61 research outputs found
The (theta, wheel)-free graphs Part II: Structure theorem
A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three paths between the same pair of distinct vertices so that the union of any two of the paths induces a hole. A wheel is a graph formed by a hole and a node that has at least 3 neighbors in the hole. In this paper we obtain a decomposition theorem for the class of graphs that do not contain an induced subgraph isomorphic to a theta or a wheel, i.e. the class of (theta, wheel)-free graphs. The decomposition theorem uses clique cutsets and 2-joins. Clique cutsets are vertex cutsets that work really well in decomposition based algorithms, but are unfortunately not general enough to decompose more complex hereditary graph classes. A 2-join is an edge cutset that appeared in decomposition theorems of several complex classes, such as perfect graphs, even-hole-free graphs and others. In these decomposition theorems 2-joins are used together with vertex cutsets that are more general than clique cutsets, such as star cutsets and their generalizations (which are much harder to use in algorithms). This is a first example of a decomposition theorem that uses just the combination of clique cutsets and 2-joins. This has several consequences. First, we can easily transform our decomposition theorem into a complete structure theorem for (theta, wheel)-free graphs, i.e. we show how every (theta, wheel)-free graph can be built starting from basic graphs that can be explicitly constructed, and gluing them together by prescribed composition operations; and all graphs built this way are (theta, wheel)-free. Such structure theorems are very rare for hereditary graph classes, only a few examples are known. Secondly, we obtain an (n⁴m)-time decomposition based recognition algorithm for (theta, wheel)-free graphs. Finally, in Parts III and IV of this series, we give further applications of our decomposition theorem
Even-hole-free graphs with bounded degree have bounded treewidth
Treewidth is a parameter that emerged from the study of minor closed classes
of graphs (i.e. classes closed under vertex and edge deletion, and edge
contraction). It in some sense describes the global structure of a graph.
Roughly, a graph has treewidth if it can be decomposed by a sequence of
noncrossing cutsets of size at most into pieces of size at most . The
study of hereditary graph classes (i.e. those closed under vertex deletion
only) reveals a different picture, where cutsets that are not necessarily
bounded in size (such as star cutsets, 2-joins and their generalization) are
required to decompose the graph into simpler pieces that are structured but not
necessarily bounded in size. A number of such decomposition theorems are known
for complex hereditary graph classes, including even-hole-free graphs, perfect
graphs and others. These theorems do not describe the global structure in the
sense that a tree decomposition does, since the cutsets guaranteed by them are
far from being noncrossing. They are also of limited use in algorithmic
applications.
We show that in the case of even-hole-free graphs of bounded degree the
cutsets described in the previous paragraph can be partitioned into a bounded
number of well-behaved collections. This allows us to prove that even-hole-free
graphs with bounded degree have bounded treewidth, resolving a conjecture of
Aboulker, Adler, Kim, Sintiari and Trotignon [arXiv:2008.05504]. As a
consequence, it follows that many algorithmic problems can be solved in
polynomial time for this class, and that even-hole-freeness is testable in the
bounded degree graph model of property testing. In fact we prove our results
for a larger class of graphs, namely the class of -free odd-signable
graphs with bounded degree
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
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
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
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