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We also show that there are strongly connected hypotraceable oriented graphs and that there are hypotraceable digraphs with precisely two strong components one of which is a source or a sink. Volume 4 , Issue 4. The full text of this article hosted at iucr.
If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account. If the address matches an existing account you will receive an email with instructions to retrieve your username. Carsten Thomassen Aarhus Universitet Search for more papers by this author.
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Font Size. Two examples are that every connected locally semicomplete digraph is traceable and every strongly connected locally semicomplete digraph has a hamiltonian cycle. Graph Theory, 14 3 —, , [ 9 ] was published in there has been a significant amount of research done on the class including several PhD theses. In this chapter we survey a number of important results, both structural and algorithmic, on locally semicomplete digraphs and illustrate various important proof-techniques.
Several of the results hold even for some superclasses of locally semicomplete digraphs.
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Many of the proofs and algorithms rely on a structural characterization of those locally semicomplete digraphs that are not semicomplete have independence number at least 2. As it turns out, these digraphs fall in two disjoint classes, called round decomposable and evil locally semicomplete digraphs, respectively. The first of these has a structure which allows many problems to be solved efficiently, whereas the second class, the evil locally semicomplete digraphs, has a structure which is much closer to that of semicomplete digraphs and hence requires much more work for many problems.
In this chapter we will consider the class of semicomplete multipartite digraphs SMD. A digraph is semicomplete multipartite if it is obtained from a complete multipartite graph by replacing every edge by an arc or a pair of opposite arcs.
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In other words, the vertex set of a semicomplete multipartite digraph can be partitioned into sets such that vertices within the same set are nonadjacent and vertices between different sets are adjacent. This chapter gives a comprehensive survey on this class of digraphs.
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Quasi-transitive digraphs generalize both tournaments and semicomplete digraphs and transitive digraphs, and share some of the nice properties of these families. In this chapter, we focus on presenting how usually difficult problems admit efficient solutions for the family of quasi-transitive digraphs and some of its generalizations. We begin with the study of the structure of quasi-transitive digraphs, given by the recursive characterization theorem known as the Canonical Decomposition Theorem; two generalizations of quasi-transitive digraphs are introduced.
Some of the problems studied for quasi-transitive digraphs and its generalizations include hamiltonicity, traceability, k -linkages weak k -linkages, existence and number of k -kings, the Path Partition Conjecture and pancyclicity. A brief section is devoted to homomorphisms in transitive digraphs.
Structural parameters for undirected graphs such as the path-width or tree-width of graphs have played a crucial role in developing a structure theory for graphs based on the minor relation and they have also found many algorithmic applications. Starting in the late s, several ideas for generalizing this theory to digraphs have appeared. Broadly, for the purpose of this chapter, we distinguish these approaches into three categories: tree-width inspired , rank-width inspired and density based.
The tree-width inspired approaches are based on the idea of generalizing the concept of undirected tree-width or path-width to digraphs.
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The various attempts, which we will discuss below, all have the goal of generalizing some natural property or some natural characterization of tree-width of undirected graphs to digraphs. In the same way as tree-width can be seen as a global connectivity measure for undirected graphs, the various versions of a directed analogue of tree-width measure global connectivity in digraphs.
However, on digraphs, connectivity can be measured in many different natural ways.
It turns out that equivalent characterizations of tree-width on undirected graphs yield different concepts on digraphs, with different properties, advantages and disadvantages. This chapter is a survey of the four standard associative digraph products, namely the Cartesian, strong, direct and lexicographic products.
Topics include metric properties, connectedness, hamiltonian properties and invariants. Special attention is given to issues of cancellation and unique prime factorization. Obviously, there are countless digraph classes, so that any attempt to give a complete overview is doomed to failure.
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