Greedy domination on biclique-free graphs

TL;DR

This paper improves the approximation ratio of the greedy dominating set algorithm on graphs excluding a complete bipartite subgraph, achieving an $O(t^2 imes ext{log} k)$-approximation, where $k$ is the size of the minimum dominating set.

Contribution

The authors introduce a small modification to the greedy algorithm that yields better approximation ratios on biclique-free graphs.

Findings

Achieves $O(t^2 imes ext{log} k)$-approximation for $K_{t,t}$-free graphs.

Shows the modified greedy algorithm outperforms the standard approach on certain graph classes.

Provides theoretical bounds for dominating set approximations in biclique-free graphs.

Abstract

The greedy algorithm for approximating dominating sets is a simple method that is known to compute an $(\ln n+1)$-approximation of a minimum dominating set on any graph with $n$ vertices. We show that a small modification of the greedy algorithm can be used to compute an $O(t^2\cdot \ln k)$-approximation, where~$k$ is the size of a minimum dominating set, on graphs that exclude the complete bipartite graph $K_{t,t}$ as a subgraph.