## Graph Treewidth and Geometric Thickness Parameters (2005)

Venue: | DISCRETE AND COMPUTATIONAL GEOMETRY |

Citations: | 13 - 7 self |

### BibTeX

@MISC{Dujmović05graphtreewidth,

author = {Vida Dujmović and David R. Wood},

title = { Graph Treewidth and Geometric Thickness Parameters },

year = {2005}

}

### OpenURL

### Abstract

Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k/2⌉. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215–221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.

### Citations

380 | The factors of graphs
- Tutte
- 1952
(Show Context)
Citation Context ... that admits an outer noncrossing drawing is outerplanar. The thickness of a graph G, denoted by θ(G), is the minimum number of planar subgraphs that partition G. Thickness was first defined by Tutte =-=[73]-=-; see the surveys [46, 60]. The outerthickness of a graph G, denoted by θo(G), is the minimum number of outerplanar subgraphs that partition G. Outerthickness was first studied by Guy [40]; also see [... |

329 |
Graph Drawing: Algorithms for the Visualization of Graphs
- Battista, Eades, et al.
- 1999
(Show Context)
Citation Context ... that f(G) ∈ N := {0,1,2,... } for every graph G. For a graph class G, let f(G) := max{f(G) : G ∈ G}. If f(G) is unbounded, we write f(G) := ∞. Our interest is in drawings of graphs in the plane; see =-=[19, 23, 51, 62, 69]-=-. A drawing φ of graph G is a pair (φV ,φE), where: • φV is an injection from the vertex set V (G) into R 2 , and • φE is a mapping from the edge set E(G) into the set of simple curves 1 in R 2 , such... |

275 |
A First Course
- Munkres, Topology
- 1975
(Show Context)
Citation Context ...is the minimum number of forests that partition G. Nash-Williams [61] proved that � � |E(H)| a(G) = max . (1) H⊆G |V (H)| − 1 1 A simple curve is a homeomorphic image of the closed unit interval; see =-=[59]-=- for background topology. 2 In the literature on crossing numbers it is customary to require that intersecting edges cross ‘properly’ and do not ‘touch’. This distinction will not be important in this... |

248 | A Partial k-Arboretum of Graphs with Bounded Treewidth
- Bodlaender
- 1998
(Show Context)
Citation Context ... concern the relationship between the above parameters and treewidth, which is a more modern graph parameter that is particularly important in structural and algorithmic graph theory; see the surveys =-=[16, 66]-=-. In particular, we determine the maximum thickness, maximum outerthickness, maximum arboricity, and maximum star-arboricity of a graph with treewidth k. These results are presented in Section 3 (foll... |

135 |
Lectures on Discrete Geometry
- Matouˇsek
(Show Context)
Citation Context ...{u1,u2,... ,u5}, Tε(v) is shaded, and v is ε-empty. (b) The new edges coloured 1 in Proposition 5. Proof. Consider the arrangement A consisting of the lines through every pair of vertices in G\v; see =-=[57]-=- for background on line arrangements. Since V (G) is in general position, v is in some cell C of A. Since C is an open set, there exists ε > 0 such that Dε(v) ⊂ C. For every neighbour u ∈ NG(v), no ve... |

123 |
On straight line representation of planar graphs
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- 1948
(Show Context)
Citation Context ...ric drawing of G with thickness k. Kainen [50] first defined geometric thickness under the name of real linear thickness, and it has also been called rectilinear thickness. By the Fáry-Wagner theorem =-=[35, 74]-=-, a graph has geometric thickness 1 if and only if it is planar. Graphs of geometric thickness 2, the so-called doubly linear graphs, were studied by Hutchinson et al. [48]. The outerthickness (respec... |

96 |
Bemerkungen zum Vierfarbenproblem. Jahresbericht der Deutschen Mathematiker-Vereinigung
- Wagner
- 1936
(Show Context)
Citation Context ...ric drawing of G with thickness k. Kainen [50] first defined geometric thickness under the name of real linear thickness, and it has also been called rectilinear thickness. By the Fáry-Wagner theorem =-=[35, 74]-=-, a graph has geometric thickness 1 if and only if it is planar. Graphs of geometric thickness 2, the so-called doubly linear graphs, were studied by Hutchinson et al. [48]. The outerthickness (respec... |

