# Parameter: rankwidth

Definition:

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 $G$.

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## Relations

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.

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## Problems

Problems in italics have no summary page and are only listed when ISGCI contains a result for the current parameter.

3-Colourability FPT [+]Details
Clique FPT [+]Details
Clique cover XP [+]Details
Colourability FPT [+]Details
Domination FPT [+]Details
Feedback vertex set FPT [+]Details
Graph isomorphism XP [+]Details
Hamiltonian cycle XP [+]Details
Hamiltonian path XP [+]Details
Independent set FPT [+]Details
Maximum cut XP,W-hard [+]Details
Monopolarity Unknown to ISGCI [+]Details
Polarity XP [+]Details
Weighted clique FPT [+]Details
Weighted feedback vertex set FPT [+]Details
Weighted independent dominating set FPT [+]Details
Weighted independent set FPT [+]Details