4,695 research outputs found
The Rainbow Prim Algorithm for Selecting Putative Orthologous Protein Sequences
We present a selection method designed for eliminating species redundancy in clusters of putative orthologous sequences, to be applied as a post-processing procedure to pre-clustered data obtained from other methods. The algorithm can always zero-out the cluster redundancy while preserving the number of species of the original cluster
Rainbow Connection Number and Connected Dominating Sets
Rainbow connection number rc(G) of a connected graph G is the minimum number
of colours needed to colour the edges of G, so that every pair of vertices is
connected by at least one path in which no two edges are coloured the same. In
this paper we show that for every connected graph G, with minimum degree at
least 2, the rainbow connection number is upper bounded by {\gamma}_c(G) + 2,
where {\gamma}_c(G) is the connected domination number of G. Bounds of the form
diameter(G) \leq rc(G) \leq diameter(G) + c, 1 \leq c \leq 4, for many special
graph classes follow as easy corollaries from this result. This includes
interval graphs, AT-free graphs, circular arc graphs, threshold graphs, and
chain graphs all with minimum degree at least 2 and connected. We also show
that every bridge-less chordal graph G has rc(G) \leq 3.radius(G). In most of
these cases, we also demonstrate the tightness of the bounds. An extension of
this idea to two-step dominating sets is used to show that for every connected
graph on n vertices with minimum degree {\delta}, the rainbow connection number
is upper bounded by 3n/({\delta} + 1) + 3. This solves an open problem of
Schiermeyer (2009), improving the previously best known bound of 20n/{\delta}
by Krivelevich and Yuster (2010). Moreover, this bound is seen to be tight up
to additive factors by a construction of Caro et al. (2008).Comment: 14 page
An Algorithmic Proof of the Lovasz Local Lemma via Resampling Oracles
The Lovasz Local Lemma is a seminal result in probabilistic combinatorics. It
gives a sufficient condition on a probability space and a collection of events
for the existence of an outcome that simultaneously avoids all of those events.
Finding such an outcome by an efficient algorithm has been an active research
topic for decades. Breakthrough work of Moser and Tardos (2009) presented an
efficient algorithm for a general setting primarily characterized by a product
structure on the probability space.
In this work we present an efficient algorithm for a much more general
setting. Our main assumption is that there exist certain functions, called
resampling oracles, that can be invoked to address the undesired occurrence of
the events. We show that, in all scenarios to which the original Lovasz Local
Lemma applies, there exist resampling oracles, although they are not
necessarily efficient. Nevertheless, for essentially all known applications of
the Lovasz Local Lemma and its generalizations, we have designed efficient
resampling oracles. As applications of these techniques, we present new results
for packings of Latin transversals, rainbow matchings and rainbow spanning
trees.Comment: 47 page
Fixed parameter tractability of crossing minimization of almost-trees
We investigate exact crossing minimization for graphs that differ from trees
by a small number of additional edges, for several variants of the crossing
minimization problem. In particular, we provide fixed parameter tractable
algorithms for the 1-page book crossing number, the 2-page book crossing
number, and the minimum number of crossed edges in 1-page and 2-page book
drawings.Comment: Graph Drawing 201
Solution discovery via reconfiguration for problems in P
In the recently introduced framework of solution discovery via
reconfiguration [Fellows et al., ECAI 2023], we are given an initial
configuration of tokens on a graph and the question is whether we can
transform this configuration into a feasible solution (for some problem) via a
bounded number of small modification steps. In this work, we study solution
discovery variants of polynomial-time solvable problems, namely Spanning Tree
Discovery, Shortest Path Discovery, Matching Discovery, and Vertex/Edge Cut
Discovery in the unrestricted token addition/removal model, the token jumping
model, and the token sliding model. In the unrestricted token addition/removal
model, we show that all four discovery variants remain in P. For the toking
jumping model we also prove containment in P, except for Vertex/Edge Cut
Discovery, for which we prove NP-completeness. Finally, in the token sliding
model, almost all considered problems become NP-complete, the exception being
Spanning Tree Discovery, which remains polynomial-time solvable. We then study
the parameterized complexity of the NP-complete problems and provide a full
classification of tractability with respect to the parameters solution size
(number of tokens) and transformation budget (number of steps) . Along
the way, we observe strong connections between the solution discovery variants
of our base problems and their (weighted) rainbow variants as well as their
red-blue variants with cardinality constraints
On the Rainbow Connection Number for Snowflake Graph
Let G be an arbitrary non-trivial connected graph. An edge-colored graph G is called a rainbow connected if any two vertices are connected by a path whose edges have distinct colors, such path is called a rainbow path. The smallest number of colors required to make G rainbow connected is called the rainbow connection number of G, denoted by rc(G). A snowflake graph is a graph obtained by resembling one of the snowflake shapes into vertices and edges so that it forms a simple graph. Let be a generalized snowflake graph, i.e., a graph with paths of the stem, pair of outer leaves, middle circles, and pairs of inner leaves. In this paper we determine the rainbow connection number for generalized snowflake graph
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