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

Nicolas Bousquet 1 Marthe Bonamy Konrad Dabrowski Matthew Johnson Daniël Paulusma Théo Pierron 
1 GOAL - Graphes, AlgOrithmes et AppLications
LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
Abstract : Abstract We resolve the computational complexity of Graph Isomorphism for classes of graphs characterized by two forbidden induced subgraphs $$ H_{1} $$ H 1 and $$H_2$$ H 2 for all but six pairs $$(H_1,H_2)$$ ( 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 $$(H_1,H_2)$$ ( H 1 , H 2 ) -free graphs to five.
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https://hal.archives-ouvertes.fr/hal-03394356
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Submitted on : Friday, October 22, 2021 - 8:47:07 AM
Last modification on : Friday, September 30, 2022 - 11:34:16 AM

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Nicolas Bousquet, Marthe Bonamy, Konrad Dabrowski, Matthew Johnson, Daniël Paulusma, et al.. Graph Isomorphism for $$(H_1,H_2)$$-Free Graphs: An Almost Complete Dichotomy. Algorithmica, 2021, 83 (3), pp.822-852. ⟨10.1007/s00453-020-00747-x⟩. ⟨hal-03394356⟩

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