A formal theory of undirected graphs in higher-order logc
This paper describes a formal theory of undirected (labeled) graphs in higher-order logic developed using the mechanical theoremproving system HOL. It formalizes and proves theorems about such notions as the empty graph, single-node graphs, finite graphs, subgraphs, adjacency relations, walks, paths, cycles, bridges, reachability, connectedness, acyclicity, trees, trees oriented with respect to roots, oriented trees viewed as family trees, top-down and bottom-up inductions in a family tree, distributing associative and commutative operations with identities recursively over subtrees of a family tree, and merging disjoint subgraphs of a graph. The main contribution of this work lies in the precise formalization of these graph-theoretic notions and the rigorous derivation of their properties in higher-order logic. This is significant because there is little tradition of formalization in graph theory due to the concreteness of graphs. A companion paper  describes the application of this formal graph theory to the mechanical verification of distributed algorithms.
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