28 research outputs found
Graphs that do not contain a cycle with a node that has at least two neighbors on it
We recall several known results about minimally 2-connected graphs, and show
that they all follow from a decomposition theorem. Starting from an analogy
with critically 2-connected graphs, we give structural characterizations of the
classes of graphs that do not contain as a subgraph and as an induced subgraph,
a cycle with a node that has at least two neighbors on the cycle. From these
characterizations we get polynomial time recognition algorithms for these
classes, as well as polynomial time algorithms for vertex-coloring and
edge-coloring
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 (e.g. 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. We also obtain polynomial χ-bounding functions for these classes
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
Graphs with polynomially many minimal separators
We show that graphs that do not contain a theta, pyramid, prism, or turtle as
an induced subgraph have polynomially many minimal separators. This result is
the best possible in the sense that there are graphs with exponentially many
minimal separators if only three of the four induced subgraphs are excluded. As
a consequence, there is a polynomial time algorithm to solve the maximum weight
independent set problem for the class of (theta, pyramid, prism, turtle)-free
graphs. Since every prism, theta, and turtle contains an even hole, this also
implies a polynomial time algorithm to solve the maximum weight independent set
problem for the class of (pyramid, even hole)-free graphs
Detecting 2-joins faster
2-joins are edge cutsets that naturally appear in the decomposition of
several classes of graphs closed under taking induced subgraphs, such as
balanced bipartite graphs, even-hole-free graphs, perfect graphs and claw-free
graphs. Their detection is needed in several algorithms, and is the slowest
step for some of them. The classical method to detect a 2-join takes
time where is the number of vertices of the input graph and the number
of its edges. To detect \emph{non-path} 2-joins (special kinds of 2-joins that
are needed in all of the known algorithms that use 2-joins), the fastest known
method takes time . Here, we give an -time algorithm for both
of these problems. A consequence is a speed up of several known algorithms
Structure and algorithms for (cap, even hole)-free graphs
A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, and more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs. We prove that every such graph G has a vertex of degree at most [View the MathML source], and hence [View the MathML source], where ω(G) denotes the size of a largest clique in G and χ(G) denotes the chromatic number of G. We give an O(nm) algorithm for q-coloring these graphs for fixed q and an O(nm) algorithm for maximum weight stable set, where n is the number of vertices and m is the number of edges of the input graph. We also give a polynomial-time algorithm for minimum coloring. Our algorithms are based on our results that triangle-free odd-signable graphs have treewidth at most 5 and thus have clique-width at most 48, and that (cap, 4-hole)-free odd-signable graphs G without clique cutsets have treewidth at most 6ω(G)−1 and clique-width at most 48
Algorithms for 3PC(.,.)-free Berge graphs
Extended abstractInternational audienc
Induced subgraphs and tree decompositions V. Small components of big vertices
Aboulker, Adler, Kim, Sintiari, and Trotignon conjectured that every graph
with bounded maximum degree and large treewidth must contain, as an induced
subgraph, a large subdivided wall, or the line graph of a large subdivided
wall. This conjecture was recently proved by Korhonen, but the problem of
identifying the obstacles to bounded treewidth in the general case (that is,
without the bounded maximum degree condition) remains wide open. Examples of
structures of large treewidth which avoid the "usual suspects" have been
constructed by Sintiari and Trotignon, and by Davies. In this note, we aim to
better isolate the features of these examples that lead to large treewidth. To
this end, we prove the following result. Let be a graph, and write
for the size of a largest connected component in the graph induced
by on the set of vertices of degree at least 3. If is small and
the treewidth of is large, then must contain a large subdivided wall or
the line graph of a large subdivided wall. This result is the best possible, in
the sense that the conclusion fails if we replace 3 by any larger number in the
definition of , as evidenced by Davies' example.Comment: We found a much quicker way of proving our result using Korhonen's
result from arXiv:2203.13233. We have included our new proof as part of the
new "Induced subgraphs and tree decompositions V. One neighbor in a hole."
That new manuscript is available at arXiv:2205.04420. The main result from
the withdrawn manuscript is Theorem 7.1 in the new manuscrip
Coloring perfect graphs with no balanced skew-partitions
International audienceWe present an algorithm that computes a maximum stable set of any perfect graph with no balanced skew-partition. We present time algorithm that colors them