84 |
The book thickness of a graph
- Bernhart, Kainen
- 1979
(Show Context)
Citation Context ...nces. A book embedding is also called a stack layout, and book thickness is also called stacknumber, pagenumber and fixed outerthickness. A graph has book thickness 1 if and only if it is outerplanar =-=[13]-=-. Bernhart and Kainen [13] proved that a graph has a book thickness at most 2 if and only if it is a subgraph of a Hamiltonian planar graph. Yannakakis [78] proved that every planar graph has book thi... |

81 |
Research problems in discrete geometry
- Brass, Moser, et al.
- 2005
(Show Context)
Citation Context ... that f(G) ∈ N := {0,1,2,... } for every graph G. For a graph class G, let f(G) := max{f(G) : G ∈ G}. If f(G) is unbounded, we write f(G) := ∞. Our interest is in drawings of graphs in the plane; see =-=[19, 23, 51, 62, 69]-=-. A drawing φ of graph G is a pair (φV ,φE), where: • φV is an injection from the vertex set V (G) into R 2 , and • φE is a mapping from the edge set E(G) into the set of simple curves 1 in R 2 , such... |

75 |
An extremal function for contractions of graphs
- Thomason
- 1984
(Show Context)
Citation Context ...lity of k-trees. Minors: Let Mℓ be the class of graphs with no Kℓ-minor. Note that M3 = T1 and M4 = T2. Jünger et al. [49] proved that θ(M5) = 2. What is θ(M5) and bt(M5)? Kostochka [53] and Thomason =-=[70]-=- independently proved that the maximum arboricity of all graphs with no Kℓ minor is Θ(ℓ √ log ℓ). In fact, Thomason [71] asymptotically determined the right constant. Thus θ(Mℓ) ∈ Θ(ℓ √ log ℓ) by Equa... |

68 |
Minimum partition of a matroid into independent subsets
- Edmonds
- 1965
(Show Context)
Citation Context ...s of ∆. For example, the best bounds on bt(D3) are Ω(n 1/6 ) and O(n 1/2 ). Computational Complexity: Arboricity can be computed in polynomial time using the matroid partitioning algorithm of Edmonds =-=[30]-=-. Computing the thickness of a graph is N P-hard [56]. Testing whether a graph has book thickness at most 2 is N Pcomplete [76]. Dillencourt et al. [26] asked what is the complexity of determining the... |

64 |
Drawing Graphs: Methods and Models
- Kaufmann, Wagner
- 1999
(Show Context)
Citation Context ... that f(G) ∈ N := {0,1,2,... } for every graph G. For a graph class G, let f(G) := max{f(G) : G ∈ G}. If f(G) is unbounded, we write f(G) := ∞. Our interest is in drawings of graphs in the plane; see =-=[19, 23, 51, 62, 69]-=-. A drawing φ of graph G is a pair (φV ,φE), where: • φV is an injection from the vertex set V (G) into R 2 , and • φE is a mapping from the edge set E(G) into the set of simple curves 1 in R 2 , such... |

54 |
The extremal function for complete minors
- Thomason
(Show Context)
Citation Context ...9] proved that θ(M5) = 2. What is θ(M5) and bt(M5)? Kostochka [53] and Thomason [70] independently proved that the maximum arboricity of all graphs with no Kℓ minor is Θ(ℓ √ log ℓ). In fact, Thomason =-=[71]-=- asymptotically determined the right constant. Thus θ(Mℓ) ∈ Θ(ℓ √ log ℓ) by Equation (2). Blankenship and Oporowski [14, 15] proved that bt(Mℓ) (and hence θ(Mℓ)) is finite. The proofsGraph Treewidth a... |

53 |
The minimum Hadwiger number for graphs with a given mean degree of vertices, Metody Diskret
- Kostochka
- 1982
(Show Context)
Citation Context ...e (k + 1)colourability of k-trees. Minors: Let Mℓ be the class of graphs with no Kℓ-minor. Note that M3 = T1 and M4 = T2. Jünger et al. [49] proved that θ(M5) = 2. What is θ(M5) and bt(M5)? Kostochka =-=[53]-=- and Thomason [70] independently proved that the maximum arboricity of all graphs with no Kℓ minor is Θ(ℓ √ log ℓ). In fact, Thomason [71] asymptotically determined the right constant. Thus θ(Mℓ) ∈ Θ(... |

