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(k,l)-kernels, (k,l)-semikernels, k-Grundy functions and duality for state splittings

Hortensia Galeana-Sánchez;  Ricardo Gómez — 2007

Discussiones Mathematicae Graph Theory

Line digraphs can be obtained by sequences of state splittings, a particular kind of operation widely used in symbolic dynamics [12]. Properties of line digraphs inherited from the source have been studied, for instance in [7] Harminc showed that the cardinalities of the sets of kernels and solutions (kernel's dual definition) of a digraph and its line digraph coincide. We extend this for (k,l)-kernels in the context of state splittings and also look at (k,l)-semikernels, k-Grundy functions and...

Independent transversals of longest paths in locally semicomplete and locally transitive digraphs

Hortensia Galeana-Sánchez;  Ricardo Gómez;  Juan José Montellano-Ballesteros — 2009

Discussiones Mathematicae Graph Theory

We present several results concerning the Laborde-Payan-Xuang conjecture stating that in every digraph there exists an independent set of vertices intersecting every longest path. The digraphs we consider are defined in terms of local semicompleteness and local transitivity. We also look at oriented graphs for which the length of a longest path does not exceed 4.

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