13 research outputs found
Vertex elimination orderings for hereditary graph classes
We provide a general method to prove the existence and compute efficiently
elimination orderings in graphs. Our method relies on several tools that were
known before, but that were not put together so far: the algorithm LexBFS due
to Rose, Tarjan and Lueker, one of its properties discovered by Berry and
Bordat, and a local decomposition property of graphs discovered by Maffray,
Trotignon and Vu\vskovi\'c. We use this method to prove the existence of
elimination orderings in several classes of graphs, and to compute them in
linear time. Some of the classes have already been studied, namely
even-hole-free graphs, square-theta-free Berge graphs, universally signable
graphs and wheel-free graphs. Some other classes are new. It turns out that all
the classes that we study in this paper can be defined by excluding some of the
so-called Truemper configurations. For several classes of graphs, we obtain
directly bounds on the chromatic number, or fast algorithms for the maximum
clique problem or the coloring problem
Few induced disjoint paths for H-free graphs
Paths P1,…,Pk in a graph G=(V,E) are mutually induced if any two distinct Pi and Pj have neither common vertices nor adjacent vertices. For a fixed integer k, the k-INDUCED DISJOINT PATHS problem is to decide if a graph G with k pairs of specified vertices (si,ti) contains k mutually induced paths Pi such that each Pi starts from si and ends at ti. Whereas the non-induced version is well-known to be polynomial-time solvable for every fixed integer k, a classical result from the literature states that even 2-INDUCED DISJOINT PATHS is NP-complete. We prove new complexity results for k-INDUCED DISJOINT PATHS if the input is restricted to H-free graphs, that is, graphs without a fixed graph H as an induced subgraph. We compare our results with a complexity dichotomy for INDUCED DISJOINT PATHS, the variant where k is part of the input
(Theta, triangle)-free and (even hole, )-free graphs. Part 1 : Layered wheels
We present a construction called layered wheel. Layered wheels are graphs of
arbitrarily large treewidth and girth. They might be an outcome for a possible
theorem characterizing graphs with large treewidth in terms of their induced
subgraphs (while such a characterization is well-understood in terms of
minors). They also provide examples of graphs of large treewidth and large
rankwidth in well-studied classes, such as (theta, triangle)-free graphs and
even-hole-free graphs with no (where a hole is a chordless cycle of
length at least four, a theta is a graph made of three internally vertex
disjoint paths of length at least two linking two vertices, and is the
complete graph on four vertices)
The (theta, wheel)-free graphs Part IV: Induced paths and cycles
A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three internally vertex-disjoint paths of length at least 2 between the same pair of distinct vertices. A wheel is a graph formed by a hole and a node that has at least 3 neighbors in the hole. In this series of papers we study the class of graphs that do not contain as an induced subgraph a theta nor a wheel. In Part II of the series we prove a decomposition theorem for this class, that uses clique cutsets and 2-joins. In this paper we use this decomposition theorem to solve several problems related to finding induced paths and cycles in our class
Exploiting structure to cope with NP-hard graph problems: Polynomial and exponential time exact algorithms
An ideal algorithm for solving a particular problem always finds an optimal solution, finds such a solution for every possible instance, and finds it in polynomial time. When dealing with NP-hard problems, algorithms can only be expected to possess at most two out of these three desirable properties. All algorithms presented in this thesis are exact algorithms, which means that they always find an optimal solution. Demanding the solution to be optimal means that other concessions have to be made when designing an exact algorithm for an NP-hard problem: we either have to impose restrictions on the instances of the problem in order to achieve a polynomial time complexity, or we have to abandon the requirement that the worst-case running time has to be polynomial. In some cases, when the problem under consideration remains NP-hard on restricted input, we are even forced to do both.
Most of the problems studied in this thesis deal with partitioning the vertex set of a given graph. In the other problems the task is to find certain types of paths and cycles in graphs. The problems all have in common that they are NP-hard on general graphs. We present several polynomial time algorithms for solving restrictions of these problems to specific graph classes, in particular graphs without long induced paths, chordal graphs and claw-free graphs. For problems that remain NP-hard even on restricted input we present exact exponential time algorithms. In the design of each of our algorithms, structural graph properties have been heavily exploited. Apart from using existing structural results, we prove new structural properties of certain types of graphs in order to obtain our algorithmic results
Three-in-a-Tree in Near Linear Time
The three-in-a-tree problem is to determine if a simple undirected graph
contains an induced subgraph which is a tree connecting three given vertices.
Based on a beautiful characterization that is proved in more than twenty pages,
Chudnovsky and Seymour [Combinatorica 2010] gave the previously only known
polynomial-time algorithm, running in time, to solve the
three-in-a-tree problem on an -vertex -edge graph. Their three-in-a-tree
algorithm has become a critical subroutine in several state-of-the-art graph
recognition and detection algorithms.
In this paper we solve the three-in-a-tree problem in time,
leading to improved algorithms for recognizing perfect graphs and detecting
thetas, pyramids, beetles, and odd and even holes. Our result is based on a new
and more constructive characterization than that of Chudnovsky and Seymour. Our
new characterization is stronger than the original, and our proof implies a new
simpler proof for the original characterization. The improved characterization
gains the first factor in speed. The remaining improvement is based on
dynamic graph algorithms.Comment: 46 pages, 12 figures, accepted to STOC 202