Étude de la conjecture de Seymour sur le second voisinage (original) (raw)
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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
The structure of digraphs with excess one
arXiv (Cornell University), 2021
A digraph G is k-geodetic if for any (not necessarily distinct) vertices u, v there is at most one directed walk from u to v with length not exceeding k. The order of a k-geodetic digraph with minimum out-degree d is bounded below by the directed Moore bound M (d, k) = 1+d+d 2 +• • •+d k. The Moore bound can be met only in the trivial cases d = 1 and k = 1, so it is of interest to look for k-geodetic digraphs with out-degree d and smallest possible order M (d, k) + ǫ, where ǫ is the excess of the digraph. Miller, Miret and Sillasen recently ruled out the existence of digraphs with excess one for k = 3, 4 and d ≥ 2 and for k = 2 and d ≥ 8. We conjecture that there are no digraphs with excess one for d, k ≥ 2 and in this paper we investigate the structure of minimal counterexamples to this conjecture. We severely constrain the possible structures of the outlier function and prove the non-existence of certain digraphs with degree three and excess one, as well closing the open cases k = 2 and d = 3, 4, 5, 6, 7 left by the analysis of Miller et al. We further show that there are no involutary digraphs with excess one, i.e. the outlier function of any such digraph must contain a cycle of length ≥ 3.
Note on in-antimagicness and out-antimagicness of digraphs
Journal of Discrete Mathematical Sciences and Cryptography, 2020
A digraph D is called (a, d)-vertex-in-antimagic ((a, d)-vertex-out-antimagic) if it is possible to label its vertices and arcs with distinct numbers from the set {1, 2, ,| ()| | ()|} V D A D … + such that the set of all in-vertex-weights (out-vertex-weights) form an arithmetic sequence with initial term a and a common difference d, where a > 0, d ≥ 0 are integers. The in-vertexweight (out-vertex-weight) of a vertex n OE V(D) is defined as the sum of the vertex label and the labels of arcs ingoing (outgoing) to n. Moreover, if the smallest numbers are used to label the vertices of D such a graph is called super. In this paper we will deal with in-antimagicness and out-antimagicness of in-regular and out-regular digraphs.
Arc-disjoint circuits in digraphs
Discrete Mathematics, 1982
Let D be a digraph of order n (n a 3) in which the irldegree and the outdegree of each verr:ex is at least in. In this paper, we shall show that n contaa IS two arc-disjoint circuits, one of whch is a Hamiltonian circuit, the other is a circuit oi lengtll at least n -1. Let 0 =-(V; A) be a. digraph of order n, V be t'le vertex set of D, A be the arc set of D. (u, U) denotes an arc from vertex u to u. Define I(u; D)=(u: (u,u)EA(D), UE V
SIAM Journal on Discrete Mathematics, 2020
We unify several seemingly different graph and digraph classes under one umbrella. These classes are all, broadly speaking, different generalizations of interval graphs, and include, in addition to interval graphs, adjusted interval digraphs, complements of threshold tolerance graphs (known as co-TT graphs), bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray bigraphs. (The last three classes coincide, but have been investigated in different contexts.) We show that all of the above classes are united by a common ordering characterization, the existence of a min ordering. However, because the presence or absence of reflexive relationships (loops) affect whether a graph or digraph has a min ordering, to obtain this result, we must define the graphs and digraphs to have those loops that are implied by their definitions. These have been largely ignored in previous work. We propose a common generalization of all these graph and digraph classes, namely signed-interval digraphs, characterized by the existence of a compact representation, a signed-interval model, which is a generalization of known representations of the graph classes. We show that the signed-interval digraphs are precisely those digraphs that are characterized by the existence of a min ordering when the loops implied by the model are considered part of the graph. We also offer an alternative geometric characterization of these digraphs. We show that co-TT graphs are the symmetric signed-interval digraphs, the adjusted interval digraphs are the reflexive signed-interval digraphs, and the interval graphs are the intersection of these two classes, namely, the reflexive and symmetric signed-interval digraphs. Similar results hold for bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray bigraphs. 1
Irregularity strength of digraphs
Discrete Mathematics, 2009
This paper is dedicated to the special occasion of Gary Chartrand's 70th birthday and in recognition of his numerous contributions to graph theory.
