Instruction set for the representation of graphs

TL;DR

The paper introduces IsalGraph, a novel compact string encoding for finite graphs that allows efficient, valid, and isomorphism-invariant representation, enabling applications in graph similarity, generation, and language modeling.

Contribution

It presents a new graph encoding method using a small instruction set and a virtual machine, with algorithms for encoding and canonicalization that are efficient and robust.

Findings

Strong correlation between string Levenshtein distance and graph edit distance

Encoding is valid for all strings over the alphabet, ensuring no invalid states

Evaluation on real-world datasets demonstrates practical effectiveness

Abstract

We present IsalGraph, a method for representing the structure of any finite, simple graph as a compact string over a nine-character instruction alphabet. The encoding is executed by a small virtual machine comprising a sparse graph, a circular doubly-linked list (CDLL) of graph-node references, and two traversal pointers. Instructions either move a pointer through the CDLL or insert a node or edge into the graph. A key design property is that every string over the alphabet decodes to a valid graph, with no invalid states reachable. A greedy \emph{GraphToString} algorithm encodes any connected graph into a string in time polynomial in the number of nodes; an exhaustive-backtracking variant produces a canonical string by selecting the lexicographically smallest shortest string across all starting nodes and all valid traversal orders. We evaluate the representation on five real-world graph benchmark datasets (IAM Letter LOW/MED/HIGH, LINUX, and AIDS) and show that the Levenshtein distance between IsalGraph strings correlates strongly with graph edit distance (GED). Together, these properties make IsalGraph strings a compact, isomorphism-invariant, and language-model-compatible sequential encoding of graph structure, with direct applications in graph similarity search, graph generation, and graph-conditioned language modelling