boolean width

equivalent to: module-width, clique-width, rank-width, NLC-width, inf-flip-width

providers: ISGCI

Relations

Other		Relation from	Relation to
acyclic chromatic number	blue■	exclusion	exclusion
arboricity	blue■	exclusion	exclusion
average degree	blue■	exclusion	exclusion
average distance	blue■	exclusion	exclusion
bandwidth	green■	upper bound	exclusion
bipartite	blue■	unbounded	exclusion
bipartite number	magenta■	exclusion	unknown to HOPS
bisection bandwidth	blue■	exclusion	exclusion
block	cyan■	unknown to HOPS	exclusion
book thickness	blue■	exclusion	exclusion
boolean width	yellow■	equal	equal
bounded components	green■	upper bound	exclusion
boxicity	blue■	exclusion	exclusion
branch width	green■	upper bound	exclusion
c-closure	blue■	exclusion	exclusion
carving-width	green■	upper bound	exclusion
chordal	cyan■	unknown to HOPS	exclusion
chordality	blue■	exclusion	exclusion
chromatic number	blue■	exclusion	exclusion
clique cover number	blue■	exclusion	exclusion
clique-tree-width	lime■	upper bound	unknown to HOPS
clique-width	yellow■	upper bound	upper bound
cluster	green■	upper bound	exclusion
co-cluster	green■	upper bound	exclusion
cograph	green■	upper bound	exclusion
complete	green■	upper bound	exclusion
connected	blue■	unbounded	exclusion
contraction complexity	green■	upper bound	exclusion
cutwidth	green■	upper bound	exclusion
cycle	green■	upper bound	exclusion
cycles	green■	upper bound	exclusion
d-path-free	green■	upper bound	exclusion
degeneracy	blue■	exclusion	exclusion
degree treewidth	green■	upper bound	exclusion
diameter	blue■	exclusion	exclusion
diameter+max degree	green■	upper bound	exclusion
disconnected	cyan■	unknown to HOPS	exclusion
disjoint cycles	green■	upper bound	exclusion
distance to bipartite	blue■	exclusion	exclusion
distance to block	cyan■	unknown to HOPS	exclusion
distance to bounded components	green■	upper bound	exclusion
distance to chordal	blue■	exclusion	exclusion
distance to cluster	green■	upper bound	exclusion
distance to co-cluster	green■	upper bound	exclusion
distance to cograph	green■	upper bound	exclusion
distance to complete	green■	upper bound	exclusion
distance to disconnected	blue■	exclusion	exclusion
distance to edgeless	green■	upper bound	exclusion
distance to forest	green■	upper bound	exclusion
distance to interval	blue■	exclusion	exclusion
distance to linear forest	green■	upper bound	exclusion
distance to maximum degree	blue■	exclusion	exclusion
distance to outerplanar	green■	upper bound	exclusion
distance to perfect	blue■	exclusion	exclusion
distance to planar	blue■	exclusion	exclusion
distance to stars	green■	upper bound	exclusion
domatic number	blue■	exclusion	exclusion
domination number	blue■	exclusion	exclusion
edge clique cover number	green■	upper bound	exclusion
edge connectivity	blue■	exclusion	exclusion
edgeless	green■	upper bound	exclusion
feedback edge set	green■	upper bound	exclusion
feedback vertex set	green■	upper bound	exclusion
forest	green■	upper bound	exclusion
genus	blue■	exclusion	exclusion
girth	blue■	exclusion	exclusion
grid	blue■	unbounded	exclusion
h-index	blue■	exclusion	exclusion
inf-flip-width	yellow■	upper bound	upper bound
interval	cyan■	unknown to HOPS	exclusion
iterated type partitions	green■	upper bound	exclusion
linear clique-width	lime■	upper bound	unknown to HOPS
linear forest	green■	upper bound	exclusion
linear NLC-width	lime■	upper bound	unknown to HOPS
linear rank-width	lime■	upper bound	unknown to HOPS
maximum clique	blue■	exclusion	exclusion
maximum degree	blue■	exclusion	exclusion
maximum independent set	blue■	exclusion	exclusion
maximum induced matching	blue■	exclusion	exclusion
maximum leaf number	green■	upper bound	exclusion
maximum matching	green■	upper bound	exclusion
maximum matching on bipartite graphs	green■	upper bound	exclusion
mim-width	orange■	unknown to HOPS	upper bound
minimum degree	blue■	exclusion	exclusion
mm-width	green■	upper bound	exclusion
modular-width	green■	upper bound	exclusion
module-width	yellow■	upper bound	upper bound
neighborhood diversity	green■	upper bound	exclusion
NLC-width	yellow■	upper bound	upper bound
NLCT-width	lime■	upper bound	unknown to HOPS
odd cycle transversal	blue■	exclusion	exclusion
outerplanar	green■	upper bound	exclusion
path	green■	upper bound	exclusion
pathwidth	green■	upper bound	exclusion
pathwidth+maxdegree	green■	upper bound	exclusion
perfect	blue■	unbounded	exclusion
planar	blue■	unbounded	exclusion
radius-r flip-width	red■	exclusion	upper bound
rank-width	yellow■	upper bound	upper bound
shrub-depth	green■	upper bound	exclusion
sim-width	orange■	unknown to HOPS	upper bound
size	green■	upper bound	exclusion
star	green■	upper bound	exclusion
stars	green■	upper bound	exclusion
topological bandwidth	green■	upper bound	exclusion
tree	green■	upper bound	exclusion
tree-independence number	gray■	unknown to HOPS	unknown to HOPS
treedepth	green■	upper bound	exclusion
treelength	magenta■	exclusion	unknown to HOPS
treewidth	green■	upper bound	exclusion
twin-cover number	green■	upper bound	exclusion
twin-width	red■	exclusion	upper bound
vertex connectivity	blue■	exclusion	exclusion
vertex cover	green■	upper bound	exclusion
vertex integrity	green■	upper bound	exclusion