Line-digraphs, arborescences, and theorems of tutte and knuth
Journal of Combinatorial Theory, Series B, 1978
The line-digraph of a digraph D with vertices V , ..... V is the digrah D* obtained from D by associating with each edge of D a vertex of D*, and then directing an edge from vertex (Vi, VYj) of D* to vertex (Vk, Vm) if and only if j = k. This paper extends a characterization given by Harary and Norman for linedigraphs. It is also possible to repeatedly contract vertices of the line-digraph (with a new contraction procedure) so as to obtain the digraph derived from D by deleting all vertices with no incoming edges. Several new identities for arborescences are presented, leading to a combinatorial proof of Knuth's formula for the number of arborescences of a line-digraph. A new proof is given for the fact that in a digraph with every vertex having indegree equal to outdegree, the number of arborescences with root VY is independent of j. Finally a new proof is presented for Tutte's Matrix Tree Theorem which shows the theorem to be a special case of the principle of inclusion-exclusion.
An infinite family of self-diclique digraphs
Applied Mathematics Letters, 2010
Let D = (V , A) be a digraph. Consider X and Y (not necessarily disjoint), nonempty subsets of vertices of D. We define a disimplex K (X, Y) of D to be the subdigraph whose vertex set is V (K (X, Y)) = X ∪ Y and in which an arc goes from every vertex of X to every vertex of Y (when X ∩ Y = ∅, loops are not considered). A disimplex K (X, Y) is called a diclique of D if K (X, Y) is not a proper subdigraph of any other disimplex of D. The diclique digraph (or diclique operator) − → k (D) of a digraph D is the digraph whose vertex set is the set of all dicliques of D and K (X, Y), K (X , Y) is an arc of − → k (D) if and only if Y ∩ X = ∅. We say that a digraph D is self-diclique if − → k (D) is isomorphic to D. In this paper we exhibit an infinite family of self-diclique circulant digraphs for which one of its members is an Eulerian orientation of the graph of the regular octahedron. This family is a natural generalization of the example given in Zelinka (2002) [5].
Self-diclique circulant digraphs
Mathematica Bohemica
We study a particular digraph dynamical system, the so called digraph diclique operator. Dicliques have frequently appeared in the literature the last years in connection with the construction and analysis of different types of networks, for instance biochemical, neural, ecological, sociological and computer networks among others. Let D = (V, A) be a reflexive digraph (or network). Consider X and Y (not necessarily disjoint) nonempty subsets of vertices (or nodes) of D. A disimplex K(X, Y) of D is the subdigraph of D with vertex set X ∪ Y and arc set {(x, y) : x ∈ X, y ∈ Y } (when X ∩ Y = ∅, loops are not considered). A disimplex K(X, Y) of D is called a diclique of D if K(X, Y) is not a proper subdigraph of any other disimplex of D. The diclique digraph k(D) of a digraph D is the digraph whose vertex set is the set of all dicliques of D and (K(X, Y), K(X ′ , Y ′)) is an arc of k(D) if and only if Y ∩ X ′ = ∅. We say that a digraph D is self-diclique if k(D) is isomorphic to D. In this paper, we provide a characterization of the self-diclique circulant digraphs and an infinite family of non-circulant self-diclique digraphs.
Discussiones Mathematicae Graph Theory, 2012
Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A digraph D is 3-transitive if the existence of the directed path (u, v, w, x) of length 3 in D implies the existence of the arc (u, x) ∈ A(D). In this article strong 3-transitive digraphs are characterized and the structure of non-strong 3-transitive digraphs is described. The results are used, e.g, to characterize 3-transitive digraphs that are transitive and to characterize 3-transitive digraphs with a kernel.