On Tractable Parameterizations of Graph Isomorphism

Bouland, Adam; Dawar, Anuj; Kopczyński, Eryk

doi:10.1007/978-3-642-33293-7_21

Cited by 14 publications

(23 citation statements)

References 29 publications

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“…Since G\A contains the ≤-maximal elements, they are all incomparable. By definition of an elimination order to degree d, this means that each vertex in G\A has at most d neighbours in G\A, so this graph has degree at most d, establishing (2).…”

Section: Proposition 43 a Graph G Has Ed D (G) ≤ K If And Only If mentioning

confidence: 96%

“…On the other hand, GI has been shown to be FPT when parameterized by eigenvalue multiplicity [5], tree distance width [21], the maximum size of a simplical component [19,20] and minimum feedback vertex set [12]. Bouland et al [2] showed that the problem is FPT when parameterized by the tree depth of a graph and extended this result to a parameter they termed generalised tree depth. In a recent advance on this, Lokshtanov et al [14] have shown that graph isomorphism is also FPT parameterized by tree width.…”

Section: Introductionmentioning

confidence: 99%

“…5. It should be noted that the parameter termed generalised tree depth in [2] can be seen as a special case of elimination distance to degree 2.…”

Section: Introductionmentioning

confidence: 99%

“…a tree-depth decomposition of C) which isminimal. We use the result of Bouland et al [2] on canonisation of bounded tree-depth graphs to bound the branching in the search and establish that canonisation is FPT.…”

Section: Introductionmentioning

confidence: 99%

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Graph Isomorphism Parameterized by Elimination Distance to Bounded Degree

Bulian

Dawar

2015

Algorithmica

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A commonly studied means of parameterizing graph problems is the deletion distance from triviality (Guo et al., Parameterized and exact computation, Springer, Berlin, pp. 162-173, 2004), which counts vertices that need to be deleted from a graph to place it in some class for which efficient algorithms are known. In the context of graph isomorphism, we define triviality to mean a graph with maximum degree bounded by a constant, as such graph classes admit polynomial-time isomorphism tests. We generalise deletion distance to a measure we call elimination distance to triviality, based on elimination trees or tree-depth decompositions. We establish that graph canonisation, and thus graph isomorphism, is FPT when parameterized by elimination distance to bounded degree, extending results of Bouland et al. (Parameterized and exact computation, Springer, Berlin, pp. 218-230, 2012).

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Section: Proposition 43 a Graph G Has Ed D (G) ≤ K If And Only If mentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

“…5. It should be noted that the parameter termed generalised tree depth in [2] can be seen as a special case of elimination distance to degree 2.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Graph Isomorphism Parameterized by Elimination Distance to Bounded Degree

Bulian

Dawar

2015

Algorithmica

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“…To prove the lemma, at first, it is not clear at all how modulo-counting quantifiers can be used to count the number of components satisfying a given FO+MOD-sentence. But it is shown in [2,Lem. 7] that the number of tree-depth roots of each component of a graph is bounded in terms of its tree-depth.…”

Section: Order-invariant Monadic Second-order Logicmentioning

confidence: 96%

Expressivity and Succinctness of Order-Invariant Logics on Depth-Bounded Structures

Eickmeyer

Elberfeld

Harwath

2014

Mathematical Foundations of Computer Science 2014

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Abstract. We study the expressive power and succinctness of orderinvariant sentences of first-order (FO) and monadic second-order (MSO) logic on graphs of bounded tree-depth. Order-invariance is undecidable in general and, therefore, in finite model theory, one strives for logics with a decidable syntax that have the same expressive power as orderinvariant sentences. We show that on graphs of bounded tree-depth, order-invariant FO has the same expressive power as FO, and orderinvariant MSO has the same expressive power as the extension of FO with modulo-counting quantifiers. Our proof techniques allow for a finegrained analysis of the succinctness of these translations. We show that for every order-invariant FO sentence there exists an FO sentence whose size is elementary in the size of the original sentence, and whose number of quantifier alternations is linear in the tree-depth. Our techniques can be adapted to obtain a similar quantitative variant of a known result that the expressive power of MSO and FO coincides on graphs of bounded tree-depth.

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Isomorphism Testing for T-graphs in FPT

Ağaoğlu

Hliněný

2022

WALCOM: Algorithms and Computation

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On Tractable Parameterizations of Graph Isomorphism

Cited by 14 publications

References 29 publications

Graph Isomorphism Parameterized by Elimination Distance to Bounded Degree

Graph Isomorphism Parameterized by Elimination Distance to Bounded Degree

Expressivity and Succinctness of Order-Invariant Logics on Depth-Bounded Structures

Isomorphism Testing for T-graphs in FPT

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