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

Abstract

We resolve the computational complexity of Graph Isomorphism for classes of graphs characterized by two forbidden induced subgraphs H1 and H2 for all but six pairs (H1, H2). 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 cliquewidth, we simultaneously reduce the number of open cases for boundedness of clique-width for (H1, H2)-free graphs to five.Later, 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 P 4 -free graphs. Classifying the case where F G has size 2 is much more difficult than the size-1 case. Kratsch and Schweitzer [22] initiated this classification. 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 (H 1 , H 2 )-free graphs 5 for all pairs H 1 , H 2 ? The analogous research question for H-induced-minor-free graphs was fully answered by Belmonte, Otachi and Schweitzer [3], who also determined all graphs H for which the class of H-induced-minor-free graphs has bounded clique-width. Similar classifications for Graph Isomorphism [28] and boundedness of clique-width [15] are also known for H-minor-free graphs. Lokshtanov et al. [23] recently gave an FPT algorithm for Graph Isomorphism with parameter k on graph classes of treewidth at most k, and this has since been improved by Grohe et al. [19]. Whether an FPT algorithm exists when parameterized by clique-width is still open. Grohe and Schweitzer [20] proved membership of XP. 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 also in XP when parameterized by clique-width. 5 A graph is (H1, H2)-free if it has no induced subgraph isomorphic to H1 or H2.

Tree pivot-minors and linear rank-width

“…Such a classification already exists for H-free graphs, as observed in [22]: for a graph H, the class of H-free graphs has bounded linear rank-width if and only if H is a subgraph of P 3 not isomorphic to 3P 1 . We note that similar classifications also exist for other width parameters: for the tree-width of pH 1 , H 2 q-free graphs [7], the rank-width of H-free graphs (see [22]), rank-width of H-free bipartite graphs [23,35,36], and up to five non-equivalent open cases, rank-width of pH 1 , H 2 q-free graphs (see [8] or [22]), mim-width of H-free graphs, whereas there is still an infinite number of open cases left for the mim-width of pH 1 , H 2 q-free graphs [13].…”

Tree Pivot-Minors and Linear Rank-Width

“…Such a classification already exists for H-free graphs, as observed in [22]: for a graph H, the class of H-free graphs has bounded linear rank-width if and only if H is a subgraph of P 3 not isomorphic to 3P 1 . We note that similar classifications also exist for other width parameters: for the tree-width of pH 1 , H 2 q-free graphs [7], which was later generalized to a classification for tree-width of H-free graphs, where H is a finite set of graphs [37], rank-width of H-free graphs (see [24]), rank-width of H-free bipartite graphs [23,36,38], and up to five non-equivalent open cases, rank-width of pH 1 , H 2 q-free graphs (see [8] or [22]), and for the mim-width of H-free graphs [13], whereas there is still an infinite number of open cases left for the mim-width of pH 1 , H 2 q-free graphs [13].…”

Clique-Width: Harnessing the Power of Atoms

“…In [7] it was shown that the graph q(G) is (P 1 + P 4 , P 1 + 2P 2 )-free and that the clique-width of such graphs is unbounded.…”