Unit and proper bitolerance digraphs

In this paper we prove that the following statements about a directed graph G→ are equivalent. (1) G→ is a unit bitolerance digraph, (2) G→ is a proper bitolerance digraph, and (3) the digraph obtained by reversing all arc directions of G→ is an interval catch digraph (also known as a point-core digraph). This result combined with known algorithms for recognizing interval catch digraphs, gives the first known polynomial-time algorithm for recognizing a class of (bi)tolerance digraphs. © 1997 John Wiley & Sons, Inc.     

Bartholdi zeta functions of digraphs

We give a determinant expression for the Bartholdi zeta function of a digraph which is not symmetric. This is a generalization of Bartholdi’s result on the Bartholdi zeta function of a graph or a symmetric digraph.     

Super-connected arc-transitive digraphs

A digraph is said to be super-connected if every minimum vertex cut is the out-neighbor set or in-neighbor set of a vertex. A digraph is said to be reducible, if there are two vertices with the same out-neighbor set or the same in-neighbor set. In this paper, we prove that a strongly connected arc-transitive oriented graph is either reducible or super-connected. Furthermore, if this digraph is also an Abelian Cayley digraph, then it is super-connected.     

Reflexive digraphs with near unanimity polymorphisms

In this paper, we prove that if a finite reflexive digraph admits Gumm operations, then it also admits a near unanimity operation. This is a generalization of similar results obtained earlier for posets and symmetric reflexive digraphs by the second author and his collaborators. In the special case of reflexive digraphs, our new result confirms a conjecture of Valeriote that states that any finite relational structure of finite signature that admits Gumm operations also admits an edge operation. We also prove that every finite reflexive digraph that admits a near unanimity operation admits totally symmetric idempotent operations of all arities. Finally, the aforementioned results yield a polynomial-time algorithm to decide whether a finite reflexive digraph admits a near unanimity operation.     

On super-edge-connected digraphs and bipartite digraphs

A maximally edge-connected digraph is called super-λ if every minimum edge disconnecting set is trivial, i.e., it consists of the edges adjacent to or from a given vertex. In this paper sufficient conditions for a digraph to be super-λ are presented in terms of parameters such as diameter and minimum degree. Similar results are also given for bipartite digraphs. As a corollary, some characterizations of super-λ graphs and bipartite graphs are obtained. © 1929 John Wiley & Sons, Inc.     

Distance and size in digraphs

We determine the maximum number of arcs in an Eulerian digraph of given order and diameter. Our bound generalises a classical result on the maximum number of edges of an undirected graph of given order and diameter by Ore (1968) and Homenko and Ostroverhii? (1970). We further determine the maximum size of a bipartite digraph of given order and radius.     

On pancyclic digraphs

We show that a strongly connected digraph with n vertices and minimum degree ? n is pancyclic unless it is one of the graphs K_p,p. This generalizes a result of A. Ghouila-Houri. We disprove a conjecture of J. A. Bondy by showing that there exist hamiltonian digraphs with n vertices and $\text{1}\text{2}\text{n(n + 1) – 3}$ edges which are not pancyclic. We show that any hamiltonian digraph with n vertices and at least $\text{1}\text{2}\text{n(n + 1) – 1}$ edges is pancyclic and we give some generalizations of this result. As applications of these results we determine the minimal number of edges required in a digraph to guarantee the existence of a cycle of length k, k ? 2, and we consider the corresponding problem where the digraphs under consideration are assumed to be strongly connected.     

Gaussian integral circulant digraphs

The spectrum of a digraph in general contains real and complex eigenvalues. A digraph is called a Gaussian integral digraph if it has a Gaussian integral spectrum that is all eigenvalues are Gaussian integers. In this paper, we consider Gaussian integral digraphs among circulant digraphs.     

Double-super-connected digraphs

A strongly connected digraph D is said to be super-connected if every minimum vertex-cut is the out-neighbor or in-neighbor set of a vertex. A strongly connected digraph D is said to be double-super-connected if every minimum vertex-cut is both the out-neighbor set of a vertex and the in-neighbor set of a vertex. In this paper, we characterize the double-super-connected line digraphs, Cartesian product and lexicographic product of two digraphs. Furthermore, we study double-super-connected Abelian Cayley digraphs and illustrate that there exist double-super-connected digraphs for any given order and minimum degree.     

On extremal weighted digraphs with no heavy paths

Bollobás and Scott proved that if the weighted outdegree of every vertex of an edge-weighted digraph is at least 1, then the digraph contains a (directed) path of weight at least 1. In this note we characterize the extremal weighted digraphs with no heavy paths. Our result extends a corresponding theorem of Bondy and Fan on weighted graphs. We also give examples to show that a result of Bondy and Fan on the existence of heavy paths connecting two given vertices in a 2-connected weighted graph does not extend to 2-connected weighted digraphs.     

Weakly distance-regular digraphs

We introduce the concept of weakly distance-regular digraph and study some of its basic properties. In particular, the (standard) distance-regular digraphs, introduced by Damerell, turn out to be those weakly distance-regular digraphs which have a normal adjacency matrix. As happens in the case of distance-regular graphs, the study is greatly facilitated by a family of orthogonal polynomials called the distance polynomials. For instance, these polynomials are used to derive the spectrum of a weakly distance-regular digraph. Some examples of these digraphs, such as the butterfly and the cycle prefix digraph which are interesting for their applications, are analyzed in the light of the developed theory. Also, some new constructions involving the line digraph and other techniques are presented.     

