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22
Structure theorem and isomorphism test for graphs with excluded topological subgraphs
 In Proc. 44th ACM Symp. on the Theory of Computing
, 2012
"... We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph H as a minor to graphs excluding H as a topological subgraph. We prove that for a fixed H, every graph excluding H as a topological subgraph has a tree decomposition where each part is either “almost emb ..."
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We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph H as a minor to graphs excluding H as a topological subgraph. We prove that for a fixed H, every graph excluding H as a topological subgraph has a tree decomposition where each part is either “almost embeddable ” to a fixed surface or has bounded degree with the exception of a bounded number of vertices. Furthermore, such a decomposition is computable by an algorithm that is fixedparameter tractable with parameter∣H ∣. We present two algorithmic applications of our structure theorem. To illustrate the mechanics of a “typical ” application of the structure theorem, we show that on graphs excluding H as a topological subgraph, Partial Dominating Set (find k vertices whose closed neighborhood has maximum size) can be solved in time f(H,k) ⋅ nO(1) time. More significantly, we show that on graphs excluding H as a topological subgraph, Graph Isomorphism can be solved in time nf(H). This result unifies and generalizes two previously known important polynomialtime solvable cases of Graph Isomorphism: boundeddegree graphs [18] and Hminor free graphs [22]. The proof of this result needs a generalization of our structure theorem to the context of invariant treelike decomposition.
Minimum Bisection is fixed parameter tractable
 THE PROCEEDINGS OF STOC
, 2013
"... In the classic Minimum Bisection problem we are given as input a graph G and an integer k. The task is to determine whether there is a partition of V (G) into two parts A and B such that A  − B  ≤ 1 and there are at most k edges with one endpoint in A and the other in B. In this paper we giv ..."
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In the classic Minimum Bisection problem we are given as input a graph G and an integer k. The task is to determine whether there is a partition of V (G) into two parts A and B such that A  − B  ≤ 1 and there are at most k edges with one endpoint in A and the other in B. In this paper we give an algorithm for Minimum Bisection with running time O(2 O(k3) n 3 log
Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask)
, 2014
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Excluded Forest Minors and the Erdős–Pósa Property
, 2013
"... A classical result of Robertson and Seymour states that the set of graphs containing a fixed planar graph H as a minor has the socalled Erdős–Pósa property; namely, there exists a function f depending only on H such that, for every graph G and every positive integer k, the graph G has k vertexdi ..."
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A classical result of Robertson and Seymour states that the set of graphs containing a fixed planar graph H as a minor has the socalled Erdős–Pósa property; namely, there exists a function f depending only on H such that, for every graph G and every positive integer k, the graph G has k vertexdisjoint subgraphs each containing H as a minor, or there exists a subset X of vertices of G with X  ! f(k) such that G−X has no Hminor (see Robertson and Seymour, J. Combin. Theory Ser. B 41 (1986) 92–114). While the best function f currently known is exponential in k, a O(k log k) bound is known in the special case where H is a forest. This is a consequence of a theorem of Bienstock, Robertson, Seymour and Thomas on the pathwidth of graphs with an excluded forestminor. In this paper we show that the function f can be taken to be linear when H is a forest. This is best possible in the sense that no linear bound is possible if H has a cycle.
A Survey on the Computational Complexity of Colouring Graphs with Forbidden Subgraphs
, 2014
"... For a positive integer k, a kcolouring of a graph G = (V,E) is a mapping c: V → {1, 2,..., k} such that c(u) 6 = c(v) whenever uv ∈ E. The COLOURING problem is to decide, for a given G and k, whether a kcolouring of G exists. If k is fixed (that is, it is not part of the input), we have the deci ..."
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For a positive integer k, a kcolouring of a graph G = (V,E) is a mapping c: V → {1, 2,..., k} such that c(u) 6 = c(v) whenever uv ∈ E. The COLOURING problem is to decide, for a given G and k, whether a kcolouring of G exists. If k is fixed (that is, it is not part of the input), we have the decision problem kCOLOURING instead. We survey known results on the computational complexity of COLOURING and kCOLOURING for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial colouring, or where lists of permissible colours are given for each vertex.
Contraction checking in graphs on surfaces
, 2012
"... The Contraction Checking problem asks, given two graphs H and G as input, whether H can be obtained from G by a sequence of edge contractions. Contraction Checking remains NPcomplete, even when H is fixed. We show that this is not the case when G is embeddable in a surface of fixed Euler genus. In ..."
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The Contraction Checking problem asks, given two graphs H and G as input, whether H can be obtained from G by a sequence of edge contractions. Contraction Checking remains NPcomplete, even when H is fixed. We show that this is not the case when G is embeddable in a surface of fixed Euler genus. In particular, we give an algorithm that solves Contraction Checking in f(h, g) · V (G)  3 steps, where h is the size of H and g is the Euler genus of the input graph G.
The planar directed kvertexdisjoint paths problem is fixedparameter tractable
 CORR
"... Given a graph G and k pairs of vertices (s1, t1),..., (sk, tk), the kVertexDisjoint Paths problem asks for pairwise vertexdisjoint paths P1,..., Pk such that Pi goes from si to ti. Schrijver [SICOMP’94] proved that the kVertexDisjoint Paths problem on planar directed graphs can be solved in ti ..."
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Cited by 2 (0 self)
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Given a graph G and k pairs of vertices (s1, t1),..., (sk, tk), the kVertexDisjoint Paths problem asks for pairwise vertexdisjoint paths P1,..., Pk such that Pi goes from si to ti. Schrijver [SICOMP’94] proved that the kVertexDisjoint Paths problem on planar directed graphs can be solved in time nO(k). We give an algorithm with running time 22 O(k2) ·nO(1) for the problem, that is, we show the fixedparameter tractability of the problem.
Forbidding kuratowski graphs as immersions
 CoRR
"... Abstract The immersion relation is a partial ordering relation on graphs that is weaker than the topological minor relation in the sense that if a graph G contains a graph H as a topological minor, then it also contains it as an immersion but not vice versa. Kuratowski graphs, namely K 5 and K 3,3 ..."
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Abstract The immersion relation is a partial ordering relation on graphs that is weaker than the topological minor relation in the sense that if a graph G contains a graph H as a topological minor, then it also contains it as an immersion but not vice versa. Kuratowski graphs, namely K 5 and K 3,3 , give a precise characterization of planar graphs when excluded as topological minors. In this note we give a structural characterization of the graphs that exclude Kuratowski graphs as immersions. We prove that they can be constructed by applying consecutive iedgesums, for i ≤ 3, starting from graphs that are planar subcubic or of branchwidth at most 10.
Graphs with no 7wheel subdivision
, 2009
"... The subgraph homeomorphism problem, SHP(H), has been shown to be polynomialtime solvable for any fixed pattern graph H, but practical algorithms have been developed only for a few specific pattern graphs. Among these are the wheels with four, five, and six spokes. This paper examines the subgraph h ..."
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The subgraph homeomorphism problem, SHP(H), has been shown to be polynomialtime solvable for any fixed pattern graph H, but practical algorithms have been developed only for a few specific pattern graphs. Among these are the wheels with four, five, and six spokes. This paper examines the subgraph homeomorphism problem where the pattern graph is a wheel with seven spokes, and gives a result that describes graphs with no W7subdivision, showing how they can be built up, using certain operations, from ‘pieces ’ of at most 37 vertices. The result leads to an efficient algorithm solving SHP(W7). This algorithm has features that are similar to those in some parameterized algorithms, and may provide useful insight in searching for a fixedparameter tractable result for SHP(Wk), with parameter k. 1