1,096 research outputs found

    The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings

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    Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings ubiquitously use iterative numerical approximations rather than constructing and then solving algebraic expressions representing their exact solutions. To explain this phenomenon, we use Galois theory to show that many variants of these problems have solutions that cannot be expressed by nested radicals or nested roots of low-degree polynomials. Hence, such solutions cannot be computed exactly even in extended computational models that include such operations.Comment: Graph Drawing 201

    Small cycles, generalized prisms and Hamiltonian cycles in the Bubble-sort graph

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    The Bubble-sort graph BSn, n⩾2BS_n,\,n\geqslant 2, is a Cayley graph over the symmetric group SymnSym_n generated by transpositions from the set {(12),(23),…,(n−1n)}\{(1 2), (2 3),\ldots, (n-1 n)\}. It is a bipartite graph containing all even cycles of length ℓ\ell, where 4⩽ℓ⩽n!4\leqslant \ell\leqslant n!. We give an explicit combinatorial characterization of all its 44- and 66-cycles. Based on this characterization, we define generalized prisms in BSn, n⩾5BS_n,\,n\geqslant 5, and present a new approach to construct a Hamiltonian cycle based on these generalized prisms.Comment: 13 pages, 7 figure
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