Graph Isomorphism for $(H_1,H_2)$-free Graphs: An Almost Complete Dichotomy

TL;DR

This paper nearly completely classifies the computational complexity of Graph Isomorphism for graph classes defined by two forbidden induced subgraphs, reducing the open cases to five by leveraging clique-width properties.

Contribution

It provides a near-complete dichotomy for Graph Isomorphism on $(H_1,H_2)$-free graphs, combining existing results with new insights to minimize unresolved cases.

Findings

Resolved all but six open cases of Graph Isomorphism for $(H_1,H_2)$-free graphs.

Reduced the number of open cases related to clique-width to five.

Established a relationship between Graph Isomorphism and clique-width for these graph classes.

Abstract

We resolve the computational complexity of Graph Isomorphism for classes of graphs characterized by two forbidden induced subgraphs $H_1$ and $H_2$ for all but six pairs $(H_1,H_2)$. Schweitzer had previously shown that the number of open cases was finite, but without specifying the open cases. Grohe and Schweitzer proved that Graph Isomorphism is polynomial-time solvable on graph classes of bounded clique-width. Our work combines known results such as these with new results. By exploiting a relationship between Graph Isomorphism and clique-width, we simultaneously reduce the number of open cases for boundedness of clique-width for $(H_1,H_2)$-free graphs to five.