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On the Parallel Parameterized Complexity of the Graph Isomorphism Problem
Ist Teil von
WALCOM: Algorithms and Computation, 2018, Vol.10755, p.252-264
Ort / Verlag
Switzerland: Springer International Publishing AG
Erscheinungsjahr
2018
Link zum Volltext
Quelle
Alma/SFX Local Collection
Beschreibungen/Notizen
In this paper, we study the parallel and the space complexity of the graph isomorphism problem (\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {GI}$$\end{document}) for several parameterizations.
Let \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {H}=\{H_1,H_2,\cdots ,H_l\}$$\end{document} be a finite set of graphs where \documentclass[12pt]{minimal}
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\begin{document}$$|V(H_i)|\le d$$\end{document} for all i and for some constant d. Let \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {G}$$\end{document} be an \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {H}$$\end{document}-free graph class i.e., none of the graphs \documentclass[12pt]{minimal}
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\begin{document}$$G\in \mathcal {G}$$\end{document} contain any \documentclass[12pt]{minimal}
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\begin{document}$$H \in \mathcal {H}$$\end{document} as an induced subgraph. We show that \documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {GI}$$\end{document} parameterized by vertex deletion distance to \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {G}$$\end{document} is in a parameterized version of \documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {AC} ^1$$\end{document}, denoted \documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {Para} $$\end{document}-\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {AC} ^1$$\end{document}, provided the colored graph isomorphism problem for graphs in \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {G}$$\end{document} is in \documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {AC} ^1$$\end{document}. From this, we deduce that \documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {GI}$$\end{document} parameterized by the vertex deletion distance to cographs is in \documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {Para} $$\end{document}-\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {AC} ^1$$\end{document}.
The parallel parameterized complexity of \documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {GI}$$\end{document} parameterized by the size of a feedback vertex set remains an open problem. Towards this direction we show that the graph isomorphism problem is in \documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {Para} $$\end{document}-\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {TC} ^0$$\end{document} when parameterized by vertex cover or by twin-cover.
Let \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {G}'$$\end{document} be a graph class such that recognizing graphs from \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {G}'$$\end{document} and the colored version of \documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {GI}$$\end{document} for \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {G}'$$\end{document} is in logspace (\documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {L} $$\end{document}). We show that \documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {GI}$$\end{document} for bounded vertex deletion distance to \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {G}'$$\end{document} is in \documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {L} $$\end{document}. From this, we obtain logspace algorithms for \documentclass[12pt]{minimal}
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\begin{document}$$\mathsf {GI}$$\end{document} for graphs with bounded vertex deletion distance to interval graphs and graphs with bounded vertex deletion distance to cographs.