1,700 research outputs found
Inductive -independent graphs and -colorable subgraphs in scheduling: A review
Inductive -independent graphs generalize chordal graphs and have recently
been advocated in the context of interference-avoiding wireless communication
scheduling. The NP-hard problem of finding maximum-weight induced -colorable
subgraphs, which is a generalization of finding maximum independent sets,
naturally occurs when selecting sets of pairwise non-conflicting jobs
(modeled as graph vertices). We investigate the parameterized complexity of
this problem on inductive -independent graphs. We show that the Independent
Set problem is W[1]-hard even on 2-simplicial 3-minoes---a subclass of
inductive 2-independent graphs. In contrast, we prove that the more general
Maximum -Colorable Subgraph problem is fixed-parameter tractable on
edge-wise unions of cluster and chordal graphs, which are 2-simplicial. In both
cases, the parameter is the solution size. Aside from this, we survey other
graph classes between inductive 1-inductive and inductive 2-inductive graphs
with applications in scheduling
On edge-sets of bicliques in graphs
A biclique is a maximal induced complete bipartite subgraph of a graph. We
investigate the intersection structure of edge-sets of bicliques in a graph.
Specifically, we study the associated edge-biclique hypergraph whose hyperedges
are precisely the edge-sets of all bicliques. We characterize graphs whose
edge-biclique hypergraph is conformal (i.e., it is the clique hypergraph of its
2-section) by means of a single forbidden induced obstruction, the triangular
prism. Using this result, we characterize graphs whose edge-biclique hypergraph
is Helly and provide a polynomial time recognition algorithm. We further study
a hereditary version of this property and show that it also admits polynomial
time recognition, and, in fact, is characterized by a finite set of forbidden
induced subgraphs. We conclude by describing some interesting properties of the
2-section graph of the edge-biclique hypergraph.Comment: This version corrects an error in Theorem 11 found after the paper
went into prin
Unit Interval Editing is Fixed-Parameter Tractable
Given a graph~ and integers , , and~, the unit interval
editing problem asks whether can be transformed into a unit interval graph
by at most vertex deletions, edge deletions, and edge
additions. We give an algorithm solving this problem in time , where , and denote respectively
the numbers of vertices and edges of . Therefore, it is fixed-parameter
tractable parameterized by the total number of allowed operations.
Our algorithm implies the fixed-parameter tractability of the unit interval
edge deletion problem, for which we also present a more efficient algorithm
running in time . Another result is an -time algorithm for the unit interval vertex deletion problem,
significantly improving the algorithm of van 't Hof and Villanger, which runs
in time .Comment: An extended abstract of this paper has appeared in the proceedings of
ICALP 2015. Update: The proof of Lemma 4.2 has been completely rewritten; an
appendix is provided for a brief overview of related graph classe
Biclique coverings, rectifier networks and the cost of -removal
We relate two complexity notions of bipartite graphs: the minimal weight
biclique covering number and the minimal rectifier network
size of a bipartite graph . We show that there exist
graphs with . As a
corollary, we establish that there exist nondeterministic finite automata
(NFAs) with -transitions, having transitions total such that
the smallest equivalent -free NFA has
transitions. We also formulate a version of previous bounds for the weighted
set cover problem and discuss its connections to giving upper bounds for the
possible blow-up.Comment: 12 pages, to appear in proceedings of DCFS 2014: 16th International
Conference on Descriptional Complexity of Finite-State System
The complexity of partitioning into disjoint cliques and a triangle-free graph
Motivated by Chudnovsky's structure theorem of bull-free graphs, Abu-Khzam,
Feghali, and M\"uller have recently proved that deciding if a graph has a
vertex partition into disjoint cliques and a triangle-free graph is NP-complete
for five graph classes. The problem is trivial for the intersection of these
five classes. We prove that the problem is NP-complete for the intersection of
two subsets of size four among the five classes. We also show NP-completeness
for other small classes, such as graphs with maximum degree 4 and line graphs
A Local Prime Factor Decomposition Algorithm for Strong Product Graphs
This work is concerned with the prime factor decomposition (PFD) of strong
product graphs. A new quasi-linear time algorithm for the PFD with respect to
the strong product for arbitrary, finite, connected, undirected graphs is
derived. Moreover, since most graphs are prime although they can have a
product-like structure, also known as approximate graph products, the practical
application of the well-known "classical" prime factorization algorithm is
strictly limited. This new PFD algorithm is based on a local approach that
covers a graph by small factorizable subgraphs and then utilizes this
information to derive the global factors. Therefore, we can take advantage of
this approach and derive in addition a method for the recognition of
approximate graph products
Confluent Drawings: Visualizing Non-planar Diagrams in a Planar Way
In this paper, we introduce a new approach for drawing diagrams that have
applications in software visualization. Our approach is to use a technique we
call confluent drawing for visualizing non-planar diagrams in a planar way.
