic Point Location in a Monotone Subdivision - Ozawa, Takahashi - 1988 |

10 |
The decomposition of complete graphs into planar subgraphs
- Beineke
- 1967
(Show Context)
Citation Context ...e thickness of the complete graph K n is `(K n ) = n + 7 6 ; for n 6= 9; 10 and `(K 9 ) = `(K 10 ) = 3: As a by-product, the proof of Theorem 3.2 yields the following corollary. Corollary 3.3 [AG76], =-=[Bei67b]-=- The thickness of the n-dimensional octahedron K n(2) is `(K n(2) ) = 1 3 n : Along with their work on complete graphs, Beineke and Harary have computed the thickness of complete bipartite graphs in m... |

10 |
Every planar graph with nine points has a nonplanar complement
- BATTLE, HARARY, et al.
- 1962
(Show Context)
Citation Context ...me to him through Selfridge: "Prove or disprove the following conjecture: For any graph G with 9 points, G or its complementary graph G is nonplanar". In the following year, Harary, Battle a=-=nd Kodoma [BHK62]-=- and Tutte [Tut63a] independently gave a proof (they checked all planar subgraphs with nine nodes!) of Selfridge's conjecture. In other words, K 9 is not biplanar, i.e., not the union of two planar su... |

9 |
Embedding graphs in books: A survey
- Bilski
- 1992
(Show Context)
Citation Context ...book embedding, i.e., an arrangement of vertices in a line along the spine of the book and edges on the pages in such a way that edges residing on the same page do not cross. A survey can be found in =-=[Bil92]-=-. The relations to outerthickness and thickness are given by T o (G)sT b (G) and T (G)sd T b (G) 2 e. Another modification of the ground-concept is made by considering the thickness on other surfaces.... |

7 |
On the genus and thickness of graphs
- Asano
- 1987
(Show Context)
Citation Context ...umber (G) and the genus fl(G) (see, e.g., Harary [Har69]) of a graph G. Whereas the simple formula `(G)s(G)+1 fulfills the first relation, the situation for the genus is not that easy. However, Asano =-=[Asa87]-=- proved that `(G)sfl(G) + 1 holds, if G contains no triangle. Furthermore, he showed that a graph of genus 1 has thickness 2. Dean and Hutchinson [DH88] strengthened Asano's result in proving that `(G... |

7 |
private communication
- Jordan
(Show Context)
Citation Context ...ndwhile (4) ` 0 (G) := t We compare the results of the following three heuristics for planarization. The ThickHT heuristic is based on the Cai, Han and Tarjan algorithm in an implementation of Jordan =-=[Jor93]-=-. The basic algorithm for the Thick PQ heuristic is by Jayakumar, Thulasiraman and Swamy in an implementation of Winter [Win93]. Finally, the Thick JM heuristic is founded on the branch and cut algori... |

5 |
The non-biplanar character of the complete 9-graph
- TUTTE
- 1963
(Show Context)
Citation Context ...Selfridge: "Prove or disprove the following conjecture: For any graph G with 9 points, G or its complementary graph G is nonplanar". In the following year, Harary, Battle and Kodoma [BHK62] =-=and Tutte [Tut63a]-=- independently gave a proof (they checked all planar subgraphs with nine nodes!) of Selfridge's conjecture. In other words, K 9 is not biplanar, i.e., not the union of two planar subgraphs. Generalizi... |

4 | On computing a maximal planar subgraph using PQ-trees
- Junger, Leipert, et al.
- 1996
(Show Context)
Citation Context ...og jV j) algorithm of Cai, Han and Tarjan [CHT91]. The history of maximal planarization 2 algorithms using the PQ-approach is quite interesting [OT81, JTS89, Kan92] and appears to be still incomplete =-=[Lei94]-=-. All these algorithms only guarantee that the resulting planar subgraph is a spanning tree together with an unknown small number of additional edges, which results in a worst-case ratio of 1 3 . An i... |

4 |
Decomposition de K 16 en trois graphes planaires
- Mayer
- 1972
(Show Context)
Citation Context ... offer of $10 to anyone who could compute the thickness of K 16 . It lasted until 1972, when Jean Mayer, surprisingly a professor of french literature (!), won the prize by proving that `(K 16 ) = 3 (=-=[May72]-=-). We now list all known formulas describing the thickness of several graph classes. It is interesting to note that in the case of complete, complete bipartite graphs and hypercubes the lower bound of... |

