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A prime factor theorem for a generalized direct product

Wilfried Imrich, Peter F. Stadler (2006)

Discussiones Mathematicae Graph Theory

We introduce the concept of neighborhood systems as a generalization of directed, reflexive graphs and show that the prime factorization of neighborhood systems with respect to the the direct product is unique under the condition that they satisfy an appropriate notion of thinness.

A Sokoban-type game and arc deletion within irregular digraphs of all sizes

Zyta Dziechcińska-Halamoda, Zofia Majcher, Jerzy Michael, Zdzisław Skupień (2007)

Discussiones Mathematicae Graph Theory

Digraphs in which ordered pairs of out- and in-degrees of vertices are mutually distinct are called irregular, see Gargano et al. [3]. Our investigations focus on the problem: what are possible sizes of irregular digraphs (oriented graphs) for a given order n? We show that those sizes in both cases make up integer intervals. The extremal sizes (the endpoints of these intervals) are found in [1,5]. In this paper we construct, with help of Sokoban-type game, n-vertex irregular oriented graphs (irregular...

A sufficient condition for the existence of k-kernels in digraphs

H. Galeana-Sánchez, H.A. Rincón-Mejía (1998)

Discussiones Mathematicae Graph Theory

In this paper, we prove the following sufficient condition for the existence of k-kernels in digraphs: Let D be a digraph whose asymmetrical part is strongly conneted and such that every directed triangle has at least two symmetrical arcs. If every directed cycle γ of D with l(γ) ≢ 0 (mod k), k ≥ 2 satisfies at least one of the following properties: (a) γ has two symmetrical arcs, (b) γ has four short chords. Then D has a k-kernel. This result generalizes some previous results...

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