On the  domination polynomial of a digraph: a  generation function approach

Alencar, Jorge ^[2] ; de Lima, Leonardo ^[1]
1. [1] Universidade Federal do Paraná
  
  Universidade Federal do Paraná
  
  Brasil
2. [2] Instituto Federal de Educação, Ciência e Tecnologia do Triângulo Mineiro
Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 40, Nº. 6, 2021, págs. 1587-1602
Idioma: inglés
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Resumen
- Let $G$ be a directed graph on $n$ vertices. The domination polynomial of $G$ is the polynomial $D(G, x) = \sum^n_{i=0} d(G, i)x^i$, where $d(G, i)$ is the number of dominating sets of $G$ with $i$ vertices. In this paper, we prove that the domination polynomial of $G$ can be obtained by using an ordinary generating function, which can be extended to undirected graphs. Besides, we show that our method is useful to obtain the minimum-weighted dominating set of a graph.