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(log * 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 + log * 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⌈log B n⌉ -ruling set in O(α · B · log B n) rounds in the CONGEST model. The algorithm can be modified to compute a 2, β -ruling set in O(β∆ 2/β + log * n) rounds in the CONGEST model, which matches the currently best known such algorithm in the more general LOCAL model. 1 A preliminary version of this paper appeared in the 25th International Colloquium on Structural Information and Communication Complexity (SIROCCO 2018) [KMW18].This paper presents fast and simple deterministic distributed algorithms, in the CONGEST model, for computing ruling sets of graphs, line graphs, line graphs of hypergraphs, and graphs of bounded diversity as introduced in [BEM17].The CONGEST Model of Distributed Computing [Pel00]. The graph is abstracted as an n-node network G = (V, E) with maximum degree at most ∆. Each node is assumed to have a unique O(log n)-bit ID. Communication happens in synchronous rounds. Per round, each node can send one message of at most O(log n) bits to each of its neighbors and perform (unbounded) local computations 3 . At the end, each node should know its own part of the output, e.g., whether it belongs to the ruling set or not. The time complexity of an algorithm is the number of rounds it requires to terminate.Ruling Sets. A α, β -ruling set of a graph G = (V, E) is a subset R ⊆ V of the nodes such that any two nodes in R are at distance at least α in G and for every node v ∈ V \ R, there is a node in R within distance β [AGLP89]. That is, R is an independent set in G α−1 , where G r denotes the graph with node set V and where two nodes u, v are connected by an edge if d G (u, v) ≤ r. Typically, α is called the independence parameter and r the domination parameter of the ruling set R. If α = 2, one often also simply calls R a β-ruling set. 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) (or a hypergraph H = (V, E)) if the corresponding nodes form an (α, β)ruling se...

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