Regular Grammars as Effective Representations of Recognizable Sets of Series-Parallel Graphs

TL;DR

This paper demonstrates that recognizable sets of series-parallel graphs can be effectively represented using regular grammars, with improved bounds on recognizer algebra size and complexity of related decision problems.

Contribution

It introduces an improved construction of finite recognizer algebras with singly-exponential size, enhancing the efficiency of recognizing series-parallel graph sets.

Findings

Finite recognizer algebras are singly-exponential in size.

Intersection and inclusion problems are ExpTime-complete.

Improved bounds enhance algorithmic effectiveness for graph recognition.

Abstract

Series-parallel (SP) graphs are binary edge-labeled graphs with a designated source and target vertex, built using serial and parallel composition. A set of graphs is recognizable if membership depends only on its image under a homomorphism into a finite algebra. For SP-graphs, and more generally, for graphs of bounded tree-width, recognizability coincides with definability in Counting Monadic Second-Order (CMSO) logic. Despite this strong logical characterization, the conciseness and algorithmic effectiveness of syntactic representations of recognizable sets of SP (and bounded-tree-width) graphs remain poorly understood. Building on previously introduced regular grammars for SP-graphs, we show that recognizable sets admit concise and effective syntactic representations. The main contribution is an improved construction of finite recognizer algebras whose size is singly-exponential in the size of a regular grammar, improving upon the previously known double-exponential bound. As a consequence, the problems of intersection and language inclusion for sets represented by regular grammars are shown to be ExpTime-complete, thus improving on a previously known 2ExpTime upper bound.