# Total 2-domination of proper interval graphs

## TL;DR

This paper improves the computational efficiency of finding total 2-dominating sets in proper interval graphs, reducing the time complexity from polynomial to linear in the number of edges.

## Contribution

It presents a new algorithm that computes total 2-dominating sets in proper interval graphs in linear time, significantly faster than previous methods.

## Key findings

- Total 2-dominating sets can be computed in O(m) time.
- The new algorithm is more efficient than previous polynomial-time algorithms.
- The approach leverages properties of proper interval graphs.

## Abstract

A set of vertices $W$ of a graph $G$ is a total $k$-dominating set when every vertex of $G$ has at least $k$ neighbors in $W$. In a recent article, Chiarelli et al.\ (Improved Algorithms for $k$-Domination and Total $k$-Domination in Proper Interval Graphs, Lecture Notes in Comput.\ Sci.\ 10856, 290--302, 2018) prove that a total $k$-dominating set can be computed in $O(n^{3k})$ time when $G$ is a proper interval graph with $n$ vertices and $m$ edges. In this note we reduce the time complexity to $O(m)$ for $k=2$.