Robust Algebraic Theories of Triangle Graphs

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.