Components and acyclicity of graphs. An exercise in combining precision with concision

• Published in: Journal of logical and algebraic methods in programming Vol. 124; p. 100730
• Main Authors: Backhouse, Roland, Doornbos, Henk, Glück, Roland, van der Woude, Jaap
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.01.2022
• Subjects: Acyclic graph; Calculational method; Point-free reasoning; Regular algebra; Relation algebra; Strongly connected component; Relation algebra; Strongly connected component; Calculational method; Regular algebra; Acyclic graph; Point-free reasoning
• ISSN: 2352-2208
• DOI: 10.1016/j.jlamp.2021.100730

Central to algorithmic graph theory are the concepts of acyclicity and strongly connected components of a graph, and the related search algorithms. This article is about combining mathematical precision and concision in the presentation of these concepts. Concise formulations are given for, for example, the reflexive-transitive reduction of an acyclic graph, reachability properties of acyclic graphs and their relation to the fundamental concept of “definiteness”, and the decomposition of paths in a graph via the identification of its strongly connected components and a pathwise homomorphic acyclic subgraph. The relevant properties are established by precise algebraic calculation. The combination of concision and precision is achieved by the use of point-free relation algebra capturing the algebraic properties of paths in graphs, as opposed to the use of pointwise reasoning about paths between nodes in graphs.