# Parameter: booleanwidth

Definition:

Consider the following decomposition of a graph $G$ which is defined as a pair $(T,L)$ where $T$ is a binary tree and $L$ is a bijection from $V(G)$ to the leaves of the tree $T$. The function $\text{cut-bool} \colon 2^{V(G)} \rightarrow R$ is defined as $\text{cut-bool}(A)$ := $\log_2|\{S \subseteq V(G) \backslash A \mid \exists X \subseteq A \colon S = (V(G) \backslash A) \cap \bigcup_{x \in X} N(x)\}|$. Every edge $e$ in $T$ partitions the vertices $V(G)$ into $\{A_e,\overline{A_e}\}$ according to the leaves of the two connected components of $T - e$. The booleanwidth of the above decomposition $(T,L)$ is $\max_{e \in E(T)\;} \{ \text{cut-bool}(A_e)\}$. The booleanwidth of a graph $G$ is the minimum booleanwidth of a decomposition of $G$ as above.

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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