J. Halton, On the thickness of graphs of given degree, Info. Sci. 54 (1991), 219--238.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....as straight linesegments [7, Lemma 2.1] Note that this construction has O(n ) area under the vertex resolution rule. Upper bounds for the thickness of an arbitrary graph G include (G) m=3 3=2c by Dean et al. 10] G) 6 2 2 by Dean and Hutchinson [9] and (G) d =2e by Halton [14]. Of these three upper bounds, the rst two are asymptotically matched by the book thickness. In particular, Malitz [19] proved that bt(G) 2 O( and since m, bt(G) 2 O( m) a result proved independently by Malitz [20] It is an open problem to decide if the third of these upper bounds ....

J. H. Halton, On the thickness of graphs of given degree. Inform. Sci., 54(3):219-238, 1991.

.... the edges can be straight line segments in such a drawing [BK79] Thus, the book thickness of a graph is the same as its geometric outerplanar thickness , and thus, G) bt(G) Other upper bounds for the thickness of arbitrary graphs are b m=3 3=2c [DHS91] 6 2 2 [DH91] and d =2e [Hal91]. Of these three upper bounds, the rst two are asymptotically matched by the book thickness. That is, bt(G) 2 O( Mal94a] and since m, bt(G) 2 O( m) a result proved independently in [Mal94b] It is an open problem to decide if the third of these upper bounds is asymptotically ....

J. H. Halton. On the thickness of graphs of given degree. Inform. Sci., 54(3):219-238, 1991.

....number of planar subgraphs into which G can be decomposed. The thickness of complete graphs and complete bipartite graphs is known [1 4] but on the other hand, very little is known about the thickness of an arbitrary graph. We consider simple graphs only. Recently, Dean et al. 6] Halton [7] and Cimikowski [5] have studied the thickness of a graph as a function of the number of edges or as a function of its maximum degree. The present note continues this study. A somewhat different line of research is followed by Jtinger et al. 9] 2. Halton s theorem We say that a graph has ....

....graph as a function of the number of edges or as a function of its maximum degree. The present note continues this study. A somewhat different line of research is followed by Jtinger et al. 9] 2. Halton s theorem We say that a graph has degree d, if d is the maximum degree of its nodes. Halton [7] has shown that any graph G of degree d has 0(G) d 2] Haltoffs proof is based on Petersefts theorem. If a graph is regular and of even degree, then it is 2 factorable [8, p. 90] Hence, in order to prove the theorem, Halton first constructs a regular graph containing the given graph as a ....

J.H. Halton. On the thickness of graphs of given degree. Info. Sci. 54:219-238 (1991).

....number of planar subgraphs into which G can be decomposed. The thickness of complete graphs and complete bipartite graphs is known [1 4] but on the other hand, very little is known about the thickness of an arbitrary graph. We consider simple graphs only. Recently, Dean et al. 6] Halton [7] and Cimikowski [5] have studied the thickness of a graph as a function of the number of edges or as a function of its maximum degree. The present note continues this study. A somewhat different line of research is followed by Jnger et al. 9] 2. Halton s theorem We say that a graph has degree ....

....a graph as a function of the number of edges or as a function of its maximum degree. The present note continues this study. A somewhat different line of research is followed by Jnger et al. 9] 2. Halton s theorem We say that a graph has degree d, if d is the maximum degree of its nodes. Halton [7] has shown that any graph G of degree d has q(G) d 2. Halton s proof is based on Petersen s theorem. If a graph is regular and of even degree, then it is 2factorable [8, p. 90] Hence, in order to prove the theorem, Halton first constructs a regular graph containing the given graph as a ....

J.H. Halton. On the thickness of graphs of given degree. ########## 54:219-238 (1991).

.... [Asa87] and a graph of orientable genus 2 has thickness either 2 or 3 [Asa94] Every graph G = V, E) has thickness at most ## E 3 3 2 # [DHS91] If # and # are the minimum and maximum vertex degree of G, respectively, then # # 1 6 # # #(G) # # # 2 # [Wes84] Independently, Hal91] presents similar results about the relation between the minimum and maximum vertex degrees of a graph and its thickness. 5.4 Variations of Thickness Bernhart and Kainen [BK79, Kai90] introduced the book thickness of a graph. A book B with n # 0 pages consists of a line L in 3 dimensional ....

