11 research outputs found
Practical and Efficient Split Decomposition via Graph-Labelled Trees
Split decomposition of graphs was introduced by Cunningham (under the name
join decomposition) as a generalization of the modular decomposition. This
paper undertakes an investigation into the algorithmic properties of split
decomposition. We do so in the context of graph-labelled trees (GLTs), a new
combinatorial object designed to simplify its consideration. GLTs are used to
derive an incremental characterization of split decomposition, with a simple
combinatorial description, and to explore its properties with respect to
Lexicographic Breadth-First Search (LBFS). Applying the incremental
characterization to an LBFS ordering results in a split decomposition algorithm
that runs in time , where is the inverse Ackermann
function, whose value is smaller than 4 for any practical graph. Compared to
Dahlhaus' linear-time split decomposition algorithm [Dahlhaus'00], which does
not rely on an incremental construction, our algorithm is just as fast in all
but the asymptotic sense and full implementation details are given in this
paper. Also, our algorithm extends to circle graph recognition, whereas no such
extension is known for Dahlhaus' algorithm. The companion paper [Gioan et al.]
uses our algorithm to derive the first sub-quadratic circle graph recognition
algorithm
Splitting cubic circle graphs
We show that every 3-regular circle graph has at least two pairs of twin
vertices; consequently no such graph is prime with respect to the split
decomposition. We also deduce that up to isomorphism, K_4 and K_{3,3} are the
only 3-connected, 3-regular circle graphs.Comment: 18 pages, 15 figure
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
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
Extending Partial Representations of Circle Graphs in Near-Linear Time
The partial representation extension problem generalizes the recognition problem for geometric intersection graphs. The input consists of a graph G, a subgraph and a representation of H . The question is whether G admits a representation whose restriction to H is . We study this question for circle graphs, which are intersection graphs of chords of a circle. Their representations are called chord diagrams. We show that for a graph with n vertices and m edges the partial representation extension problem can be solved in time, thereby improving over an -time algorithm by Chaplick et al. (J Graph Theory 91(4), 365–394, 2019). The main technical contributions are a canonical way of orienting chord diagrams and a novel compact representation of the set of all canonically oriented chord diagrams that represent a given circle graph G, which is of independent interest
Practical and Efficient Split Decomposition via Graph-Labelled Trees
International audienceSplit decomposition of graphs was introduced by Cunningham (under the name join decomposition) as a generalization of the modular decomposition. This paper undertakes an investigation into the algorithmic properties of split decompo- sition. We do so in the context of graph-labelled trees (GLTs), a new combinatorial object designed to simplify its consideration. GLTs are used to derive an incremental characterization of split decomposition, with a simple combinatorial description, and to explore its properties with respect to Lexicographic Breadth-First Search (LBFS). Applying the incremental characterization to an LBFS ordering results in a split de- composition algorithm that runs in time O(n + m)α(n + m), where α is the inverse Ackermann function, whose value is smaller than 4 for any practical graph. Com- pared to Dahlhaus' linear time split decomposition algorithm (Dahlhaus in J. Algo- rithms 36(2):205-240, 2000), which does not rely on an incremental construction, our algorithm is just as fast in all but the asymptotic sense and full implementatio