The recognizability of sets of graphs is a robust property

TL;DR

This paper investigates the robustness of recognizability properties of graph sets across different algebraic signatures and shows their equivalence under certain combinatorial conditions, enhancing understanding of graph recognition frameworks.

Contribution

It demonstrates that recognizability notions are stable across various algebraic signatures and establishes conditions under which different recognizability classes coincide.

Findings

Recognizability is robust across multiple graph algebra signatures.

HR- and VR-recognizability coincide for graphs without large complete bipartite subgraphs.

Results are applicable in the broader context of relational structures.

Abstract

Once the set of finite graphs is equipped with an algebra structure (arising from the definition of operations that generalize the concatenation of words), one can define the notion of a recognizable set of graphs in terms of finite congruences. Applications to the construction of efficient algorithms and to the theory of context-free sets of graphs follow naturally. The class of recognizable sets depends on the signature of graph operations. We consider three signatures related respectively to Hyperedge Replacement (HR) context-free graph grammars, to Vertex Replacement (VR) context-free graph grammars, and to modular decompositions of graphs. We compare the corresponding classes of recognizable sets. We show that they are robust in the sense that many variants of each signature (where in particular operations are defined by quantifier-free formulas, a quite flexible framework) yield the same notions of recognizability. We prove that for graphs without large complete bipartite subgraphs, HR-recognizability and VR-recognizability coincide. The same combinatorial condition equates HR-context-free and VR-context-free sets of graphs. Inasmuch as possible, results are formulated in the more general framework of relational structures.