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

Marthe Bonamy (1) , Nicolas Bousquet (2) , Konrad K. Dabrowski (3) , Matthew Johnson (3) , Daniël Paulusma (3) , Théo Pierron (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 EDF R&D - EDF R&D
3 Department of Computer Science

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.

Domaines: Algorithme et structure de données [cs.DS]; Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Marthe Bonamy, Nicolas Bousquet, Konrad K. Dabrowski, Matthew Johnson, Daniël Paulusma, et al.. Graph Isomorphism for $(H_1,H_2)$-free Graphs: An Almost Complete Dichotomy. 2019. ⟨hal-02402919⟩