46 | The linear arboricity of graphs
- Alon
- 1988
(Show Context)
Citation Context ...ch component is a star. The star-arboricity of a graph G, denoted by sa(G), is the minimum number of star-forests that partition G. Star arboricity was first studied by Akiyama and Kano [1]; also see =-=[3, 4, 5, 39, 43, 47]-=-. It is well known that thickness, outerthickness, arboricity, and star-arboricity are within a constant factor of each other. In particular, Gonçalves [38] recently proved a longstanding conjecture t... |

46 |
Nash-Williams, Decomposition of finite graphs into forests
- A
- 1964
(Show Context)
Citation Context ...tion G. Outerthickness was first studied by Guy [40]; also see [31, 38, 41, 42, 52, 65]. The arboricity of a graph G, denoted by a(G), is the minimum number of forests that partition G. Nash-Williams =-=[61]-=- proved that � � |E(H)| a(G) = max . (1) H⊆G |V (H)| − 1 1 A simple curve is a homeomorphic image of the closed unit interval; see [59] for background topology. 2 In the literature on crossing numbers... |

38 | A successful concept for measuring non-planarity of graphs: the crossing number
- Székely
(Show Context)
Citation Context |

31 |
Embedding planar graphs in four pages
- Yannakakis
- 1989
(Show Context)
Citation Context ...hickness 1 if and only if it is outerplanar [13]. Bernhart and Kainen [13] proved that a graph has a book thickness at most 2 if and only if it is a subgraph of a Hamiltonian planar graph. Yannakakis =-=[78]-=- proved that every planar graph has book thickness at most 4. The book arboricity (respectively, book star-arboricity) of a graph G, denoted by ba(G) (bsa(G)), is the minimum k ∈ N such that there is ... |

30 | On linear layouts of graphs
- Dujmovic, Wood
- 2004
(Show Context)
Citation Context ...e of a book and each noncrossing subgraph being drawn without crossings on a single page. Book embeddings, first defined by Ollmann [63], are ubiquitous structures with a variety of applications; see =-=[28]-=- for a survey with over 50 references. A book embedding is also called a stack layout, and book thickness is also called stacknumber, pagenumber and fixed outerthickness. A graph has book thickness 1 ... |

30 | Embedding planar graphs at fixed vertex locations
- Pach, Wenger
(Show Context)
Citation Context ...k + 1.sGraph Treewidth and Geometric Thickness Parameters 8 one of k colours such that edges with same colour do not cross. Every planar graph can be drawn with its vertices at prespecified locations =-=[44, 64]-=-. Thus a graph with thickness k has a drawing with thickness k [44]. However, in such a drawing the edges might be highly curved. This motivates the notion of geometric thickness. A drawing (φV ,φE) o... |

28 | On simultaneous planar graph embeddings
- Braß, Cenek, et al.
(Show Context)
Citation Context ...hickness? Whether all graphs with arboricity 2 have bounded geometric thickness is also interesting. It is easily seen that graphs with star arboricity 2 have geometric star arboricity at most 2 (cf. =-=[18]-=-). Book Arboricity: Bernhart and Kainen [13] proved that every graph G with book thickness t satisfies |E(G)| ≤ (t+1)|V (G)| −3t. Thus Equation (1) implies that a(G) ≤ bt(G) + 1 for every graph G, as ... |

27 | Geometric thickness of complete graphs
- Dillencourt, Eppstein, et al.
- 2000
(Show Context)
Citation Context ...of Kn has arboricity (and thus thickness) at most n − � n/12. It is unknown whether for some constant ε > 0, every geometric drawing of Kn has thickness at most (1 − ε)n; see [17]. Dillencourt et al. =-=[26]-=- studied the geometric thickness of Kn, and proved that 5 ⌈(n/5.646) + 0.342⌉ ≤ θ(Kn) ≤ ⌈n/4⌉ . What is θ(Kn)? It seems likely that the answer is closer to ⌈n/4⌉ rather than the above lower bound. Asy... |

