We introduce the class of bi-arc digraphs, and show they coincide with the class of digraphs that admit a conservative semi-lattice polymorphism, i.e., a min ordering. Surprisingly this turns out to be also the class of digraphs that admit totally symmetric conservative polymorphisms of all arities. We give an obstruction characterization of, and a polynomial time recognition algorithm for, thi...

We show that the independent spanning tree conjecture on digraphs is true if we restrict ourselves to line digraphs. Also, we construct independent spanning trees with small depths in iterated line digraphs. From the results, we can obtain independent spanning trees with small depths in de Bruijn and Kautz digraphs that improve the previously known upper bounds on the depths.

In this paper we define the removable cycle that, if $Im$ is a class of graphs, $Gin Im$, the cycle $C$ in $G$ is called removable if $G-E(C)in Im$. The removable cycles in Eulerian graphs have been studied. We characterize Eulerian graphs which contain two edge-disjoint removable cycles, and the necessary and sufficient conditions for Eulerian graph to have removable cycles h...

A classical result by Erdős and Pósa[3] states that there is a function f : N → N such that for every k, every graph G contains k pairwise vertex disjoint cycles or a set T of at most f(k) vertices such that G− T is acyclic. The generalisation of this result to directed graphs is known as Younger’s conjecture and was proved by Reed, Robertson, Seymour and Thomas in 1996. This so-called Erdős-Pó...

Thomassen proved that there is no degree of strong connectivity which guarantees a cycle through two given vertices in a digraph (Combinatorica 11 (1991) 393-395). In this paper we consider a large family of digraphs, including symmetric digraphs (i.e. digraphs obtained from undirected graphs by replacing each edge by a directed cycle of length two), semicomplete bipartite digraphs, locally sem...

We study the list homomorphism and retraction problems for the class of reflexive digraphs (digraphs in which each vertex has a loop). These problems have been intensively studied in the case of undirected graphs, but the situation seems more complex for digraphs. We also focus on an intermediate ‘subretraction’ problem. It turns out that the complexity of the subretraction problem can be class...

The Moore bound for a diregular digraph of degree d and diameter k is M d;k = 1 + d + : : : + d k. It is known that digraphs of order M d;k do not exist for d > 1 and k > 1. In this paper we study digraphs of order M d;k ? 1, that is, digraphs with defect 1, denoted by (d; k)-digraphs. If G is a (d; k)-digraph, then for each vertex v of G there exists a vertex w (called the repeat of v) such th...

A digraph D is k-transitive if the existence of a directed path (v0, v1, . . . , vk), of length k implies that (v0, vk) ∈ A(D). Clearly, a 2-transitive digraph is a transitive digraph in the usual sense. Transitive digraphs have been characterized as compositions of complete digraphs on an acyclic transitive digraph. Also, strong 3 and 4-transitive digraphs have been characterized. In this work...

let $d$ be a digraph with vertex set $v(d)$. for vertex $vin v(d)$, the degree of $v$,denoted by $d(v)$, is defined as the minimum value if its out-degree and its in-degree.now let $d$ be a digraph with minimum degree $deltage 1$ and edge-connectivity$lambda$. if $alpha$ is real number, then the zeroth-order general randic index is definedby $sum_{xin v(d)}(d(x))^{alpha}$. a digraph is maximall...