Complexity of Phylogenetic Networks: Counting Cubes in Median Graphs and Related Problems

Kovše, Matjaž

doi:10.1002/9783527627981.ch13

Cited by 7 publications

(22 citation statements)

References 46 publications

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“…The cube polynomial was generalized to a Hamming polynomial in two different ways in [3] and [7], respectively. For a survey on the cube polynomial, its extensions and related results see [16]. We add here that, interestingly, the sequence {c k (G)} k≥0 turned out to be important in human genetics, see [1] for its use in studies of the so-called phantom mutations.…”

mentioning

confidence: 98%

Cube Polynomial of Fibonacci and Lucas Cubes

Klavžar

Mollard

2011

Acta Appl Math

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The cube polynomial of a graph is the counting polynomial for the number of induced k-dimensional hypercubes (k ≥ 0). We determine the cube polynomial of Fibonacci cubes and Lucas cubes, as well as the generating functions for the sequences of these cubes. Several explicit formulas for the coefficients of these polynomials are obtained, in particular they can be expressed with convolved Fibonacci numbers. Zeros of the studied cube polynomials are explicitly determined. Consequently, the coefficients sequences of cube polynomials of Fibonacci and Lucas cubes are unimodal.

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mentioning

confidence: 98%

Cube Polynomial of Fibonacci and Lucas Cubes

Klavžar

Mollard

2011

Acta Appl Math

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“…For example, Woodall [51] examined the zeros of chromatic and flow polynomials and determined zero-free regions thereof. When giving [50] a survey on counting hypercubes, Kovše [50] also sketched some results concerning the zeros of cube polynomials. Jackson [49] surveyed results and conjectures dealing with the zero distribution of chromatic and flow polynomials of graphs, and characteristic polynomials of matroids.…”

Section: Resultsmentioning

confidence: 99%

New Polynomial-Based Molecular Descriptors with Low Degeneracy

2010

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In this paper, we introduce a novel graph polynomial called the ‘information polynomial’ of a graph. This graph polynomial can be derived by using a probability distribution of the vertex set. By using the zeros of the obtained polynomial, we additionally define some novel spectral descriptors. Compared with those based on computing the ordinary characteristic polynomial of a graph, we perform a numerical study using real chemical databases. We obtain that the novel descriptors do have a high discrimination power.

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“…We denote by L the set of POFs of the median graph G. The notion of POF is strongly related to the induced hypercubes in median graphs. First, observe that all Θ-classes of a median graph form a POF if and only if the graph is an hypercube of dimension log n [33,34]. Secondly, the next lemma precises the relationship between POFs and hypercubes.…”

Section: Orthogonal θ-Classes and Hypercubesmentioning

confidence: 96%

“…We present now another important notion on median graphs: orthogonality. In [33], Kovse studied a relationship between splits which refer to the halfspaces of Θ-classes. It says that two splits Lemma 8 (Squares [11,14]).…”

Section: Orthogonal θ-Classes and Hypercubesmentioning

confidence: 99%

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Subquadratic-time algorithm for the diameter and all eccentricities on median graphs

Bergé¹,

Ducoffe²,

Habib³

2021

Preprint

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On sparse graphs, Roditty and Williams [2013] proved that no O(n 2−ε )-time algorithm achieves an approximation factor smaller than 3 2 for the diameter problem unless SETH fails. In this article, we solve a longstanding question: can we use the structural properties of median graphs to break this global quadratic barrier?We propose the first combinatiorial algorithm computing exactly all eccentricities of a median graph in truly subquadratic time. Median graphs constitute the family of graphs which is the most studied in metric graph theory because their structure represent many other discrete and geometric concepts, such as CAT(0) cube complexes. Our result generalizes a recent one, stating that there is a linear-time algorithm for all eccentricities in median graphs with bounded dimension d, i.e. the dimension of the largest induced hypercube. This prerequisite on d is not necessarily anymore to determine all eccentricities in subquadratic time. The execution time of our algorithm is O(n 1.6456 log O(1) n).We provide also some satellite outcomes related to this general result. In particular, restricted to simplex graphs, this algorithm enumerate all eccentricities with a quasilinear running time. Moreover, an algorithm is proposed to compute exactly all reach centralities in time O(2 3d n log O(1) n).

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Complexity of Phylogenetic Networks: Counting Cubes in Median Graphs and Related Problems

Cited by 7 publications

References 46 publications

Cube Polynomial of Fibonacci and Lucas Cubes

Cube Polynomial of Fibonacci and Lucas Cubes

New Polynomial-Based Molecular Descriptors with Low Degeneracy

Subquadratic-time algorithm for the diameter and all eccentricities on median graphs

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