# Some Bounds on the Double Domination of Signed Generalized Petersen Graphs and Signed I-Graphs

## TL;DR

This paper investigates bounds on the double domination number in signed graphs, focusing on signed cubic graphs, generalized Petersen graphs, and I-graphs, revealing new theoretical limits for these structures.

## Contribution

It provides new bounds for the double domination number in signed cubic graphs, generalized Petersen graphs, and I-graphs, advancing understanding of domination in signed graph theory.

## Key findings

- Established bounds for the double domination number in signed cubic graphs.
- Derived bounds for signed generalized Petersen graphs.
- Analyzed bounds for signed I-graphs.

## Abstract

In a graph $G$, a vertex dominates itself and its neighbors. A subset $D \subseteq V(G)$ is a double dominating set of $G$ if $D$ dominates every vertex of $G$ at least twice. A signed graph $\Sigma = (G,\sigma)$ is a graph $G$ together with an assignment $\sigma$ of positive or negative signs to all its edges. A cycle in a signed graph is positive if the product of its edge signs is positive. A signed graph is balanced if all its cycles are positive. A subset $D \subseteq V(\Sigma)$ is a double dominating set of $\Sigma$ if it satisfies the following conditions: (i) $D$ is a double dominating set of $G$, and (ii) $\Sigma[D:V \setminus D]$ is balanced, where $\Sigma[D:V \setminus D]$ is the subgraph of $\Sigma$ induced by the edges of $\Sigma$ with one end point in $D$ and the other end point in $V \setminus D$. The cardinality of a minimum double dominating set of $\Sigma$ is the double domination number $\gamma_{\times 2}(\Sigma)$. In this paper, we give bounds for the double domination number of signed cubic graphs. We also obtain some bounds on the double domination number of signed generalized Petersen graphs and signed I-graphs.