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Planarizing Graphs  A Survey and Annotated Bibliography
, 1999
"... Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results abo ..."
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Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with online algorithms.
A Proposed Algorithm for Calculating the Minimum Crossing Number of a Graph
 Western Michigan University
, 1995
"... In this paper we present a branchandbound algorithm for finding the minimum crossing number of a graph. We begin with the vertex set and add edges by selecting every legal option for creating a crossing or not. After each edge is added we determine if the resulting partial graph is planar. We cont ..."
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In this paper we present a branchandbound algorithm for finding the minimum crossing number of a graph. We begin with the vertex set and add edges by selecting every legal option for creating a crossing or not. After each edge is added we determine if the resulting partial graph is planar. We continue adding edges until either all edges have been added or we reach a point where the graph cannot be completed as started. At this point we backtrack to see if the graph can be drawn with fewer crossings by selecting other options when adding edges. keywords: Crossing Number, Algorithm 1 Introduction Determining the crossing number of a graph is an important problem with applications in areas such as circuit design and network configuration [17]. It is this importance that has driven our work in finding the minimum crossing number of a graph. Informally, the crossing number of a graph G, denoted (G), is the minimum number of crossings among all good drawings of G in the plane, where a g...
Parallel Computation and Graphical Visualization of the Minimum Crossing Number of a Graph
, 1998
"... Finding the minimum crossing number of a graph is an interesting and challenging problem in graph theory and applied mathematics. Real world applications of this problem, such as circuit layout and network design, are becoming more and more important. This thesis presents a parallel algorithm for f ..."
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Finding the minimum crossing number of a graph is an interesting and challenging problem in graph theory and applied mathematics. Real world applications of this problem, such as circuit layout and network design, are becoming more and more important. This thesis presents a parallel algorithm for finding the minimum crossing number of a graph, based on the first sequential algorithm presented in [20]. This parallel algorithm was tested on various architectures and a comparison of the corresponding results is given, including running time, efficiency, and speedup. Implementation of the algorithm gives us an ability to verify conjectures proposed for various families of graphs, as well as apply the algorithm to real world applications. Another important aspect of the problem is ability to draw the solution on the 2 \Gamma D plane. This thesis gives an overview of graph drawing algorithms, starting with the famous result by F'ary, and presents a new algorithm for drawing complete graphs...
On the crossing number of Cm x Cn
, 1998
"... We show that the Mcrossing number cr M (CmC n ) of Cm C n behaves asymptotically according to lim n!1 fcr M (Cm C n )=((m 2)n)g = 1, for each m 3. This result reinforces the conjecture cr(Cm C n ) = (m 2)n if 3 m n, which has been proved only for m 6. ..."
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We show that the Mcrossing number cr M (CmC n ) of Cm C n behaves asymptotically according to lim n!1 fcr M (Cm C n )=((m 2)n)g = 1, for each m 3. This result reinforces the conjecture cr(Cm C n ) = (m 2)n if 3 m n, which has been proved only for m 6.
Open Problems 16
, 1994
"... to the disjoint union G 0 + G 0 a path P of length k that has one endpoint in each copy of G 0 . Since only k \Gamma 1 colors can appear on interior vertices of P , for every proper kcoloring of G there is some color class disconnected in G k\Gamma1 . For such a graph G, we cannot reduce t ..."
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to the disjoint union G 0 + G 0 a path P of length k that has one endpoint in each copy of G 0 . Since only k \Gamma 1 colors can appear on interior vertices of P , for every proper kcoloring of G there is some color class disconnected in G k\Gamma1 . For such a graph G, we cannot reduce the dispersion below k \Gamma 1. Chen, Schelp, and Shreve [3] have studied the extremal problem that results. Given k, we seek the minimum f(k) such that, for sufficiently large n 0 , all kchromatic graphs with at least n 0 vertices have kcolorings with dispersion less than f(k). T
Crossing Number Bounds for the Twisted Cube
, 2001
"... The twisted cube TQ d is formed from the ddimensional hypercube Q d by `twisting' a pair of independent edges of any 4cycle of Q d . Asymptotic upper and lower bounds for the crossing numbers of the twisted cube and generalized twisted cube (GTQ d ) graphs are derived. We also determine the skewne ..."
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The twisted cube TQ d is formed from the ddimensional hypercube Q d by `twisting' a pair of independent edges of any 4cycle of Q d . Asymptotic upper and lower bounds for the crossing numbers of the twisted cube and generalized twisted cube (GTQ d ) graphs are derived. We also determine the skewness of TQ d . 1
Drawings of C_m × C_n with one disjoint family
, 1998
"... We show that every drawing of Cm C n with either the m n{cycles pairwise disjoint or the n m{cycles pairwise disjoint has at least (m 2)n crossings, for every m;n satisfying n m 3. This supports the long standing conjecture by Harary et al. that the crossing number of Cm C n is (m 2)n. ..."
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We show that every drawing of Cm C n with either the m n{cycles pairwise disjoint or the n m{cycles pairwise disjoint has at least (m 2)n crossings, for every m;n satisfying n m 3. This supports the long standing conjecture by Harary et al. that the crossing number of Cm C n is (m 2)n.
A Lower Bound for the Crossing Number of C_m × C_n
, 1999
"... We prove that the crossing number of Cm C n is at least (m 2)n=3, for all m;n such that n m. This is the best general lower bound known for the crossing number of Cm C n , whose exact value has been long conjectured to be (m 2)n, for 3 m n. ..."
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We prove that the crossing number of Cm C n is at least (m 2)n=3, for all m;n such that n m. This is the best general lower bound known for the crossing number of Cm C n , whose exact value has been long conjectured to be (m 2)n, for 3 m n.
The Splitting Number and Skewness of . . .
"... The skewness of a graph G is the minimum number of edges that need to be deleted from G to produce a planar graph. The splitting number of a graph G is the minimum number of splitting steps needed to turn G into a planar graph; where each step replaces some of the edges fu; vg incident to a sele ..."
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The skewness of a graph G is the minimum number of edges that need to be deleted from G to produce a planar graph. The splitting number of a graph G is the minimum number of splitting steps needed to turn G into a planar graph; where each step replaces some of the edges fu; vg incident to a selected vertex u by edges fu ; vg, where u is a new vertex. We show that the splitting number of the toroidal grid graph Cn \Theta Cm is minfn; mg \Gamma 2ffi n;3 ffi m;3 \Gamma ffi n;4 ffi m;3 \Gamma ffi n;3 ffi m;4 and its skewness is minfn; mg \Gamma ffi n;3 ffi m;3 \Gamma ffi n;4 ffi m;3 \Gamma ffi n;3 ffi m;4 . Here, ffi is the Kronecker symbol, i.e., ffi i;j is 1 if i = j, and 0 if i 6= j.