# Signed edge domination numbers of complete tripartite graphs: Part 2

## TL;DR

This paper determines the signed edge domination number for complete tripartite graphs, completing the classification for all such graphs and advancing understanding of their domination properties.

## Contribution

It provides the exact signed edge domination number for all complete tripartite graphs $K_{m,n,p}$ with specified parameters, filling a gap in the existing literature.

## Key findings

- Exact signed edge domination numbers for $K_{m,n,p}$ with $p \\geq m+n$
- Completes the classification of signed edge domination for all complete tripartite graphs
- Advances understanding of domination functions in complex graph structures

## Abstract

The closed neighborhood $N_G[e]$ of an edge $e$ in a graph $G$ is the set consisting of $e$ and of all edges having an end-vertex in common with $e$. Let $f$ be a function on $E(G)$, the edge set of $G$, into the set $\{-1,1\}$. If $\sum_{x\in{N[e]}}f(x)\geq 1$ for each edge $e \in E(G)$, then $f$ is called a signed edge dominating function of $G$. The signed edge domination number of $G$ is the minimum weight of a signed edge dominating function of $G$. In this paper, we find the signed edge domination number of the complete tripartite graph $K_{m,n,p}$, where $1\leq m\leq n$ and $p\geq m+n$. This completes the search for the signed edge domination numbers of the complete tripartite graphs.