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The concept of (k,l)-kernels of digraphs was introduced in [2]. Next, H. Galeana-Sanchez [4] proved a sufficient condition for a digraph to have a (k,l)-kernel. The result generalizes the well-known theorem of P. Duchet and it is formulated in terms of symmetric pairs of arcs. Our aim is to give necessary and sufficient conditions for digraphs without symmetric pairs of arcs to have a (k,l)-kernel. We restrict our attention to special superdigraphs of digraphs Pₘ and Cₘ.
We show that the pairs where T is a tree and its dual are the only maximal antichains of size 2 in the category of directed graphs endowed with its natural homomorphism ordering.
Consider an arc-colored digraph. A set of vertices N is a kernel by monochromatic paths if all pairs of distinct vertices of N have no monochromatic directed path between them and if for every vertex v not in N there exists n ∈ N such that there is a monochromatic directed path from v to n. In this paper we prove different sufficient conditions which imply that an arc-colored tournament has a kernel by monochromatic paths. Our conditions concerns to some subdigraphs of T and its quasimonochromatic...
We define digraphs minimal, critical, and maximal by three types of radii. Some of these classes are completely characterized, while for the others it is shown that they are large in terms of induced subgraphs.
The paper gives an overview of results for radially minimal, critical, maximal and stable graphs and digraphs.
We assign to each pair of positive integers and a digraph whose set of vertices is and for which there is a directed edge from to if . The digraph is semiregular if there exists a positive integer such that each vertex of the digraph has indegree or 0. Generalizing earlier results of the authors for the case in which , we characterize all semiregular digraphs when is arbitrary.
An orientation of a simple graph is referred to as an oriented graph. Caccetta and Häggkvist conjectured that any digraph on vertices with minimum outdegree contains a directed cycle of length at most . In this paper, we consider short cycles in oriented graphs without directed triangles. Suppose that is the smallest real such that every -vertex digraph with minimum outdegree at least contains a directed triangle. Let be a positive real. We show that if is an oriented graph without...
In the present paper we generalize a few algebraic concepts to graphs. Applying this graph language we solve some problems on subalgebra lattices of unary partial algebras. In this paper three such problems are solved, other will be solved in papers [Pió I], [Pió II], [Pió III], [Pió IV]. More precisely, in the present paper first another proof of the following algebraic result from [Bar1] is given: for two unary partial algebras and , their weak subalgebra lattices are isomorphic if and only...
The (directed) distance from a vertex to a vertex in a strong digraph is the length of a shortest - (directed) path in . The eccentricity of a vertex of is the distance from to a vertex furthest from in . The radius rad is the minimum eccentricity among the vertices of and the diameter diam is the maximum eccentricity. A central vertex is a vertex with eccentricity and the subdigraph induced by the central vertices is the center . For a central vertex in a strong digraph...
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