Relational Characterisations of Paths

TL;DR

This paper introduces an algebraic framework for reasoning about paths in graphs using relation algebras, enabling formal verification of graph algorithms within a logical setting.

Contribution

It develops a purely algebraic method to specify and analyze different types of paths in graphs, bridging relation algebras with path reasoning.

Findings

Verified correctness of three basic graph algorithms

Established algebraic characterizations of paths with and without root vertices

Demonstrated formal proof development in Isabelle/HOL

Abstract

Binary relations are one of the standard ways to encode, characterise and reason about graphs. Relation algebras provide equational axioms for a large fragment of the calculus of binary relations. Although relations are standard tools in many areas of mathematics and computing, researchers usually fall back to point-wise reasoning when it comes to arguments about paths in a graph. We present a purely algebraic way to specify different kinds of paths in relation algebras. We study the relationship between paths with a designated root vertex and paths without such a vertex. Since we stay in first-order logic this development helps with mechanising proofs.To demonstrate the applicability of the algebraic framework we verify the correctness of three basic graph algorithms. All results of this paper are formally verified using the interactive proof assistant Isabelle/HOL.