Let $M$ be the $|V| \times |V|$ adjacency matrix of a graph $G$. The cut rank of a set $A \subseteq V(G)$ is the rank of the submatrix of $M$ induced by the rows of $A$ and the columns of $V(G) \backslash A$. A rank decomposition of a graph $G$ is a pair $(T,L)$ where $T$ is a binary tree and $L$ is a bijection from $V(G)$ to the leaves of the tree $T$. Any edge $e$ in the tree $T$ splits $V(G)$ into two parts $A_e, B_e$ corresponding to the leaves of the two connected components of $T - e$. The width of an edge $e \in E(T)$ is the cutrank of $A_e$. The width of the rank-decomposition $(T,L)$ is the maximum width of an edge in $T$. The rankwidth of the graph $G$ is the minimum width of a rank-decomposition of $G$.
Minimal/maximal is with respect to the contents of ISGCI. Only references for direct bounds are given. Where no reference is given, check equivalent parameters.
Problems in italics have no summary page and are only listed when ISGCI contains a result for the current parameter.
3-Colourability
[?]
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FPT | [+]Details | |||||
Clique
[?]
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FPT | [+]Details | |||||
Clique cover
[?]
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XP | [+]Details | |||||
Colourability
[?]
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FPT | [+]Details | |||||
Domination
[?]
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FPT | [+]Details | |||||
Feedback vertex set
[?]
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FPT | [+]Details | |||||
Graph isomorphism
[?]
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XP | [+]Details | |||||
Hamiltonian cycle
[?]
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XP | [+]Details | |||||
Hamiltonian path
[?]
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XP | [+]Details | |||||
Independent set
[?]
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FPT | [+]Details | |||||
Maximum cut
[?]
(decision variant)
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XP,W-hard | [+]Details | |||||
Monopolarity
[?]
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Unknown to ISGCI | [+]Details | |||||
Polarity
[?]
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XP | [+]Details | |||||
Weighted clique
[?]
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FPT | [+]Details | |||||
Weighted feedback vertex set
[?]
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FPT | [+]Details | |||||
Weighted independent dominating set
[?]
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FPT | [+]Details | |||||
Weighted independent set
[?]
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FPT | [+]Details |