John H. Halton. On the Thickness of Graphs of Given Degree. Information Sciences, 54:219--238, 1991.

....2 (called doubly linear graphs) have been studied by Hutchinson et al. 13] where the connection with certain types of visibility graphs was explored. A notion related to geometrical thickness is that of (graph theoretical) thickness of a graph, #(G) which has been studied extensively [1, 3, 8, 9, 10, 14, 16] and has been defined as the minimum number of planar graphs into which a graph can be decomposed. The key di#erence between geometric thickness and graph theoretical thickness is that geometric thickness requires that the vertex placements be consistent at all layers and that straight line edges ....

J. H. Halton. On the thickness of graphs of given degree. Information Sciences, 54:219--238, 1991.

.... und ihre Anwendungen y Institut fur Informatik, Pohligstrae 1, 50969 Koln, Germany z Max Planck Institut fur Informatik, Im Stadtwald, 66123 Saarbrucken, Germany graphs, see, e.g. BW78] In some cases, there are (often relatively poor) bounds on the thickness of a graph ( DHS91] and [Hal91]) The thickness problem has applications in VLSI design. In electronic circuits, components are joined by means of conducting strips. These may not cross, since this would lead to undesirable signals. In this case, an insulated wire must be used. For that reason, circuits with a large number of ....

Halton, J., On the thickness of graphs of given degree, Info. Sci. 54 (1991), 219--238.

....graph, since the formula 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 summarize these three approaches. Theorem 3.8 If G = V; E) is a graph with jV j = n (n 10) jEj = m and maximal degree d, then i) AG76] K n ) n 7 6 , ii) DHS91] ....

....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 = n (n 10) jEj = m and maximal degree d, then i) AG76] K n ) n 7 6 , ii) DHS91] G) q m 3 7 6 , iii) [Hal91] (G) d 2 . Since graphs arising in practice are usually sparse and have a small maximal degree, Halton s attempt to relate the thickness of a graph to the maximal degree of the graph seems to be the most appropriate approach (see section 5) Halton also makes the following conjecture, which ....

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Halton, J., On the thickness of graphs of given degree, Info. Sci., 54 (1991), 219--238.

....a line segment, and assign each edge to one of k layers so that no two edges on the same layer cross. This corresponds to the notion of real linear thickness introduced by Kainen [10] A related notion is that of (graph theoretical) thickness of a graph, G) which has been studied extensively [1, 5, 6, 7, 9, 11] and has been defined as the Supported by NSF Grants CDA 9617349 and CCR 9703572. Supported by NSF Grant CCR 9258355 and matching funds from Xerox Corp. minimum number of planar graphs into which a graph can be decomposed. The key difference between geometric thickness and ....

J. H. Halton. On the thickness of graphs of given degree. Information Sciences, 54:219--238, 1991.

.... Algorithmen fur diskrete Probleme und ihre Anwendungen y Institut fur Informatik, Pohligstra e 1, 50969 Koln, Germany z Max Planck Institut fur Informatik, Im Stadtwald, 66123 Saarbrucken, Germany cases, there are (often relatively poor) bounds on the thickness of a graph ( DHS91] and [Hal91]) The thickness problem has applications in VLSI design. In electronic circuits, components are joined by means of conducting strips. These may not cross, since this would lead to undesirable signals. In this case, an insulated wire must be used. For that reason, circuits with a large number of ....

J. Halton, On the thickness of graphs of given degree, Info. Sci. 54 (1991), 219--238.

....graphs, almost all) complete bipartite graphs, hypercubes, and graphs of orientable genus 1 and 2. Computing the thickness is NP hard [4] For further results on thickness, see the survey papers [2, 5] Wessel [8] proved that ; 1) where d is the maximum vertex degree of the graph. Halton [3] independently proved the same upper bound, and in addition, he conjectured a stronger bound: 4 This research was supported in part by the EPSRC grant COJN8. This research was supported in part by the NSF contract 007 2187. This research was supported in part by the VEGA grant ....

Halton, J., On the thickness of graphs of given degree, Information Sciences 54 (1991), 219-238.

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J. Halton, On the thickness of graphs of given degree, Info. Sci. 54 (1991), 219--238.

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