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Unit Disk Graph Recognition is NPHard
 Computational Geometry. Theory and Applications
, 1993
"... Unit disk graphs are the intersection graphs of unit diameter closed disks in the plane. This paper reduces SATISFIABILITY to the problem of recognizing unit disk graphs. Equivalently, it shows that determining if a graph has sphericity 2 or less, even if the graph is planar or is known to have s ..."
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Cited by 78 (1 self)
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Unit disk graphs are the intersection graphs of unit diameter closed disks in the plane. This paper reduces SATISFIABILITY to the problem of recognizing unit disk graphs. Equivalently, it shows that determining if a graph has sphericity 2 or less, even if the graph is planar or is known to have sphericity at most 3, is NPhard. We show how this reduction can be extended to 3 dimensions, thereby showing that unit sphere graph recognition, or determining if a graph has sphericity 3 or less, is also NPhard. We conjecture that Ksphericity is NPhard for all fixed K greater than 1. 1 Introduction A unit disk graph is the intersection graph of a set of unit diameter closed disks in the plane. That is, each vertex corresponds to a disk in the plane, and two vertices are adjacent in the graph if the corresponding disks intersect. The set of disks is said to realize the graph. Of course, the unit of distance is not critical, since the disks realize the same graph even if the coordina...
Global Optimum Protein Threading with Gapped Alignment and Empirical Pair Score Functions
 J. Mol. Biol
, 1996
"... We describe a branchandbound search algorithm for finding the exact global optimum gapped sequencestructure alignment ("threading") between a protein sequence and a protein core or structural model, using an arbitrary amino acid pair score function (e.g., contact potentials, knowledgebased potent ..."
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Cited by 53 (4 self)
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We describe a branchandbound search algorithm for finding the exact global optimum gapped sequencestructure alignment ("threading") between a protein sequence and a protein core or structural model, using an arbitrary amino acid pair score function (e.g., contact potentials, knowledgebased potentials, potentials of mean force, etc.). The search method imposes minimal conditions on how structural environments are defined or the form of the score function, and allows arbitrary sequencespecific functions for scoring loops and active site residues. Consequently the search method can be used with many different score functions and threading methodologies; this paper illustrates five from the literature. On a desktop workstation running LISP, we have found the global optimum protein sequencestructure alignment in NPhard search spaces as large as 9:6 \Theta 10 31 , at rates ranging as high as 6:8 \Theta 10 28 equivalent threadings per second (most of which are pruned before they eve...
A global representation of the protein fold space
 Proc. Natl Acad
, 2003
"... One of the principal goals of the structural genomics initiative is to identify the total repertoire of protein folds and to obtain a global view of the “protein structure universe”. Here, we present a threedimensional map of the protein fold space in which structurally related folds are represente ..."
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Cited by 22 (0 self)
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One of the principal goals of the structural genomics initiative is to identify the total repertoire of protein folds and to obtain a global view of the “protein structure universe”. Here, we present a threedimensional map of the protein fold space in which structurally related folds are represented by spatially adjacent points. Such a representation reveals a highlevel organization of the fold space that is intuitively interpretable. The shape of the fold space and the overall distribution of the folds are defined by three dominant trends: secondary structure class, chain topology, and protein domain size. Random coillike structures of small proteins and peptides are mapped to a region where the three trends converge, offering interesting perspective on both the demography of fold space and the evolution of protein structures.
Index Terms
, 2010
"... protein structure alignment, structural bioinformatics, contact maps, spectral methods We present two algorithms that use spectral methods to align protein folds. One of the algorithms is suitable for database searches, the other for difficult alignments. We present computational results for 780 pai ..."
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protein structure alignment, structural bioinformatics, contact maps, spectral methods We present two algorithms that use spectral methods to align protein folds. One of the algorithms is suitable for database searches, the other for difficult alignments. We present computational results for 780 pairwise alignments used to classify 40 proteins as well as results for a separate set of 36 protein alignments used for comparison to four other alignment algorithms. We also provide a mathematically rigorous development of the intrinsic geometry underlying our spectral approach. 1