3 |
Complete bipartite graphs: Decomposition into planar subgraphs, A seminar on Graph Theory
- Beineke
- 1967
(Show Context)
Citation Context ... non-constructive, Sir'an and Hor'ak [SH87] gave an explicit construction of an infinite number of t-minimal graphs of connectivity two, edge-connectivity t and minimal degree t. In addition, Beineke =-=[Bei67a]-=- proved that K 2t\Gamma1;4t 2 \Gamma10t+7 while Bouwer and Broere [BB68] proved that K 4t\Gamma5;4t\Gamma5 are t-minimal. The thickness of a graph can be related to two other topological invariants of... |

3 |
Outerthickness and outercoarseness of graphs
- Guy
- 1974
(Show Context)
Citation Context ...iction on a layer is the outer-planarity. The outerthickness ` o (G) 6 of a graph G is the minimal number of outerplanar graphs whose union is G. In addition to the trivial relation `(G)s` o (G), Guy =-=[Guy74]-=- has derived some results for the outerthickness of a graph. The tripartite and bipartite thickness of a graph are defined almost analogously to the other modifications. Walther [Wal68] and Wessel [We... |

3 |
Die Dicke des n-dimensionalen Wurfel-Graphen
- Kleinert
- 1967
(Show Context)
Citation Context ...of the complete bipartite graph K n;n is `(K n;n ) = n + 5 4 : 4 A further graph class, for which the thickness can be determined, is that of the hypercubes, whose thickness was evaluated by Kleinert =-=[Kle67]-=-. Theorem 3.6 [Kle67] The thickness of the hypercube Q n is `(Q n ) = n + 1 4 : Recently, Junger et al. [JMOS95] have shown that the thickness of a certain minor-excluded class of graphs is less than ... |

2 |
Minimal decompositions of complete graphs into subgraphs with embeddability properties, Canad
- Beineke
- 1969
(Show Context)
Citation Context ...-embeddable graphs whose union is G. Using Euler's generalized polyhedron-formula (see, e.g., Beineke and Wilson [BW78]), one derives a lower bound for the S-thickness similar to Theorem 3.1. Beineke =-=[Bei69] has repor-=-ted formulas for some surfaces by extending his constructions of the planar cases. Independently, Ringel [Rei65] found the "toroidal" thickness. 7 Theorem 4.3 [Bei69] The S-thickness of the ... |

2 |
Thickness of graphs with degree constrained vertices
- Bose, Prabhu
- 1977
(Show Context)
Citation Context ...its, the degree-4 thicknesss` 4 (G) of a graph G has been defined as the minimal number of planar subgraphs with maximal degree four, whose union is G. Using an explicit construction, Bose and Prabhu =-=[BP77]-=- computed the degree-4 thickness of complete graphs in almost all cases. Theorem 4.1 [BP77] The degree-4 thickness of the complete graph K n is i) n ? 5: ` 4 (K n ) = b n+3 4 c, except if n = 4p + 1 (... |

2 | Solving the Maximum Planar Subgraph Problem by Branch and Cut - Junger, Mutzel - 1993 |

1 |
Note on t-minimal bipartite graphs
- Bouwer, Broere
- 1968
(Show Context)
Citation Context ...of an infinite number of t-minimal graphs of connectivity two, edge-connectivity t and minimal degree t. In addition, Beineke [Bei67a] proved that K 2t\Gamma1;4t 2 \Gamma10t+7 while Bouwer and Broere =-=[BB68]-=- proved that K 4t\Gamma5;4t\Gamma5 are t-minimal. The thickness of a graph can be related to two other topological invariants of a graph: the crossing number (G) and the genus fl(G) (see, e.g., Harary... |

1 |
A Better Approximation Algorithm for the Maximum Planar Subgraph Problem
- Calinescu, Fernandes, et al.
- 1996
(Show Context)
Citation Context ...ree together with an unknown small number of additional edges, which results in a worst-case ratio of 1 3 . An improved worst-case ratio of 4 9 can be achieved using the algorithm of Calinescu et al. =-=[CFFK96]-=-, which uses a socalledstriangle cactus approach. Although determining a maximum planar subgraph is NP-hard ([LG77]), the branch and cut algorithm of Junger and Mutzel ([JM93a, JM93b]) solves many pro... |

