32,544 research outputs found
An elementary chromatic reduction for gain graphs and special hyperplane arrangements
A gain graph is a graph whose edges are labelled invertibly by "gains" from a
group. "Switching" is a transformation of gain graphs that generalizes
conjugation in a group. A "weak chromatic function" of gain graphs with gains
in a fixed group satisfies three laws: deletion-contraction for links with
neutral gain, invariance under switching, and nullity on graphs with a neutral
loop. The laws lead to the "weak chromatic group" of gain graphs, which is the
universal domain for weak chromatic functions. We find expressions, valid in
that group, for a gain graph in terms of minors without neutral-gain edges, or
with added complete neutral-gain subgraphs, that generalize the expression of
an ordinary chromatic polynomial in terms of monomials or falling factorials.
These expressions imply relations for chromatic functions of gain graphs.
We apply our relations to some special integral gain graphs including those
that correspond to the Shi, Linial, and Catalan arrangements, thereby obtaining
new evaluations of and new ways to calculate the zero-free chromatic polynomial
and the integral and modular chromatic functions of these gain graphs, hence
the characteristic polynomials and hypercubical lattice-point counting
functions of the arrangements. We also calculate the total chromatic polynomial
of any gain graph and especially of the Catalan, Shi, and Linial gain graphs.Comment: 31 page
Disjoint compatibility graph of non-crossing matchings of points in convex position
Let be a set of labeled points in convex position in the plane.
We consider geometric non-intersecting straight-line perfect matchings of
. Two such matchings, and , are disjoint compatible if they do
not have common edges, and no edge of crosses an edge of . Denote by
the graph whose vertices correspond to such matchings, and two
vertices are adjacent if and only if the corresponding matchings are disjoint
compatible. We show that for each , the connected components of
form exactly three isomorphism classes -- namely, there is a
certain number of isomorphic small components, a certain number of isomorphic
medium components, and one big component. The number and the structure of small
and medium components is determined precisely.Comment: 46 pages, 30 figure
A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings
Combining tree decomposition and transfer matrix techniques provides a very
general algorithm for computing exact partition functions of statistical models
defined on arbitrary graphs. The algorithm is particularly efficient in the
case of planar graphs. We illustrate it by computing the Potts model partition
functions and chromatic polynomials (the number of proper vertex colourings
using Q colours) for large samples of random planar graphs with up to N=100
vertices. In the latter case, our algorithm yields a sub-exponential average
running time of ~ exp(1.516 sqrt(N)), a substantial improvement over the
exponential running time ~ exp(0.245 N) provided by the hitherto best known
algorithm. We study the statistics of chromatic roots of random planar graphs
in some detail, comparing the findings with results for finite pieces of a
regular lattice.Comment: 5 pages, 3 figures. Version 2 has been substantially expanded.
Version 3 shows that the worst-case running time is sub-exponential in the
number of vertice
Optimal Recombination in Genetic Algorithms
This paper surveys results on complexity of the optimal recombination problem
(ORP), which consists in finding the best possible offspring as a result of a
recombination operator in a genetic algorithm, given two parent solutions. We
consider efficient reductions of the ORPs, allowing to establish polynomial
solvability or NP-hardness of the ORPs, as well as direct proofs of hardness
results
Simultaneous Embeddability of Two Partitions
We study the simultaneous embeddability of a pair of partitions of the same
underlying set into disjoint blocks. Each element of the set is mapped to a
point in the plane and each block of either of the two partitions is mapped to
a region that contains exactly those points that belong to the elements in the
block and that is bounded by a simple closed curve. We establish three main
classes of simultaneous embeddability (weak, strong, and full embeddability)
that differ by increasingly strict well-formedness conditions on how different
block regions are allowed to intersect. We show that these simultaneous
embeddability classes are closely related to different planarity concepts of
hypergraphs. For each embeddability class we give a full characterization. We
show that (i) every pair of partitions has a weak simultaneous embedding, (ii)
it is NP-complete to decide the existence of a strong simultaneous embedding,
and (iii) the existence of a full simultaneous embedding can be tested in
linear time.Comment: 17 pages, 7 figures, extended version of a paper to appear at GD 201
Computing the blocks of a quasi-median graph
Quasi-median graphs are a tool commonly used by evolutionary biologists to
visualise the evolution of molecular sequences. As with any graph, a
quasi-median graph can contain cut vertices, that is, vertices whose removal
disconnect the graph. These vertices induce a decomposition of the graph into
blocks, that is, maximal subgraphs which do not contain any cut vertices. Here
we show that the special structure of quasi-median graphs can be used to
compute their blocks without having to compute the whole graph. In particular
we present an algorithm that, for a collection of aligned sequences of
length , can compute the blocks of the associated quasi-median graph
together with the information required to correctly connect these blocks
together in run time , independent of the size of the
sequence alphabet. Our primary motivation for presenting this algorithm is the
fact that the quasi-median graph associated to a sequence alignment must
contain all most parsimonious trees for the alignment, and therefore
precomputing the blocks of the graph has the potential to help speed up any
method for computing such trees.Comment: 17 pages, 2 figure
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