New bounds on the decycling number of generalized de Bruijn digraphs

A set of vertices of a graph whose removal leaves an acyclic graph is referred as a decycling set, or a feedback vertex set, of the graph. The minimum cardinality of a decycling set of a graph [Formula: see text] is referred to as the decycling number of [Formula: see text]. For [Formula: see text], the generalized de Bruijn digraph [Formula: see text] is defined by congruence equations as follows: [Formula: see text] and [Formula: see text]. In this paper, we give a systematic method to find a decycling set of [Formula: see text] and give a new upper bound that improve the best known results. By counting the number of vertex-disjoint cycles with the idea of constrained necklaces, we obtain new lower bounds on the decycling number of generalized de Bruijn digraphs.

ON THE TWIN DOMINATION NUMBER IN GENERALIZED DE BRUIJN AND GENERALIZED KAUTZ DIGRAPHS

Let V and A denote the vertex and edge sets of a digraph G. A set T ⊆V is a twin dominating set of G if for every vertex v ∈ V - T, there exist u, w ∈ T (possibly u = w) such that arcs (u, v), (v, w) ∈ A. The twin domination number γ*(G) of G is the cardinality of a minimum twin dominating set of G. In this note, we investigate the twin domination numbers of generalized de Bruijn digraph and generalized Kautz digraph. The bounds of twin domination number of special generalized Kautz digraphs are given.