Results

2021 Twin-width I: Tractable FO Model Checking by Bonnet, Kim, Thomassé, Watrigant
- page 14 : boolean width upper bounds twin-width by an exponential function – Theorem 3: Every graph with boolean-width $k$ has twin-width at most $2^{k+1}-1$.
2012 New Width Parameters of Graphs by Vatshelle
- page 42 : boolean width upper bounds mim-width by a linear function – Theorem 4.2.10. Let $G$ be a graph, then $mimw(G) \le boolw(G) \le mimw(G) \log_2(n)$
2011 Boolean-width of graphs by Bui-Xuan, Telle, Vatshelle
- boolean width – \textbf{Definition 1.} A decomposition tree of a graph $G$ is a pair $(T,\delta)$ where $T$ is a tree having internal nodes of degree three and $\delta$ a bijection between the leaf set of $T$ and the vertex set of $G$. Removing an edge from $T$ results in two subtrees, and in a cut ${A,\overline{A}}$ of $G$ given by the two subsets of $V(G)$ in bijection $\delta$ with the leaves of the two subtrees. Let $f\colon w^V \to \mathbb{R}$ be a symmetric function that is also called a cut function: $f(A)=f(\overline{A})$ for all $A\subseteq V(G)$. The $f$-width of $(T,\delta)$ is the maximum value of $f(A)$ over all cuts ${A,\overline{A}}$ of $G$ given by the removal of an edge of $T$. … \textbf{Definition 2.} Let $G$ be a graph and $A \subseteq V(G)$. Define the set of unions of neighborhoods of $A$ across the cut ${A,\overline{A}}$ as $U(A) = {Y \subseteq \overline{A} \mid \exists X \subseteq A \land Y=N(X)\cap \overline{A}}$. The \emph{bool-dim}$\colon 2^{V(G)} \to \mathbb{R}$ function of a graph $G$ is defined as $\mathrm{bool-dim}(A)=\log_2 |U(A)|$. Using Definition 1 with $f=\mathrm{bool-dim}$ we define the boolean-width of a decomposition tree, denoted by $boolw(T,\delta)$, and the boolean-width of a graph, denoted by $boolw(G)$.
- boolean width upper bounds rank-width by an exponential function – \textbf{Corollary 1.} For any graph $G$ and decomposition tree $(T,\gamma)$ of $G$ it holds that … $\log_2 rw(G) \le boolw(G)$ …
- rank-width upper bounds boolean width by a polynomial function – \textbf{Corollary 1.} For any graph $G$ and decomposition tree $(T,\gamma)$ of $G$ it holds that … $boolw(G) \le \frac 14 rw^2(G)+O(rw(G))$.
assumed
- boolean width is equivalent to boolean width – assumed
unknown source
- clique-width upper bounds boolean width by a linear function
- boolean width upper bounds clique-width by an exponential function
- branch width upper bounds boolean width by a linear function
- treewidth upper bounds boolean width by a linear function