A. Mansfield, Determining the thickness of graphs is NP-hard, Math. Proc. Cambridge Philos. Soc. 9 (1983), 9--23.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....union is G. By definition, cr 2 (G) 0 if and only if Theta(G) 2; i.e. G is biplanar. The nature of the crossing number and the biplanar crossing number problems seems different, since testing whether cr(G) 0 can be done in linear time, while testing biplanarity is an NP complete problem [12]. Asano s result [3] implies that if a graph is toroidal, then cr 2 (G) 0: Surveys on biplanar graphs and the thickness problem can be found in [5, 13] A k book embedding of a graph G consists of placing vertices of G on the spine of a book and drawing each edge on one of the k pages. The book ....

Mansfield, A., Determining the thickness of graphs is NP-hard, Mathematical Proceedings of the Cambridge Philosophical Society 9 (1983), 9--23.

....establishes many results about the thickness of graphs in one of the earliest papers about this topic. Surveys about thickness are [WB78] Bei88] and [MOS98] The following sections give a brief summary of the known results about thickness: Section 5. 1 describes the result of Mansfield [Man83] that says that determining the thickness of a graph is NP hard, and mentions heuristic approaches for finding the thickness. Thickness minimal graphs are discussed in Section 5.2, and Section 5.3 lists results about the thickness of graphs belonging A. Liebers, Planarizing Graphs , JGAA, 5(1) ....

....in Section 5.2, and Section 5.3 lists results about the thickness of graphs belonging A. Liebers, Planarizing Graphs , JGAA, 5(1) 1 74 (2001) 35 to particular classes of graphs. Finally, Section 5.4 mentions two variations of the thickness. 5. 1 Finding the Thickness of a Graph Mansfield [Man83] defines the following problem (that was already mentioned in [GJ79, Problem OPEN3] Problem 37 (Thickness [Man83] Given a graph G and a positive integer K, does the thickness of G satisfy #(G) # K Mansfield shows that this problem is NP complete for the fixed value K = 2, thus ....

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Anthony Mansfield. Determining the thickness of graphs is NP-hard. Math. Proc. Camb. Phil. Soc., 93:9--23, 1983.

....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 ....

....with arbitrarily many vertices for which the standard thickness and geometric thickness coincide. We show that the lower bound in Theorem 5.1 is not a tight bound for geometric thickness by showing that #(K 6,8 ) 3. A pair of planar drawings demonstrating that #(K 6,8 ) 2 can be found in [16]. Finally we show that the upper bound in Theorem 5.1 is also not tight, since #(K 6,6 ) 2 while Theorem 5.1 only implies that #(K 6,6 ) # 3. Theorem 5.1 For the complete bipartite graph K a,b , # ab 2a 2b 4 # # #(K a,b ) # #(K a,b ) # # min(a, b) 2 # . 5.1) Proof The ....

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A. Mansfield. Determining the thickness of a graph is NP-hard. Mathematical Proceedings of the Cambridge Philosophical Society, 93(9):9--23, 1983.

....of bounded treewidth, and we prove that for such graphs PrExt is polynomial even if the color bound is a part of the input. 2. Planar Bipartite Graphs We shall prove Theorem 1 in this section. We show a reduction from Planar 1 in 3 Satisfiability, a problem which is proven to be NP complete in [10]: PRECOLORING EXTENSION WITH FIXED COLOR BOUND 141 Instance: A formula Phi with a set C of clauses over a set X of variables in conjunctive normal form such that 1. every clause contains exactly 3 distinct variables; 2. the graph G Phi = X [ C; fxc j (x 2 c 2 C) x 2 c 2 C)g is planar. ....