1 |
Thickness and connectivity in graphs
- Hobbs, Grossman
- 1968
(Show Context)
Citation Context ...n t. Furthermore, in the same paper, Tutte derived the first results on t-minimal graphs and proved the existence of infinitely many t-minimal graphs satisfying several properties. Hobbs and Grossman =-=[HG68]-=- extended these results. They derived the existence of another class of t-minimal graphs and showed that each t-minimal graph (ts2) is at least t-edge-connected. While the proofs of the existence of t... |

1 |
The Thickness of a Minor-Excluded Class of Graphs, to appear
- Junger, Mutzel, et al.
- 1995
(Show Context)
Citation Context ... can be determined, is that of the hypercubes, whose thickness was evaluated by Kleinert [Kle67]. Theorem 3.6 [Kle67] The thickness of the hypercube Q n is `(Q n ) = n + 1 4 : Recently, Junger et al. =-=[JMOS95]-=- have shown that the thickness of a certain minor-excluded class of graphs is less than or equal to two. As a special case they obtained the following result. Theorem 3.7 [JMOS95] If G is a graph with... |

1 |
Determining the thickness of graphs is NP-hard
- Mansfiled
- 1983
(Show Context)
Citation Context ...r graph class has been settled yet. Moreover, we cannot expect to find a nice formula describing the thickness of an arbitrary graph, since the thickness problem was proven to be NP-hard by Mansfield =-=[Man83]-=-. Hence, we turn to upper bounds. A simple consideration gives the order O(n) of the thickness of a graph, since the formula of the thickness of complete graphs operates as an upper bound for arbitrar... |

1 |
Die torodiale Dicke des vollstandigen Graphen
- Ringel
- 1965
(Show Context)
Citation Context ...one derives a lower bound for the S-thickness similar to Theorem 3.1. Beineke [Bei69] has reported formulas for some surfaces by extending his constructions of the planar cases. Independently, Ringel =-=[Rei65] found the-=- "toroidal" thickness. 7 Theorem 4.3 [Bei69] The S-thickness of the complete graph K n (n ? 2) is projective plane : b n+5 6 c, torus : b n+4 6 c, double-torus : b n+3 6 c. Cases which still... |

1 |
A construction of thickness-minimal graphs
- Sir'an, Hor'ak
- 1987
(Show Context)
Citation Context ...howed that each t-minimal graph (ts2) is at least t-edge-connected. While the proofs of the existence of the classes of t-minimal graphs in [Tut63b] and [HG68] are non-constructive, Sir'an and Hor'ak =-=[SH87]-=- gave an explicit construction of an infinite number of t-minimal graphs of connectivity two, edge-connectivity t and minimal degree t. In addition, Beineke [Bei67a] proved that K 2t\Gamma1;4t 2 \Gamm... |

1 |
die Zerlegung des vollstandigen Graphen in paare planare Graphen, in: Beitrage zur Graphentheorie
- Walther, Uber
- 1968
(Show Context)
Citation Context ...)s` o (G), Guy [Guy74] has derived some results for the outerthickness of a graph. The tripartite and bipartite thickness of a graph are defined almost analogously to the other modifications. Walther =-=[Wal68]-=- and Wessel [Wes83] gave some results for complete graphs. Due to the application in the design of integrated circuits, the degree-4 thicknesss` 4 (G) of a graph G has been defined as the minimal numb... |

1 |
On some variations of the thickness of a graph connected with colouring, in: Graphs and other combinatorial topics, Teubner Texte Math. 59
- Wessel
- 1983
(Show Context)
Citation Context ...74] has derived some results for the outerthickness of a graph. The tripartite and bipartite thickness of a graph are defined almost analogously to the other modifications. Walther [Wal68] and Wessel =-=[Wes83]-=- gave some results for complete graphs. Due to the application in the design of integrated circuits, the degree-4 thicknesss` 4 (G) of a graph G has been defined as the minimal number of planar subgra... |

1 |
Heuristiken fur nicht-planare Graphen, Diplomarbeit Universit at Passau
- Winter
- 1993
(Show Context)
Citation Context ...ased on the Cai, Han and Tarjan algorithm in an implementation of Jordan [Jor93]. The basic algorithm for the Thick PQ heuristic is by Jayakumar, Thulasiraman and Swamy in an implementation of Winter =-=[Win93]-=-. Finally, the Thick JM heuristic is founded on the branch and cut algorithm of Junger and Mutzel [JM93b]. First, the three heuristics are compared on the complete and complete bipartite graphs, since... |