26 |
On representation of some thickness-two graphs
- Hutchinson, Shermer, et al.
- 1995
(Show Context)
Citation Context ...the Fáry-Wagner theorem [35, 74], a graph has geometric thickness 1 if and only if it is planar. Graphs of geometric thickness 2, the so-called doubly linear graphs, were studied by Hutchinson et al. =-=[48]-=-. The outerthickness (respectively, arboricity, star-arboricity) of a graph drawing is the minimum k ∈ N such that the edges of the drawing can be partitioned into k outer noncrossing subdrawings (non... |

25 | Star coloring of graphs
- Fertin, Raspaud, et al.
(Show Context)
Citation Context ... be concluded from a result by Hakimi et al. [43]. A vertex colouring with no bichromatic edge and no bichromatic cycle is acyclic. It is folklore that every k-tree G has an acyclic (k + 1)-colouring =-=[36]-=-. (Proof. If G ≃ Kk+1 then the result is trivial. Otherwise, let v be a k-simplicial vertex. By induction, G \ v has an acyclic (k +1)-colouring. One colour is not present on the k neighbours of v. Gi... |

25 |
Genus g graphs have pagenumber O
- Malitz
- 1994
(Show Context)
Citation Context ...at most γ. Dean and Hutchinson [21] proved that θ(Sγ) ≤ 6 + √ 2γ − 2; also see [7, 8]. What is the minimum c such that θ(Sγ) ≤ (c + o(1)) √ γ? Building on prior work by Heath and Istrail [45], Malitz =-=[54]-=- proved using a probabilistic argument that bt(Sγ) ∈ O( √ γ), and thus θ(Sγ) ∈ O( √ γ). Is there a constructive proof that bt(Sγ) ∈ O( √ γ) or θ(Sγ) ∈ O( √ γ). What is the minimum c such that bt(Sγ) ≤... |

24 |
The geometric thickness of low degree graphs
- Duncan, Eppstein, et al.
- 2004
(Show Context)
Citation Context ...at θ(D∆) ≤ ⌈∆/2⌉, and S´ykora et al. [68] proved that θ(D∆) ≥ ⌈∆/2⌉. Thus θ(D∆) = ⌈∆/2⌉. Eppstein [34] asked whether θ(D∆) is finite. A positive result in this direction was obtained by Duncan et al. =-=[29]-=-, who proved that θ(D4) ≤ 2. On the other hand, Barát et al. [9] recently proved that θ(D∆) = ∞ for all ∆ ≥ 9; in particular, there exists ∆-regular n-vertex graphs with geometric thickness Ω( √ ∆n 1/... |

24 |
Determining the thickness of a graph is NP-hard
- Mansfield
- 1983
(Show Context)
Citation Context ...n 1/6 ) and O(n 1/2 ). Computational Complexity: Arboricity can be computed in polynomial time using the matroid partitioning algorithm of Edmonds [30]. Computing the thickness of a graph is N P-hard =-=[56]-=-. Testing whether a graph has book thickness at most 2 is N Pcomplete [76]. Dillencourt et al. [26] asked what is the complexity of determining the geometric thickness of a given graph? The same quest... |

23 |
Graph Theory, vol. 173 of Graduate Texts in Mathematics
- Diestel
- 2000
(Show Context)
Citation Context ...roofs of our geometric results are in Sections 7–9. Section 10 concludes with numerous open problems. 2 Background Graph Theory For undefined graph-theoretic terminology, see the monograph by Diestel =-=[23]-=-. We consider graphs G that are simple, finite, and undirected. Let V (G) and E(G) respectively denote the vertex and edge sets of G. For A,B ⊆ V (G), let G[A;B] denote the bipartite subgraph of G wit... |

18 | Thethickness of graphs: a survey
- Mutzel, Odenthal, et al.
- 1998
(Show Context)
Citation Context ...noncrossing drawing is outerplanar. The thickness of a graph G, denoted by θ(G), is the minimum number of planar subgraphs that partition G. Thickness was first defined by Tutte [73]; see the surveys =-=[46, 60]-=-. The outerthickness of a graph G, denoted by θo(G), is the minimum number of outerplanar subgraphs that partition G. Outerthickness was first studied by Guy [40]; also see [31, 38, 41, 42, 52, 65]. T... |

