Robust Algebraic Theories of Triangle Graphs

2026-08-11

AI Summary

This work addresses the lack of algebraic characterizations and formal language-theoretic foundations for graph classes of treewidth at most three in which every edge belongs to a triangle, such as Apollonian networks. By naturally extending series-parallel graph algebras from treewidth two to treewidth three, the paper introduces two graph algebras based on parallel composition and ternary serial composition that precisely generate triangular graphs and fan graphs, respectively. Building upon this, it establishes a complete algebraic theory for these graph classes, proving that their recognizable graph languages coincide with those definable in counting monadic second-order logic (CMSO). Furthermore, it demonstrates the decidability of CMSO logic over the corresponding context-free graph languages.

Abstract

Triangle graphs are graphs of tree-width at most three in which every edge belongs to a triangle. This class encompasses well-known graph families such as Apollonian networks. We also consider fan graphs, a subclass of triangle graphs closely related to the 3-connected triangle graphs. Our main result is an algebraic characterization of both classes. We introduce two graph algebras based on parallel composition and a ternary serial composition, and show that they generate exactly the triangle and fan graphs, respectively. These algebras provide a natural extension of the classical algebra of series-parallel graphs from tree-width two to tree-width three. Building on these characterizations, we investigate context-free, recognizable, and logically-definable graph languages. We show that counting monadic second-order logic (CMSO) is decidable over the context-free sets of triangle and fan graphs. Moreover, we prove that recognizable graph languages coincide with languages definable in CMSO for both algebras.

Problem

• triangle graphs
• fan graphs
• graph algebras
• tree-width three
• CMSO logic

Innovation

• triangle graphs
• graph algebras
• tree-width three
• CMSO logic
• fan graphs