8,190 research outputs found
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
A survey on algorithmic aspects of modular decomposition
The modular decomposition is a technique that applies but is not restricted
to graphs. The notion of module naturally appears in the proofs of many graph
theoretical theorems. Computing the modular decomposition tree is an important
preprocessing step to solve a large number of combinatorial optimization
problems. Since the first polynomial time algorithm in the early 70's, the
algorithmic of the modular decomposition has known an important development.
This paper survey the ideas and techniques that arose from this line of
research
On colouring point visibility graphs
In this paper we show that it can be decided in polynomial time whether or
not the visibility graph of a given point set is 4-colourable, and such a
4-colouring, if it exists, can also be constructed in polynomial time. We show
that the problem of deciding whether the visibility graph of a point set is
5-colourable, is NP-complete. We give an example of a point visibility graph
that has chromatic number 6 while its clique number is only 4
On Almost Well-Covered Graphs of Girth at Least 6
We consider a relaxation of the concept of well-covered graphs, which are
graphs with all maximal independent sets of the same size. The extent to which
a graph fails to be well-covered can be measured by its independence gap,
defined as the difference between the maximum and minimum sizes of a maximal
independent set in . While the well-covered graphs are exactly the graphs of
independence gap zero, we investigate in this paper graphs of independence gap
one, which we also call almost well-covered graphs. Previous works due to
Finbow et al. (1994) and Barbosa et al. (2013) have implications for the
structure of almost well-covered graphs of girth at least for . We focus on almost well-covered graphs of girth at least . We show
that every graph in this class has at most two vertices each of which is
adjacent to exactly leaves. We give efficiently testable characterizations
of almost well-covered graphs of girth at least having exactly one or
exactly two such vertices. Building on these results, we develop a
polynomial-time recognition algorithm of almost well-covered
-free graphs
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