Hamiltonian circuits determining the order of chromosomes

Dorninger, Dietmar

doi:10.1016/0166-218x(92)00171-h

Cited by 74 publications

(43 citation statements)

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“…On the other hand it was shown in [5] that S n is 3-admissible for all even n, from which one can easily derive that S n is 3-admissible for all n.…”

Section: Dorninger B Huebermentioning

confidence: 99%

“…For n odd it is an open question whether S n is 2-admissible for all n. (Computer calculations suggest this to be true.) For even n it has been proved in [5] that S> n is 2-admissible if and only if n ^ 12.That S n is not 2-admissible for even n > 14 could be due to the fact that the following situation (which is reasonable to rule out from the biologial point of view) was not excluded, namely that two different chromosomes whose short arms are immediate neighbours within S have also their long arms as immediate neighbours within L, more precisely: there exist i,j € {1,2,..., LfJ} with i ± j such that [tt(2i -1),ir(2i)] = [2j -1,2j]. ([a,b} denotes the unordered pair of a and b).…”

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confidence: 99%

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On Hamiltonian Graphs Arising From Spatial Orders of Chromosomes of Even Number

Dorninger¹,

Hueber²

2001

Demonstratio Mathematica

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Abstract. According to Bennett's model of cytogenetics the spatial order in haploid chromosome complements is based on a similarity relation which gives rise to a multigraph G which is the edge-disjoint union of two of its subgraphs G\ and G2. For even chromosome numbers n Bennett's model postulates that the order of the n chromosomes is given by a Hamiltonian circuit of G alternating in the edges of Gi and G2. However, such a Hamiltonian circuit does not always exist. We impose a weak condition on the similarity relation and prove that under this condition the assumed Hamiltonian circuit does exist for all even chromosome numbers n < 50, which settles the case for all biological relevant species with n pairs of chromosomes. Moreover we study the structure of the graph G in respect to cycle decompositions and possible generalizations of our results. IntroductionAn individual chromosome in a cell of a eucaryotic organism consists of a short arm and of a long arm which are linked at the so-called centromere. During a certain stage of cell division, called metaphase, the centromeres of the n chromosomes of a haploid complement have approximately the form of a plane regular n-gon in which the arms of the chromosomes axe stretched to the outside. According to Bennett's model ([1], [2], [3]) the arms are arranged in such a way that always two short arms and two long arms are adjacent with one possible exception, if n is odd. Moreover, adjacent arms are assumed to be of "most similar size". However, preparing the cell for the electron microscope often destroys the assumed order, but usually allows to identify the individual arms of the chromosomes and to measure their lengths. In order to understand the mechanism of pairing (in areas like medicine, plant breeding, and genetic engineering) it may be important to know the original order of the chromosomes. To solve the problem of reconstructing the order of the chromosomes on the basis of arm lengths, Bennett recommended a procedure which in terms of mathematics runs as follows: Let s" and l" denote the short arm and the long arm of chromosome v, u = 1,2,..., n. All short arms s v , the lengths of which are assumed to be pairwise different, and all long arms l v which are also assumed to be pairwise different in length, are separately ranked in descending order of size. This ranking gives rise to two chains of indices S and L referring to the short and long arms, respectively. Without loss of generality we assume that S = (1,2,..., n) (which means that we will have to change the numbers usually assigned to the chromosomes in biology), so that L = (7rl,7r2,... ,-rm) for an appropriate permutation 7r € S n (symmetric group on n letters). Hence any measurement of the arm-lengths corresponds to a unique permutation 7T € S n -Next we interpret the notion "most similar size" of arms: For p, q € {1,2,..., n},p / <7 we say that two short arms s p and s q are k-similar, if \p -q\ < k, and analogously we call two long arms l np , l nq fc-similar, if \p -q\ < k. li k = 1, the arms ...

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“…On the other hand it was shown in [5] that S n is 3-admissible for all even n, from which one can easily derive that S n is 3-admissible for all n.…”

Section: Dorninger B Huebermentioning

confidence: 99%

mentioning

confidence: 99%

On Hamiltonian Graphs Arising From Spatial Orders of Chromosomes of Even Number

Dorninger¹,

Hueber²

2001

Demonstratio Mathematica

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“…, p − 2} and, in addition, if W is closed then edges e p−1 and e 1 are of different colors. PC walks are of interest in graph theory applications, e.g., in molecular biology [5,6,7,18] and in VLSI for compacting programmable logical arrays [12]. They are also of interest in graph theory itself as generalizations of walks in undirected and directed graphs.…”

Section: Introductionmentioning

confidence: 99%

Acyclicity in edge-colored graphs

Gutin

Jones

Sheng

et al. 2017

Discrete Mathematics

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A walk W in edge-colored graphs is called properly colored (PC) if every pair of consecutive edges in W is of different color. We introduce and study five types of PC acyclicity in edge-colored graphs such that graphs of PC acyclicity of type i is a proper superset of graphs of acyclicity of type i + 1, i = 1, 2, 3, 4. The first three types are equivalent to the absence of PC cycles, PC closed trails, and PC closed walks, respectively. While graphs of types 1, 2 and 3 can be recognized in polynomial time, the problem of recognizing graphs of type 4 is, somewhat surprisingly, NP-hard even for 2-edge-colored graphs (i.e., when only two colors are used). The same problem with respect to type 5 is polynomial-time solvable for all edge-colored graphs. Using the five types, we investigate the border between intractability and tractability for the problems of finding the maximum number of internally vertex-disjoint PC paths between two vertices and the minimum number of vertices to meet all PC paths between two vertices.

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“…There are several other generalizations of directed graphs such as arc-coloured digraphs, hypertournaments and star hypergraphs, but the class of edgecoloured multigraphs has been given the main attention in graph theory literature because many concepts and results on directed graphs can be extended to edge-coloured multigraphs and there are several important applications of edge-coloured multigraphs. For instance, in [10,11] Dorninger considers chromosome arrangement in a cell of an eukaryotic organism by using the 2-edge-coloured multigraphs. For a more extensive treatment of this topic, see [6,7].…”

Section: Introductionmentioning

confidence: 99%

Properly Coloured Cycles and Paths: Results and Open Problems

Gutin

Kim

2009

Graph Theory, Computational Intelligence and Thought

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In this paper, we consider a number of results and six conjectures on properly coloured (PC) paths and cycles in edge-coloured multigraphs. We overview some known results and prove new ones. In particular, we consider a family of transformations of an edge-coloured multigraph G into an ordinary graph that allow us to check the existence of PC cycles and PC (s, t)-paths in G and, if they exist, to find shortest ones among them. We raise a problem of finding the optimal transformation and consider a possible solution to the problem.

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Hamiltonian circuits determining the order of chromosomes

Cited by 74 publications

References 3 publications

On Hamiltonian Graphs Arising From Spatial Orders of Chromosomes of Even Number

On Hamiltonian Graphs Arising From Spatial Orders of Chromosomes of Even Number

Acyclicity in edge-colored graphs

Properly Coloured Cycles and Paths: Results and Open Problems

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