Distributed distance domination in graphs with no $K_{2,t}$-minor

TL;DR

This paper presents distributed algorithms for finding near-optimal distance-$k$ dominating sets in graphs excluding a $K_{2,t}$-minor, achieving constant approximation and arbitrarily close solutions efficiently.

Contribution

It introduces distributed algorithms that approximate and optimize distance-$k$ dominating sets in $K_{2,t}$-minor-free graphs, including outerplanar graphs, with constant and near-optimal guarantees.

Findings

Constant approximation in a constant number of rounds.

Near-optimal solutions within $(1+ ext{epsilon})$ factor.

Algorithms applicable to outerplanar graphs.

Abstract

We prove that a simple distributed algorithm finds a constant approximation of an optimal distance-$k$ dominating set in graphs with no $K_{2,t}$-minor. The algorithm runs in a constant number of rounds. We further show how this procedure can be used to give a distributed algorithm which given $\epsilon>0$ and $k,t\in \mathbb{Z}^+$ finds in a graph $G=(V,E)$ with no $K_{2,t}$-minor a distance-$k$ dominating set of size at most $(1+\epsilon)$ of the optimum. The algorithm runs in $O(\log^*{|V|})$ rounds in the Local model. In particular, both algorithms work in outerplanar graphs.