Mansfield A., Determining the thickness of graphs is NP-hard, Proc. Math. Cambridge Phil. Soc. 93 (1983), 9--23. J. Kratochv#l, Charles University, Prague, Czech Republic

....E can be partitioned into k sets so that the graph induced by each set is planar) Therefore, the thickness is one measure of the degree of nonplanarity of a graph. Clearly, G) 1 if and only if G is planar. The thickness problem, asking for the thickness of a given graph G, is NP hard ([Man83]) so there is little hope to find a polynomial time algorithm for the thickness problem on general graphs. However, for some graph classes, the thickness can be determined in polynomial time. For example, the thickness is known for complete and complete bipartite Partially supported by DFG Grant ....

Mansfield, A., Determining the thickness of graphs is NP-hard, Math. Proc. Cambridge Philos. Soc. 9 (1983), 9--23.

....and a collection C of clauses over U with jcj = 3 for all c 2 C. Furthermore the bipartite graph G = V; E) where V = U [ C and E = ffx; cg : x or x occurs in cg is planar. Problem: Is there a satisfying truth assignment for C The NP completeness proof for this problem can be found in [9]. We use the planarity of the underlying graph of P3SAT to construct a planar graph in which we can easily determine the minimum t spanner. 3.1 Forcing Edges into a Minimum t Spanner For the construction of the instance of MinS t , we use the fact that we can force edges to be in every minimum ....

Anthony Mansfield. Determining the thickness of graphs is NP-hard. Math. Proc. Camb. Phil. Soc., 93:9--23, 1983.

....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 ....

....Conjecture 2.4 of [10] 3. Is it true that g (G) O ( G) for all graphs G It follows from Theorem 1 that this is true for complete graphs. 4. What is the complexity of computing g (G) for a given graph G In particular, is it NP complete (Computing (G) is known to be NP complete [11]. Table 1. Upper and lower bounds on g (Kn) established in this paper. n LB UB 1 4 1 1 5 8 2 2 9 12 3 3 13 14 3 4 15 16 4 4 17 20 4 5 21 24 5 6 25 26 5 7 27 28 6 7 29 31 6 8 32 7 8 33 36 7 9 37 7 10 n LB UB 38 40 8 10 41 43 8 11 44 9 11 45 48 9 12 49 52 10 13 53 54 10 14 55 56 11 14 57 60 ....

A. Mansfield. Determining the thickness of a graph is NP-hard. Mathematical Proceedings of the Cambridge Philosophical Society, 93(9):9--23, 1983.

....E can be partitioned into k sets so that the graph induced by each set is planar) Therefore, the thickness is one measure of the degree of nonplanarity of a graph. Clearly, G) 1 if and only if G is planar. The thickness problem, asking for the thickness of a given graph G, is NP hard ([Man83]) so there is little hope to find a polynomial time algorithm for the thickness problem on general graphs. However, for some graph classes, the thickness can be determined in polynomial time. For example, the thickness is known for complete and complete bipartite graphs [BW78] In some ....

A. Mansfield, Determining the thickness of graphs is NP-hard, Math. Proc. Cambridge Philos. Soc. 9 (1983), 9--23.

....and a collection C of clauses over U with jcj = 3 for all c 2 C. Furthermore the bipartite graph G = V; E) where V = U [ C and E = ffx; cg : x or x occurs in cg is planar. Problem: Is there a satisfying truth assignment for C The NP completeness proof for this problem can be found in [Man83] We use the planarity of the underlying graph of P3SAT to construct a planar graph in which we can easily determine the minimum spanner. 3.2 Forcing edges into a spanner To force an edge into a spanner, we construct certain auxiliary edges such that every minimum t spanner of the new graph ....

Anthony Mansfield. Determining the thickness of graphs is NP-hard. Math. Proc. Camb. Phil. Soc., 93:9--23, 1983.

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A. Mansfield, Determining the thickness of graphs is NP-hard, Math. Proc. Cambridge Philos. Soc. 9 (1983), 9--23.

No context found.

Mansfield, A., Determining the thickness of graphs is NP-hard, Mathematical Proceedings of the Cambridge Philosophical Society 9 (1983), 9--23.

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A. Mansfield. Determining the thickness of graphs is NP-hard. Mathematical Proceedings of the Cambridge Philosophical Society, 93 (1983), pp. 9--23.

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