Set Graphs. V. On representing graphs as membership digraphs

An undirected graph is commonly represented as a set of vertices and a set of doubletons of vertices; but one can also represent vertices by finite sets so as to ensure that membership mimics, over those sets, the edge relation of the graph. This alternative modeling, applied to connected claw-free graphs, recently gave crucial clues for obtaining simpler proofs of some of their properties (e.g., Hamiltonicity of the square of the graph). This paper adds a computer-checked contribution. On the one hand we discuss our development, by means of the Ref verifier, of two theorems on representing graphs by families of finite sets: a weaker theorem pertains to general graphs, and a stronger one to connected claw-free graphs. Before proving those theorems, we must show that every graph admits an acyclic, weakly extensional orientation, which becomes fully extensional when connectivity and claw-freeness are met. This preliminary work enables injective decoration, à la Mostowski, of the vertices by the sought-for finite sets. By this new scenario, we complement our earlier formalization with Ref of two classical properties of connected claw-free graphs. On the other hand, our present work provides another example of the ease with which graph-theoretic results are proved with the Ref verifier. For example, we managed to define and exploit the notion of connected graph without resorting to the notion of path.