Longest paths in bipartite digraphs

In this paper we give a sufficient condition on the degrees of the vertices of a digraph to insure the existence of a path of length greater than or equal to a given value, when the digraph is bipartite and when the digraph is dipartite and oriented. These conditions are shown to be best possible.     

Aparallel digraphs and splicing machines

The concepts of a splicing machine and of an aparalled digraph are introduced. A splicing machine is basically a means to uniquely obtain all circular sequences on a finite alphabet by splicing together circular sequences from a small finite set of “generators”. The existence and uniqueness of the central object related to an aparallel digraph, the strong component, is proved, and this strong component is shown to be the unique fixed point of a natural operator defined on sets of vertices of the digraph. A digraph is shown to be a splicing machine if and only if it is the strong component of an aparallel digraph. Motivation comes, on the applied side, from the splicing of circular sequences on a finite alphabet and, on the theoretical side, from the Banach fixed point theorem.     

Locally-finite connected-homogeneous digraphs

A digraph is connected-homogeneous if any isomorphism between finite connected induced subdigraphs extends to an automorphism of the digraph. We consider locally-finite connected-homogeneous digraphs with more than one end. In the case that the digraph embeds a triangle we give a complete classification, obtaining a family of tree-like graphs constructed by gluing together directed triangles. In the triangle-free case we show that these digraphs are highly arc-transitive. We give a classification in the two-ended case, showing that all examples arise from a simple construction given by gluing along a directed line copies of some fixed finite directed complete bipartite graph. When the digraph has infinitely many ends we show that the descendants of a vertex form a tree, and the reachability graph (which is one of the basic building blocks of the digraph) is one of: an even cycle, a complete bipartite graph, the complement of a perfect matching, or an infinite semiregular tree. We give examples showing that each of these possibilities is realised as the reachability graph of some connected-homogeneous digraph, and in the process we obtain a new family of highly arc-transitive digraphs without property Z.     

(p,q)-odd digraphs

A digraph D is (p,q)-odd if and only if any subdivision of D contains a directed cycle of length different from p mod q. A characterization of (p,q)-odd digraphs analogous to the Seymour-Thomassen characterization of (1, 2)-odd digraphs is provided. In order to obtain this characterization we study the lattice generated by the directed cycles of a strongly connected digraph. We show that the sets of directed cycles obtained from an ear decomposition of the digraph in a natural way are bases of this lattice. A similar result does not hold for undirected graphs. However we construct, for each undirected 2-connected graph G, a set of cycles of G which form a basis of the lattice generated by the cycles of G. © 1996 John Wiley & Sons, Inc.     

Powerful digraphs

We introduce the concept of a powerful digraph and establish that a powerful digraph structure is included into the saturated structure of each nonprincipal powerful type p possessing the global pairwise intersection property and the similarity property for the theories of graph structures of type p and some of its first-order definable restrictions (all powerful types in the available theories with finitely many (> 1) pairwise nonisomorphic countable models have this property). We describe the structures of the transitive closures of the saturated powerful digraphs that occur in the models of theories with nonprincipal powerful 1-types provided that the number of nonprincipal 1-types is finite. We prove that a powerful digraph structure, considered in a model of a simple theory, induces an infinite weight, which implies that the powerful digraphs do not occur in the structures of the available classes of the simple theories (like the supersimple or finitely based theories) that do not contain theories with finitely many (> 1) countable models.     

Randomly antitraceable digraphs

An antipath in a digraph is a semipath containing no (directed) path of length 2. A digraph D is randomly antitraceable if for each vertex v of D, any antipath beginning at v can be extended to a hamiltonian antipath beginning at v. In this paper randomly antitraceable digraphs are characterized.     

On critically connected digraphs

A digraph is called critically connected if it is connected, but the deletion of any vertex destroys the connectivity. We prove that every critically connected finite digraph has at least two vertices of outdegree one. As an application, we show that for n ≧ 2, there is no n-connected, non-complete, finite digraph such that the deletion of any n vertices results in a disconnected digraph.     

Independent sets and non-augmentable paths in arc-locally in-semicomplete digraphs and quasi-arc-transitive digraphs

A digraph is arc-locally in-semicomplete if for any pair of adjacent vertices x,y, every in-neighbor of x and every in-neighbor of y either are adjacent or are the same vertex. A digraph is quasi-arc-transitive if for any arc xy, every in-neighbor of x and every out-neighbor of y either are adjacent or are the same vertex. Laborde, Payan and Xuong proposed the following conjecture: Every digraph has an independent set intersecting every non-augmentable path (in particular, every longest path). In this paper, we shall prove that this conjecture is true for arc-locally in-semicomplete digraphs and quasi-arc-transitive digraphs.     

Bigraphs versus digraphs via matrices

It was observed by Dulmage and Mendelsohn in their work on matrix reducibility that there is a one-to-one correspondence between bigraphs and digraphs determined by the utilization of the adjacency matrix. In this semiexpository paper we explore the interaction between this correspondence and a theory of matrix decomposability that is developed in several different articles. These results include: (a) a characterization of those bipartite graphs that can be labeled so that the resulting digraph is symmetric; (b) a criterion for the bigraph of a symmetric digraph to be connected; (c) a necessary and sufficient condition for a square binary matrix to be fully indecomposable in terms of its associated bigraph, and (d) matrix criteria for a digraph to be strongly, unilaterally, or weakly connected. We close with an unsolved extermal problem on the number of components of the bigraph of various orientations of a given graph. This leads to new amusing characterizations of trees and bigraphs. Dedicated to the graph-theoretic partnership of Lloyd Dulmage and Nathan Mendelsohn.