On k-strong and k-cyclic digraphs (original) (raw)
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On a generalization of transitivity for digraphs
Discrete Mathematics, 1988
In this paper we investigate the foliowing generalization of transitivity: A digraph D is (m, n)-transitive whenever there is a path of length m from x to y there is a subset of n + 1 vertices of these m + 1 vertices which contain a path of length n from x to y.
Some remarks on cycles in graphs and digraphs
Discrete Mathematics, 2001
We survey several recent results on cycles of graphs and directed graphs of the following form: 'Does there exist a set of cycles with a property P that generates all the cycles by operation O?'.
A note on the second neighborhood problem for kkk-anti-transitive and mmm-free digraphs
arXiv (Cornell University), 2024
Seymour's Second Neighborhood Conjecture (SSNC) asserts that every finite oriented graph has a vertex whose second out-neighborhood is at least as large as its first outneighborhood. Such a vertex is called a Seymour vertex. A digraph D = (V, E) is k-anti-transitive if for every pair of vertices u, v ∈ V , the existence of a directed path of length k from u to v implies that (u, v) / ∈ E. An m-free digraph is digraph having no directed cycles with length at most m. In this paper, we prove that if D is k-anti-transitive and (k − 4)-free digraph, then D has a Seymour vertex. As a consequence, a special case of Caccetta-Haggkvist Conjecture holds on 7-anti-transitive oriented graphs. This work extends recently known results.
A classification of locally semicomplete digraphs
Discrete Mathematics, 1997
In Huang gave a characterization of local tournaments. His characterization involves arc-reversals and therefore may not be easily used to solve other structural problems on locally semicomplete digraphs (where one deals with a fixed locally semicomplete digraph). In this paper we derive a classification of locally semicomplete digraphs which is very useful for studying structural properties of locally semicomplete digraphs and which does not depend on Huang's characterization. An advantage of this new classification of locally semicomplete digraphs is that it allows one to prove results for locally semicomplete digraphs without reproving the same statement for tournaments.
Paths and cycles in extended and decomposable digraphs
Discrete Mathematics, 1997
We consider digraphs-called extended locally semicomplete digraphs, or extended LSD's, for short-that can be obtained from locally semicomplete digraphs by substituting independent sets for vertices. We characterize Hamiltonian extended LSD's as well as extended LSD's containing Hamiltonian paths. These results as well as some additional ones imply polynomial algorithms for finding a longest path and a longest cycle in an extended LSD. Our characterization of Hamiltonian extended LSD's provides a partial solution to a problem posed by R. Häggkvist in [14]. Combining results from this paper with some general results derived for so-called totally Φ-decomposable digraphs in [3], we prove that the longest path problem is polynomially solvable for totally Φ 0-decomposable digraphs-a fairly wide family of digraphs which is a common generalization of acyclic digraphs, semicomplete multipartite digraphs, extended LSD's and quasi-transitive digraphs. Similar results are obtained for the longest cycle problem and other problems on cycles in subfamilies of totally Φ 0-decomposable digraphs. These polynomial algorithms are a natural and fairly deep generalization of algorithms obtained for quasi-transitive digraphs in [3] in order to solve a problem posed by N. Alon.
Some Conditions for the Existence of ( d, k)-Digraphs
2003
A (d,k)-digraph is a diregular digraph of degree d ≥ 4, diameter k ≥ 3 and the number of vertices d + d 2 + ... + d k . The existence problem of (d,k)-digraphs is one of difficult problem. In this paper, we will present some new necessary conditions for the existence of such digraphs.
