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.
Highlights
The Graph Isomorphism problem, which is that of deciding whether two given graphs are isomorphic, is a central problem in Computer Science
Schweitzer [30] pointed out great similarities between proving unboundedness of clique-width of some graph class G and proving that Graph Isomorphism stays GI-complete for G
We will illustrate these similarities by showing that our construction demonstrating that Graph Isomorphism is GI-complete for-free graphs can be used to show that this class has unbounded clique-width
Summary
The Graph Isomorphism problem, which is that of deciding whether two given graphs are isomorphic, is a central problem in Computer Science. In order to increase understanding of the computational complexity of Graph Isomorphism, it is natural to place restrictions on the input This approach has established that on many graph classes Graph Isomorphism is polynomial-time solvable, but that on many others the problem remains GI-complete. Colbourn [10] proved that Graph Isomorphism is polynomial-time solvable even for the class of permutation graphs, which form a superclass of the class of P4-free graphs. Schweitzer [30] later extended the results of [22] and proved that only a finite number of cases remain open This leads to our research question: Is it possible to determine the computational complexity of Graph Isomorphism for (H1, H2)-free graphs for all pairs H1, H2?. Theorem 2 [20] For every constant c, Graph Isomorphism is polynomial-time solvable on graphs of clique-width at most c. Grohe and Neuen [17] have since improved this result by showing that the more general Canonisation problem is in XP when parameterized by clique-width
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