17 |
The thickness of the complete graph
- Beineke, Harary
- 1965
(Show Context)
Citation Context ...Treewidth and Geometric Thickness Parameters 27 10 Open Problems Complete Graphs: The thickness of the complete graph Kn was intensely studied in the 1960’s and 1970’s. Results by a number of authors =-=[2, 10, 11, 58]-=- together prove that θ(Kn) = ⌈(n + 2)/6⌉, unless n = 9 or 10, in which case θ(K9) = θ(K10) = 3. Bernhart and Kainen [13] proved that bt(Kn) = ⌈n/2⌉. In fact, it is easily seen that a(Kn) = a(Kn) = bt(... |

17 |
Algorithmic aspects of tree width
- REED
- 2003
(Show Context)
Citation Context ... concern the relationship between the above parameters and treewidth, which is a more modern graph parameter that is particularly important in structural and algorithmic graph theory; see the surveys =-=[16, 66]-=-. In particular, we determine the maximum thickness, maximum outerthickness, maximum arboricity, and maximum star-arboricity of a graph with treewidth k. These results are presented in Section 3 (foll... |

16 | Separating thickness from geometric thickness
- Eppstein
- 2004
(Show Context)
Citation Context ...] (also see [14]) constructed n-vertex graphs Gn with sa(Gn) = a(Gn) = θ(Gn) = θ(Gn) = 2 and bt(Gn) → ∞. Thus book thickness is not bounded by any function of geometric thickness. Similarly, Eppstein =-=[34]-=- constructed n-vertex graphs Hn with sa(Hn) = a(Hn) = θ(Hn) = 3 and θ(Hn) → ∞. Thus geometric thickness is not bounded by any function of thickness (or arboricity). Eppstein [34] asked whether graphs ... |

16 |
Graphs with E edges have pagenumber O
- Malitz
- 1994
(Show Context)
Citation Context .../3 + 3/2. What is the minimum c such that θ(Em) ≤ (c + o(1)) √ m? Dean et al. [22] conjectured that the answer is c = 1/16, which would be tight for the balanced complete bipartite graph [12]. Malitz =-=[55]-=- proved using a probabilistic argument that bt(Em) ≤ 72 √ m. Is there a constructive proof that bt(Em) ∈ O( √ m) or θ(Em) ∈ O( √ m)? What is the minimum c such that θ(Em) ≤ (c + o(1)) √ m or bt(Em) ≤ ... |

15 |
The thickness of an arbitrary complete graph
- Alekseev, Goncakov
- 1976
(Show Context)
Citation Context ...Treewidth and Geometric Thickness Parameters 27 10 Open Problems Complete Graphs: The thickness of the complete graph Kn was intensely studied in the 1960’s and 1970’s. Results by a number of authors =-=[2, 10, 11, 58]-=- together prove that θ(Kn) = ⌈(n + 2)/6⌉, unless n = 9 or 10, in which case θ(K9) = θ(K10) = 3. Bernhart and Kainen [13] proved that bt(Kn) = ⌈n/2⌉. In fact, it is easily seen that a(Kn) = a(Kn) = bt(... |

15 | On the thickness of graphs of given degree
- Halton
- 1991
(Show Context)
Citation Context ...k + 1.sGraph Treewidth and Geometric Thickness Parameters 8 one of k colours such that edges with same colour do not cross. Every planar graph can be drawn with its vertices at prespecified locations =-=[44, 64]-=-. Thus a graph with thickness k has a drawing with thickness k [44]. However, in such a drawing the edges might be highly curved. This motivates the notion of geometric thickness. A drawing (φV ,φE) o... |

15 |
The pagenumber of genus g graphs is O(g
- Heath, Istrail
- 1992
(Show Context)
Citation Context ...s with genus at most γ. Dean and Hutchinson [21] proved that θ(Sγ) ≤ 6 + √ 2γ − 2; also see [7, 8]. What is the minimum c such that θ(Sγ) ≤ (c + o(1)) √ γ? Building on prior work by Heath and Istrail =-=[45]-=-, Malitz [54] proved using a probabilistic argument that bt(Sγ) ∈ O( √ γ), and thus θ(Sγ) ∈ O( √ γ). Is there a constructive proof that bt(Sγ) ∈ O( √ γ) or θ(Sγ) ∈ O( √ γ). What is the minimum c such ... |

