33 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
On hereditary graph classes defined by forbidding Truemper configurations: recognition and combinatorial optimization algorithms, and χ-boundedness results
Truemper configurations are four types of graphs that helped us understand the structure of several well-known hereditary graph classes. The most famous examples are perhaps the class of perfect graphs and the class of even-hole-free graphs: for both of them, some Truemper configurations are excluded (as induced subgraphs), and this fact appeared to be useful, and played some role in the proof of the known decomposition theorems for these classes.
The main goal of this thesis is to contribute to the systematic exploration of hereditary graph classes defined by forbidding Truemper configurations. We study many of these classes, and we investigate their structure by applying the decomposition method. We then use our structural results to analyze the complexity of the maximum clique, maximum stable set and optimal coloring problems restricted to these classes. Finally, we provide polynomial-time recognition algorithms for all of these classes, and we obtain χ-boundedness results
Detecting wheels
A \emph{wheel} is a graph made of a cycle of length at least~4 together with
a vertex that has at least three neighbors in the cycle. We prove that the
problem whose instance is a graph and whose question is "does contains
a wheel as an induced subgraph" is NP-complete. We also settle the complexity
of several similar problems
The world of hereditary graph classes viewed through Truemper configurations
In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms
Algorithms for square-3PC(·, ·)-free Berge graphs
We consider the class of graphs containing no odd hole, no odd antihole, and no configuration consisting of three paths between two nodes such that any two of the paths
induce a hole, and at least two of the paths are of length 2. This class generalizes clawfree Berge graphs and square-free Berge graphs. We give a combinatorial algorithm of
complexity O(n7) to find a clique of maximum weight in such a graph. We also consider several subgraph-detection problems related to this class
Clique-cutsets beyond chordal graphs
Truemper configurations (thetas, pyramids, prisms, and wheels) have played an important role in the study of complex hereditary graph classes (eg, the class of perfect graphs and the class of even‐hole‐free graphs), appearing both as excluded configurations, and as configurations around which graphs can be decomposed. In this paper, we study the structure of graphs that contain (as induced subgraphs) no Truemper configurations other than (possibly) universal wheels and twin wheels. We also study several subclasses of this class. We use our structural results to analyze the complexity of the recognition, maximum weight clique, maximum weight stable set, and optimal vertex coloring problems for these classes. Furthermore, we obtain polynomial x-bounding functions for these classes
Algorithms for square-3PC(.,.)-free Berge graphs
We consider the class of graphs containing no odd hole, no odd antihole and no configuration consisting of three paths between two nodes such that any two of the paths induce a hole and at least two of the paths are of length 2. This class generalizes claw-free Berge graphs and square-free Berge graphs. We give a combinatorial algorithm of complexity O(n7) to find a clique of maximum weight in such a graph. We also consider several subgraph-detection problems related to this class.Recognition algorithm, maximum weight clique algorithm, combinatorial algorithms, perfect graphs, star decompositions.
Isometric path complexity of graphs
A set of isometric paths of a graph is "-rooted", where is a
vertex of , if is one of the end-vertices of all the isometric paths in
. The isometric path complexity of a graph , denoted by , is the
minimum integer such that there exists a vertex satisfying the
following property: the vertices of any isometric path of can be
covered by many -rooted isometric paths.
First, we provide an -time algorithm to compute the isometric path
complexity of a graph with vertices and edges. Then we show that the
isometric path complexity remains bounded for graphs in three seemingly
unrelated graph classes, namely, hyperbolic graphs, (theta, prism,
pyramid)-free graphs, and outerstring graphs. Hyperbolic graphs are extensively
studied in Metric Graph Theory. The class of (theta, prism, pyramid)-free
graphs are extensively studied in Structural Graph Theory, e.g. in the context
of the Strong Perfect Graph Theorem. The class of outerstring graphs is studied
in Geometric Graph Theory and Computational Geometry. Our results also show
that the distance functions of these (structurally) different graph classes are
more similar than previously thought.
There is a direct algorithmic consequence of having small isometric path
complexity. Specifically, we show that if the isometric path complexity of a
graph is bounded by a constant, then there exists a polynomial-time
constant-factor approximation algorithm for ISOMETRIC PATH COVER, whose
objective is to cover all vertices of a graph with a minimum number of
isometric paths. This applies to all the above graph classes.Comment: A preliminary version appeared in the proceedings of the MFCS 2023
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