Isomorphism for Graphs of Bounded Feedback Vertex Set Number

Kratsch, Stefan; Schweitzer, Pascal

doi:10.1007/978-3-642-13731-0_9

Cited by 30 publications

(28 citation statements)

References 30 publications

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“…Though the result in this section is subsumed by the more general one in Sect. 6 and also by a result of Kratsch and Schweitzer [12], it provides a useful warm-up and a tighter bound, exponential in the parameter. In the present warm-up we only give an algorithm for the graph isomorphism problem, though the result easily holds for canonisation as well (and this follows from the more general result in Sect.…”

Section: Deletion Distance To Bounded Degreementioning

confidence: 85%

“…For each of these parameters, it remains an open question whether the problem is FPT. 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.…”

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: Deletion Distance To Bounded Degreementioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Graph Isomorphism Parameterized by Elimination Distance to Bounded Degree

Bulian

Dawar

2015

Algorithmica

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“…Very recently, it is shown in [14] that Graph Isomorphism for graphs with feedback vertex sets of size k is fixed parameter tractable, with k as the parameter.…”

Section: Parametrized Complexity and Isomorphism Testingmentioning

confidence: 99%

Colored Hypergraph Isomorphism is Fixed Parameter Tractable

et al. 2013

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We describe a fixed parameter tractable (fpt) algorithm for Colored Hypergraph Isomorphism which has running time 2 O(b) N O(1) , where the parameter b is the maximum size of the color classes of the given hypergraphs and N is the input size. We also describe fpt algorithms for certain permutation group problems that are used as subroutines in our algorithm. Keywords and phrases IntroductionA hypergraph is an ordered pair X = (V, E) where V is the vertex set and E ⊆ 2 V is the edge set. Two hypergraphs X = (V, E) and X = (V , E ) are said to be isomorphic,Given two hypergraphs X and X , the decision problem Hypergraph Isomorphism (HI) asks whether X ∼ = X . Graph Isomorphism (GI) is obviously polynomial-time reducible to HI. Conversely, HI is also known to be polynomial-time reducible to GI: Given a pair of hypergraphs X = (V, E) and X = (V , E ) as instance for HI, the reduced instance of GI consists of two corresponding bipartite graphs Y and Y defined as follows. The graph Y has vertex set V E and edge set E(Y ) = {{v, e} | v ∈ V, e ∈ E and v ∈ e}, and Y is defined similarly. Here, C D denotes the disjoint union of the sets C and D. It is easy to verify that Y ∼ = Y if and only if X ∼ = X assuming that V can be mapped only to V and E can be mapped only to E . This latter condition is easy to enforce. However, since the above reduction blows up the size of the vertex set in the bipartite encoding, the Zemlyachenko-Luks-Babai graph isomorphism algorithm [3,5,6,25] that runs in time c √ n log n , where n is the size of the vertex set of the graph, does not yield a similar algorithm for hypergraph isomorphism. We note here that the best known hypergraph isomorphism test due to Luks [16] has running time c n . Recently, Babai and Codenotti [4] have shown a 2Õ (k 2 √ n) isomorphism testing algorithm for hypergraphs with hyperedges of size bounded by k.Motivated by this situation, we explore the same algorithmic problem for bounded color class hypergraphs. The input to Colored Hypergraph Isomorphism (CHI) is a pair

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“…For the known bounded degree algorithm and the known bounded genus algorithms, the degree of the polynomial bounding the running time increases with increasing parameter (i.e., they have a running time of O(n f (k) )). Algorithms with uniformly polynomial running time (i.e., having a running time of O(f (k) · n d ) with d fixed) have only been devised for the parameters eigenvalue multiplicity [9], color multiplicity [12], feedback vertex set number [15], and rooted tree distance width [21]. In parametrized complexity theory such algorithms are called fixed-parameter tractable.…”

Section: Introductionmentioning

confidence: 99%

“…In parametrized complexity theory such algorithms are called fixed-parameter tractable. See [8] or [11] for general parameterized complexity theory, and the introduction of [15] for a graph isomorphism specific overview.…”

Section: Introductionmentioning

confidence: 99%

Isomorphism of (mis)Labeled Graphs

Schweitzer

2011

Algorithms – ESA 2011

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Abstract. For similarity measures of labeled and unlabeled graphs, we study the complexity of the graph isomorphism problem for pairs of input graphs which are close with respect to the measure. More precisely, we show that for every fixed integer k we can decide in quadratic time whether a labeled graph G can be obtained from another labeled graph H by relabeling at most k vertices. We extend the algorithm solving this problem to an algorithm determining the number ℓ of vertices that must be deleted and the number k of vertices that must be relabeled in order to make the graphs equivalent. The algorithm is fixed-parameter tractable in k + ℓ. Contrasting these tractability results, we also show that for those similarity measures that change only by finite amount d whenever one edge is relocated, the problem of deciding isomorphism of input pairs of bounded distance d is equivalent to solving graph isomorphism in general.

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Isomorphism for Graphs of Bounded Feedback Vertex Set Number

Cited by 30 publications

References 30 publications

Graph Isomorphism Parameterized by Elimination Distance to Bounded Degree

Graph Isomorphism Parameterized by Elimination Distance to Bounded Degree

Colored Hypergraph Isomorphism is Fixed Parameter Tractable

Isomorphism of (mis)Labeled Graphs

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