On the domination number of $t$-constrained de Bruijn graphs

TL;DR

This paper introduces $t$-constrained de Bruijn graphs, a generalization of classical de Bruijn graphs, and investigates their domination number, providing bounds and exact values that could impact network design and bioinformatics.

Contribution

The paper defines $t$-constrained de Bruijn graphs and analyzes their domination number, offering bounds and exact solutions, extending the understanding of graph domination in generalized structures.

Findings

Established upper and lower bounds for domination numbers.

Identified cases with exact domination number values.

Connected structural properties to potential applications.

Abstract

Motivated by the work on the domination number of directed de Bruijn graphs and some of its generalizations, in this paper we introduce a natural generalization of de Bruijn graphs (directed and undirected), namely $t$-constrained de Bruijn graphs, where $t$ is a positive integer, and then study the domination number of these graphs. Within the definition of $t$-constrained de Bruijn graphs, de Bruijn and Kautz graphs correspond to 1-constrained and 2-constrained de Bruijn graphs, respectively. This generalization inherits many structural properties of de Bruijn graphs and may have similar applications in interconnection networks or bioinformatics. We establish upper and lower bounds for the domination number on $t$-constrained de Bruijn graphs both in the directed and in the undirected case. These bounds are often very close and in some cases we are able to find the exact value.