This approach allows us to draw, in a crossing-free manner, graphs--such as
software interaction diagrams--that would normally have many crossings. The
main idea of this approach is quite simple: we allow groups of edges to be
merged together and drawn as "tracks" (similar to train tracks). Producing such
confluent diagrams automatically from a graph with many crossings is quite
challenging, however, so we offer two heuristic algorithms to test if a
non-planar graph can be drawn efficiently in a confluent way. In addition, we
identify several large classes of graphs that can be completely categorized as
being either confluently drawable or confluently non-drawable.Comment: 10 pages, 18 figure
Forbidden Induced Subgraphs of Normal Helly Circular-Arc Graphs: Characterization and Detection
A normal Helly circular-arc graph is the intersection graph of arcs on a
circle of which no three or less arcs cover the whole circle. Lin, Soulignac,
and Szwarcfiter [Discrete Appl. Math. 2013] characterized circular-arc graphs
that are not normal Helly circular-arc graphs, and used it to develop the first
recognition algorithm for this graph class. As open problems, they ask for the
forbidden induced subgraph characterization and a direct recognition algorithm
for normal Helly circular-arc graphs, both of which are resolved by the current
paper. Moreover, when the input is not a normal Helly circular-arc graph, our
recognition algorithm finds in linear time a minimal forbidden induced subgraph
as certificate.Comment: Preliminary results of this paper appeared in the proceedings of SBPO
2012 and FAW 201
Graphs with Plane Outside-Obstacle Representations
An \emph{obstacle representation} of a graph consists of a set of polygonal
obstacles and a distinct point for each vertex such that two points see each
other if and only if the corresponding vertices are adjacent. Obstacle
representations are a recent generalization of classical polygon--vertex
visibility graphs, for which the characterization and recognition problems are
long-standing open questions.
In this paper, we study \emph{plane outside-obstacle representations}, where
all obstacles lie in the unbounded face of the representation and no two
visibility segments cross. We give a combinatorial characterization of the
biconnected graphs that admit such a representation. Based on this
characterization, we present a simple linear-time recognition algorithm for
these graphs. As a side result, we show that the plane vertex--polygon
visibility graphs are exactly the maximal outerplanar graphs and that every
chordal outerplanar graph has an outside-obstacle representation.Comment: 12 pages, 7 figure
Bounded Search Tree Algorithms for Parameterized Cograph Deletion: Efficient Branching Rules by Exploiting Structures of Special Graph Classes
Many fixed-parameter tractable algorithms using a bounded search tree have
been repeatedly improved, often by describing a larger number of branching
rules involving an increasingly complex case analysis. We introduce a novel and
general search strategy that branches on the forbidden subgraphs of a graph
class relaxation. By using the class of -sparse graphs as the relaxed
graph class, we obtain efficient bounded search tree algorithms for several
parameterized deletion problems. We give the first non-trivial bounded search
tree algorithms for the cograph edge-deletion problem and the trivially perfect
edge-deletion problems. For the cograph vertex deletion problem, a refined
analysis of the runtime of our simple bounded search algorithm gives a faster
exponential factor than those algorithms designed with the help of complicated
case distinctions and non-trivial running time analysis [21] and computer-aided
branching rules [11].Comment: 23 pages. Accepted in Discrete Mathematics, Algorithms and
Applications (DMAA
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