Springer Monographs in Mathematics, 2009
In this chapter we introduce several classes of digraphs. We study these classes with respect to their properties, characterization, recognition and decomposition. Further properties of the classes are studied in the following chapters of this book. In Section 2.1 we study basic properties of acyclic digraphs. Acyclic digraphs form a very important family of digraphs and the reader will often encounter them in this book. Multipartite digraphs and extended digraphs are introduced in Section 2.2. These digraphs are studied in many other sections of our book. In Section 2.3, we introduce and study the transitive closure and a transitive reduction of a digraph. We use the notion of transitive reduction already in Section 2.6. Several characterizations and a recognition algorithm for line digraphs are given in Section 2.4. We investigate basic properties of de Bruijn and Kautz digraphs and their generalizations in Section 2.5. These digraphs are extreme or almost extreme with respect to their diameter and vertex-strong connectivity. Series-parallel digraphs are introduced and studied in Section 2.6. These digraphs are of interest due to various applications such as scheduling. In the study of series-parallel digraphs we use notions and results of Sections 2.3 and 2.4. An interesting generalization of transitive digraphs, the class of quasitransitive digraphs, is considered in Section 2.7. The path-merging property of digraphs which is quite important for investigation of some classes of digraphs including tournaments is introduced and studied in Section 2.8. Two classes of path-mergeable digraphs, locally in-semicomplete and locally outsemicomplete digraphs, both generalizing the class of tournaments, are defined and investigated with respect to their basic properties in Section 2.9. The intersection of these two classes forms the class of locally semicomplete digraphs, which are studied in Section 2.10. There we give a very useful classification of locally semicomplete digraphs, which is applied in several proofs in other chapters. A characterization of a special subclass of locally semicomplete digraphs, called round digraphs, is also proved. In Section 2.11, we study three classes of totally decomposable digraphs forming important generalizations of quasi-transitive digraphs as well as some other classes of digraphs. The aim of Section 2.11 is to investigate recognition
Arc-disjoint strong spanning subdigraphs in compositions and products of digraphs
Discrete Mathematics, 2019
A digraph D = (V, A) has a good decomposition if A has two disjoint sets A 1 and A 2 such that both (V, A 1) and (V, A 2) are strong. Let T be a digraph with t vertices u 1 ,. .. , u t and let H 1 ,. .. H t be digraphs such that H i has vertices u i,ji , 1 ≤ j i ≤ n i. Then the composition Q = T [H 1 ,. .. , H t ] is a digraph with vertex set {u i,ji | 1 ≤ i ≤ t, 1 ≤ j i ≤ n i } and arc set A(Q) = ∪ t i=1 A(H i)∪{u iji u pqp | u i u p ∈ A(T), 1 ≤ j i ≤ n i , 1 ≤ q p ≤ n p }. For digraph compositions Q = T [H 1 ,. .. H t ], we obtain sufficient conditions for Q to have a good decomposition and a characterization of Q with a good decomposition when T is a strong semicomplete digraph and each H i is an arbitrary digraph with at least two vertices. For digraph products, we prove the following: (a) if k ≥ 2 is an integer and G is a strong digraph which has a collection of arc-disjoint cycles covering all vertices, then the Cartesian product digraph G k (the kth powers with respect to Cartesian product) has a good decomposition; (b) for any strong digraphs G, H, the strong product G ⊠ H has a good decomposition.
On the Pathwidth of Almost Semicomplete Digraphs
Lecture Notes in Computer Science, 2015
We call a digraph h-semicomplete if each vertex of the digraph has at most h non-neighbors, where a non-neighbor of a vertex v is a vertex u = v such that there is no edge between u and v in either direction. This notion generalizes that of semicomplete digraphs which are 0-semicomplete and tournaments which are semicomplete and have no anti-parallel pairs of edges. Our results in this paper are as follows. (1) We give an algorithm which, given an h-semicomplete digraph G on n vertices and a positive integer k, in (h + 2k + 1) 2k n O(1) time either constructs a path-decomposition of G of width at most k or concludes correctly that the pathwidth of G is larger than k. (2) We show that there is a function f (k, h) such that every h-semicomplete digraph of pathwidth at least f (k, h) has a semicomplete subgraph of pathwidth at least k. One consequence of these results is that the problem of deciding if a fixed digraph H is topologically contained in a given h-semicomplete digraph G admits a polynomial-time algorithm for fixed h.