65 research outputs found
A Characterization of Mixed Unit Interval Graphs
We give a complete characterization of mixed unit interval graphs, the
intersection graphs of closed, open, and half-open unit intervals of the real
line. This is a proper superclass of the well known unit interval graphs. Our
result solves a problem posed by Dourado, Le, Protti, Rautenbach and
Szwarcfiter (Mixed unit interval graphs, Discrete Math. 312, 3357-3363 (2012)).Comment: 17 pages, referees' comments adde
On Symbolic Ultrametrics, Cotree Representations, and Cograph Edge Decompositions and Partitions
Symbolic ultrametrics define edge-colored complete graphs K_n and yield a
simple tree representation of K_n. We discuss, under which conditions this idea
can be generalized to find a symbolic ultrametric that, in addition,
distinguishes between edges and non-edges of arbitrary graphs G=(V,E) and thus,
yielding a simple tree representation of G. We prove that such a symbolic
ultrametric can only be defined for G if and only if G is a so-called cograph.
A cograph is uniquely determined by a so-called cotree. As not all graphs are
cographs, we ask, furthermore, what is the minimum number of cotrees needed to
represent the topology of G. The latter problem is equivalent to find an
optimal cograph edge k-decomposition {E_1,...,E_k} of E so that each subgraph
(V,E_i) of G is a cograph. An upper bound for the integer k is derived and it
is shown that determining whether a graph has a cograph 2-decomposition, resp.,
2-partition is NP-complete
Finding Connected Dense -Subgraphs
Given a connected graph on vertices and a positive integer ,
a subgraph of on vertices is called a -subgraph in . We design
combinatorial approximation algorithms for finding a connected -subgraph in
such that its density is at least a factor
of the density of the densest -subgraph
in (which is not necessarily connected). These particularly provide the
first non-trivial approximations for the densest connected -subgraph problem
on general graphs
Space Efficient Algorithms for Breadth-Depth Search
Continuing the recent trend, in this article we design several
space-efficient algorithms for two well-known graph search methods. Both these
search methods share the same name {\it breadth-depth search} (henceforth {\sf
BDS}), although they work entirely in different fashion. The classical
implementation for these graph search methods takes time and bits of space in the standard word RAM model (with word size being
bits), where and denotes the number of edges and
vertices of the input graph respectively. Our goal here is to beat the space
bound of the classical implementations, and design space
algorithms for these search methods by paying little to no penalty in the
running time. Note that our space bounds (i.e., with bits of
space) do not even allow us to explicitly store the required information to
implement the classical algorithms, yet our algorithms visits and reports all
the vertices of the input graph in correct order.Comment: 12 pages, This work will appear in FCT 201
Between Treewidth and Clique-width
Many hard graph problems can be solved efficiently when restricted to graphs
of bounded treewidth, and more generally to graphs of bounded clique-width. But
there is a price to be paid for this generality, exemplified by the four
problems MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set that
are all FPT parameterized by treewidth but none of which can be FPT
parameterized by clique-width unless FPT = W[1], as shown by Fomin et al [7,
8]. We therefore seek a structural graph parameter that shares some of the
generality of clique-width without paying this price. Based on splits, branch
decompositions and the work of Vatshelle [18] on Maximum Matching-width, we
consider the graph parameter sm-width which lies between treewidth and
clique-width. Some graph classes of unbounded treewidth, like
distance-hereditary graphs, have bounded sm-width. We show that MaxCut, Graph
Coloring, Hamiltonian Cycle and Edge Dominating Set are all FPT parameterized
by sm-width
Partial Homology Relations - Satisfiability in terms of Di-Cographs
Directed cographs (di-cographs) play a crucial role in the reconstruction of
evolutionary histories of genes based on homology relations which are binary
relations between genes. A variety of methods based on pairwise sequence
comparisons can be used to infer such homology relations (e.g.\ orthology,
paralogy, xenology). They are \emph{satisfiable} if the relations can be
explained by an event-labeled gene tree, i.e., they can simultaneously co-exist
in an evolutionary history of the underlying genes. Every gene tree is
equivalently interpreted as a so-called cotree that entirely encodes the
structure of a di-cograph. Thus, satisfiable homology relations must
necessarily form a di-cograph. The inferred homology relations might not cover
each pair of genes and thus, provide only partial knowledge on the full set of
homology relations. Moreover, for particular pairs of genes, it might be known
with a high degree of certainty that they are not orthologs (resp.\ paralogs,
xenologs) which yields forbidden pairs of genes. Motivated by this observation,
we characterize (partial) satisfiable homology relations with or without
forbidden gene pairs, provide a quadratic-time algorithm for their recognition
and for the computation of a cotree that explains the given relations
The Effect of Planarization on Width
We study the effects of planarization (the construction of a planar diagram
from a non-planar graph by replacing each crossing by a new vertex) on
graph width parameters. We show that for treewidth, pathwidth, branchwidth,
clique-width, and tree-depth there exists a family of -vertex graphs with
bounded parameter value, all of whose planarizations have parameter value
. However, for bandwidth, cutwidth, and carving width, every graph
with bounded parameter value has a planarization of linear size whose parameter
value remains bounded. The same is true for the treewidth, pathwidth, and
branchwidth of graphs of bounded degree.Comment: 15 pages, 6 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Solving Problems on Graphs of High Rank-Width
A modulator of a graph G to a specified graph class H is a set of vertices
whose deletion puts G into H. The cardinality of a modulator to various
tractable graph classes has long been used as a structural parameter which can
be exploited to obtain FPT algorithms for a range of hard problems. Here we
investigate what happens when a graph contains a modulator which is large but
"well-structured" (in the sense of having bounded rank-width). Can such
modulators still be exploited to obtain efficient algorithms? And is it even
possible to find such modulators efficiently?
We first show that the parameters derived from such well-structured
modulators are strictly more general than the cardinality of modulators and
rank-width itself. Then, we develop an FPT algorithm for finding such
well-structured modulators to any graph class which can be characterized by a
finite set of forbidden induced subgraphs. We proceed by showing how
well-structured modulators can be used to obtain efficient parameterized
algorithms for Minimum Vertex Cover and Maximum Clique. Finally, we use
well-structured modulators to develop an algorithmic meta-theorem for deciding
problems expressible in Monadic Second Order (MSO) logic, and prove that this
result is tight in the sense that it cannot be generalized to LinEMSO problems.Comment: Accepted at WADS 201
Fast approximation of centrality and distances in hyperbolic graphs
We show that the eccentricities (and thus the centrality indices) of all
vertices of a -hyperbolic graph can be computed in linear
time with an additive one-sided error of at most , i.e., after a
linear time preprocessing, for every vertex of one can compute in
time an estimate of its eccentricity such that
for a small constant . We
prove that every -hyperbolic graph has a shortest path tree,
constructible in linear time, such that for every vertex of ,
. These results are based on an
interesting monotonicity property of the eccentricity function of hyperbolic
graphs: the closer a vertex is to the center of , the smaller its
eccentricity is. We also show that the distance matrix of with an additive
one-sided error of at most can be computed in
time, where is a small constant. Recent empirical studies show that
many real-world graphs (including Internet application networks, web networks,
collaboration networks, social networks, biological networks, and others) have
small hyperbolicity. So, we analyze the performance of our algorithms for
approximating centrality and distance matrix on a number of real-world
networks. Our experimental results show that the obtained estimates are even
better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author
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