# Deterministic Distributed Ruling Sets of Line Graphs

An (α,β)-ruling set of a graph G=(V,E) is a set R⊆ V such that for any node v∈ V there is a node u∈ R in distance at most β from v and such that any two nodes in R are at distance at least α from each other. The concept of ruling sets can naturally be extended to edges, i.e., a subset F⊆ E is an (α,β)-ruling edge set of a graph G=(V,E) if the corresponding nodes form an (α,β)-ruling set in the line graph of G. This paper presents a simple deterministic, distributed algorithm, in the CONGEST model, for computing (2,2)-ruling edge sets in O(^* n) rounds. Furthermore, we extend the algorithm to compute ruling sets of graphs with bounded diversity. Roughly speaking, the diversity of a graph is the maximum number of maximal cliques a vertex belongs to. We devise (2,O(D))-ruling sets on graphs with diversity D in O(D+^* n) rounds. This also implies a fast, deterministic (2,O(ℓ))-ruling edge set algorithm for hypergraphs with rank at most ℓ. Furthermore, we provide a ruling set algorithm for general graphs that for any B≥ 2 computes an (α, α_B n )-ruling set in O(α· B ·_B n) rounds in the CONGEST model. The algorithm can be modified to compute a (2, β)-ruling set in O(βΔ^2/β + ^* n) rounds in the CONGEST model, which matches the currently best known such algorithm in the more general LOCAL model.

READ FULL TEXT