
Biclique Graphs of K_3free Graphs and Bipartite Graphs
A biclique of a graph is a maximal complete bipartite subgraph. The bicl...
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Characterization by forbidden induced subgraphs of some subclasses of chordal graphs
Chordal graphs are the graphs in which every cycle of length at least fo...
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Coloring graph classes with no induced fork via perfect divisibility
For a graph G, χ(G) will denote its chromatic number, and ω(G) its cliqu...
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IntervalLike Graphs and Digraphs
We unify several seemingly different graph and digraph classes under one...
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Characterization and Lineartime Recognition of Paired Threshold Graphs
In a paired threshold graph, each vertex has a weight, and two vertices ...
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A recognition algorithm for adjusted interval digraphs
Min orderings give a vertex ordering characterization, common to some gr...
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The use of a pruned modular decomposition for Maximum Matching algorithms on some graph classes
We address the following general question: given a graph class C on whic...
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Graph classes and forbidden patterns on three vertices
This paper deals with graph classes characterization and recognition. A popular way to characterize a graph class is to list a minimal set of forbidden induced subgraphs. Unfortunately this strategy usually does not lead to an efficient recognition algorithm. On the other hand, many graph classes can be efficiently recognized by techniques based on some interesting orderings of the nodes, such as the ones given by traversals. We study specifically graph classes that have an ordering avoiding some ordered structures. More precisely, we consider what we call patterns on three nodes, and the recognition complexity of the associated classes. In this domain, there are two key previous works. Damashke started the study of the classes defined by forbidden patterns, a set that contains interval, chordal and bipartite graphs among others. On the algorithmic side, Hell, Mohar and Rafiey proved that any class defined by a set of forbidden patterns can be recognized in polynomial time. We improve on these two works, by characterizing systematically all the classes defined sets of forbidden patterns (on three nodes), and proving that among the 23 different classes (up to complementation) that we find, 21 can actually be recognized in linear time. Beyond this result, we consider that this type of characterization is very useful, leads to a rich structure of classes, and generates a lot of open questions worth investigating.
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