15 |
Thickness and coarseness of graphs
- Kainen
- 1973
(Show Context)
Citation Context ... vertices. We thus refer to φV as a geometric drawing. The geometric thickness of a graph G, denoted by θ(G), is the minimum k ∈ N such that there is a geometric drawing of G with thickness k. Kainen =-=[50]-=- first defined geometric thickness under the name of real linear thickness, and it has also been called rectilinear thickness. By the Fáry-Wagner theorem [35, 74], a graph has geometric thickness 1 if... |

15 |
The complexity of the hamiltonian circuit problem for maximal planar graphs
- Wigderson
- 1982
(Show Context)
Citation Context ...d in polynomial time using the matroid partitioning algorithm of Edmonds [30]. Computing the thickness of a graph is N P-hard [56]. Testing whether a graph has book thickness at most 2 is N Pcomplete =-=[76]-=-. Dillencourt et al. [26] asked what is the complexity of determining the geometric thickness of a given graph? The same question can be asked for all of the other parameters discussed in this paper. ... |

13 | Bounded-degree graphs have arbitrarily large geometric thickness
- Barát, Matouˇsek, et al.
- 1999
(Show Context)
Citation Context ...2⌉. Thus θ(D∆) = ⌈∆/2⌉. Eppstein [34] asked whether θ(D∆) is finite. A positive result in this direction was obtained by Duncan et al. [29], who proved that θ(D4) ≤ 2. On the other hand, Barát et al. =-=[9]-=- recently proved that θ(D∆) = ∞ for all ∆ ≥ 9; in particular, there exists ∆-regular n-vertex graphs with geometric thickness Ω( √ ∆n 1/2−4/∆−ε ). It is unknown whether θ(D∆) is finite for ∆ ∈ {5,6,7,... |

13 |
On the thickness and arboricity of a graph
- Dean, Hutchinson, et al.
- 1991
(Show Context)
Citation Context ...ga in Latvia. He gave a construction that showed θ(Kn) ≤ ⌈n/4⌉.”sGraph Treewidth and Geometric Thickness Parameters 28 Number of Edges: Let Em be the class of graphs with at most m edges. Dean et al. =-=[22]-=- proved that θ(Em) ≤ � m/3 + 3/2. What is the minimum c such that θ(Em) ≤ (c + o(1)) √ m? Dean et al. [22] conjectured that the answer is c = 1/16, which would be tight for the balanced complete bipar... |

11 |
On the thickness of the complete bipartite graph
- BEINEKE, HARARY, et al.
- 1964
(Show Context)
Citation Context ...t θ(Em) ≤ � m/3 + 3/2. What is the minimum c such that θ(Em) ≤ (c + o(1)) √ m? Dean et al. [22] conjectured that the answer is c = 1/16, which would be tight for the balanced complete bipartite graph =-=[12]-=-. Malitz [55] proved using a probabilistic argument that bt(Em) ≤ 72 √ m. Is there a constructive proof that bt(Em) ∈ O( √ m) or θ(Em) ∈ O( √ m)? What is the minimum c such that θ(Em) ≤ (c + o(1)) √ m... |

11 |
The pagenumber of k-trees is
- Ganley, Heath
(Show Context)
Citation Context ... ≤ 2. Note that bt(T2) = 2 since there are series-parallel graphs that are not outerplanar, K2,3 being the primary example. We prove the stronger result that ba(T2) = 2 in Section 7. Ganley and Heath =-=[37]-=- proved that every k-tree has a book embedding with thickness at most k + 1. In their proof, each noncrossing subgraph is in fact a star-forest. Thus bt(Tk) ≤ ba(Tk) ≤ bsa(Tk) ≤ k + 1 . (3) We give an... |

