Set Graphs. III. Proof Pearl: Claw-Free Graphs Mirrored into Transitive Hereditarily Finite Sets

Abstract

We report on the formalization of two classical results about claw-free graphs, which have been verified correct by Jacob T. Schwartz’s proof-checker Referee. We have proved formally that every connected claw-free graph admits (1) a near-perfect matching, (2) Hamiltonian cycles in its square. To take advantage of the set-theoretic foundation of Referee, we exploited set equivalents of the graph-theoretic notions involved in our experiment: edge, source, square, etc. To ease some\ud proofs, we have often resorted to weak counterparts of well-established notions such as cycle, claw-freeness, longest directed path, etc

Set Graphs. V. On representing graphs as membership digraphs

“…Can we, with equal ease, formalize in Ref the Milanič-Tomescu representation result per se? This paper provides a positive answer, thus achieving one of the continuations of [10] envisaged in [9,Sec. A.10].…”

“…Moreover, the fact that the class of claw-free graphs is the largest class of graphs, closed under taking induced subgraphs, with the property that every connected member G of it admits such a transitive ν G (since the claw does not admit one), makes this representation theorem rather worthy. Further evidence of the close kinship between connected claw-free graphs and membership digraphs comes from the observation that the transitivity property is actually crucial to obtain the two simple proofs presented in [8,9,10]. For example, since the removal of an ∈-maximal element from a transitive set ν G leads to another transitive set, this representation gives an immediate hook for inductive arguments (see the details in [10]).…”

“…Specifically, the facilitation stems from transferring those results to the special class of the membership digraphs, whose set of vertices is a hereditarily finite set and whose arcs precisely reflect the membership relation between vertices. Under this change of perspective, a fully formal reconstruction of the first two results became affordable and, once carried out, was certified correct with the Ref proof-checker [8,9,10].…”

“…One usually views the edges of a graph 1 as vertex doubletons; but various ways of representing graphs can be devised (as quickly surveyed in [10,end of Sec. 2]).…”

Set graphs. II. Complexity of set graph recognition and similar problems

“…The study of extensional acyclic digraphs inspired short proofs of two classical results on claw-free graphs (a strengthening of the fact that squares of connected claw-free graphs are Hamiltonian, and the fact that every connected claw-free graph of even order has a perfect matching) [2]. A formalization of these two results in the proof-checker Referee was carried out in [3]. Since Referee deals only with Zermelo-Fraenkel sets, representing a connected claw-free graph by a transitive 'claw-free' set turned out to require the minimal formalism.…”