10 |
Book Embeddings of Graphs
- Blankenship
- 2003
(Show Context)
Citation Context ...um arboricity of all graphs with no Kℓ minor is Θ(ℓ √ log ℓ). In fact, Thomason [71] asymptotically determined the right constant. Thus θ(Mℓ) ∈ Θ(ℓ √ log ℓ) by Equation (2). Blankenship and Oporowski =-=[14, 15]-=- proved that bt(Mℓ) (and hence θ(Mℓ)) is finite. The proofsGraph Treewidth and Geometric Thickness Parameters 29 depends on Robertson and Seymour’s deep structural characterisation of the graphs in Mℓ... |

10 |
Planar Graph Drawing
- Nishizeki, Rahman
- 2004
(Show Context)
Citation Context |

9 |
Path factors of a graph
- Akiyama, Kano
- 1984
(Show Context)
Citation Context ...ph in which each component is a star. The star-arboricity of a graph G, denoted by sa(G), is the minimum number of star-forests that partition G. Star arboricity was first studied by Akiyama and Kano =-=[1]-=-; also see [3, 4, 5, 39, 43, 47]. It is well known that thickness, outerthickness, arboricity, and star-arboricity are within a constant factor of each other. In particular, Gonçalves [38] recently pr... |

9 |
The decomposition of complete graphs into planar subgraphs
- Beineke
- 1967
(Show Context)
Citation Context ...Treewidth and Geometric Thickness Parameters 27 10 Open Problems Complete Graphs: The thickness of the complete graph Kn was intensely studied in the 1960’s and 1970’s. Results by a number of authors =-=[2, 10, 11, 58]-=- together prove that θ(Kn) = ⌈(n + 2)/6⌉, unless n = 9 or 10, in which case θ(K9) = θ(K10) = 3. Bernhart and Kainen [13] proved that bt(Kn) = ⌈n/2⌉. In fact, it is easily seen that a(Kn) = a(Kn) = bt(... |

9 |
Book embeddability of SeriesParallel digraphs
- Giacomo, Didimo, et al.
(Show Context)
Citation Context ...tes that every tree has a 1-page book embedding, as proved by Bernhart and Kainen [13]. Rengarajan and Veni Madhavan [67] proved that every series-parallel graph has a 2-page book embedding (also see =-=[24]-=-); that is, bt(T2) ≤ 2. Note that bt(T2) = 2 since there are series-parallel graphs that are not outerplanar, K2,3 being the primary example. We prove the stronger result that ba(T2) = 2 in Section 7.... |

9 |
Veni Madhavan, Stack and queue number of 2-trees
- Rengarajan, E
- 1995
(Show Context)
Citation Context ...atisfy bt(Tk) = ba(Tk) = � k for k ≤ 2 , k + 1 for k ≥ 3 . Theorem 3 with k = 1 states that every tree has a 1-page book embedding, as proved by Bernhart and Kainen [13]. Rengarajan and Veni Madhavan =-=[67]-=- proved that every series-parallel graph has a 2-page book embedding (also see [24]); that is, bt(T2) ≤ 2. Note that bt(T2) = 2 since there are series-parallel graphs that are not outerplanar, K2,3 be... |

8 |
Relations among embedding parameters for graphs
- Dean, Hutchinson
- 1991
(Show Context)
Citation Context ... Kainen [13] proved that every graph G with book thickness t satisfies |E(G)| ≤ (t+1)|V (G)| −3t. Thus Equation (1) implies that a(G) ≤ bt(G) + 1 for every graph G, as observed by Dean and Hutchinson =-=[21]-=-. Is ba(G) ≤ bt(G) + 1? 5 Archdeacon [6] writes, “The question (of the value of θ(Kn)) was apparently first raised by Greenberg in some unpublished work. I read some of his personal notes in the libra... |

8 |
Graph theory, vol. 173 of Graduate Texts
- DIESTEL
(Show Context)
Citation Context ...roofs of our geometric results are in Sections 7–9. Section 10 concludes with numerous open problems. 2 Background Graph Theory For undefined graph-theoretic terminology, see the monograph by Diestel =-=[25]-=-. We consider graphs G that are simple, finite, and undirected. Let V (G) and E(G) respectively denote the vertex and edge sets of G. For A,B ⊆ V (G), let G[A;B] denote the bipartite subgraph of G wit... |