branch width

functionally equivalent to: mm-width, strong inf-coloring number, treewidth

providers: ISGCI

Definition: Minimum over branch decompositions, maximum over decomposition edges, number of vertices that are incident to edges in both parts of the bi-partition.

Relations

Other		Relation from	Relation to
acyclic chromatic number	red■	exclusion	upper bound
admissibility	red■	exclusion	upper bound
arboricity	red■	exclusion	upper bound
average degree	red■	exclusion	upper bound
average distance	blue■	exclusion	exclusion
bandwidth	green■	upper bound	exclusion
bipartite	blue■	unbounded	exclusion
bipartite number	blue■	exclusion	exclusion
bisection bandwidth	blue■	exclusion	exclusion
block	blue■	unbounded	exclusion
book thickness	red■	exclusion	upper bound
boolean width	red■	exclusion	upper bound
bounded components	green■	upper bound	exclusion
bounded expansion	red■	exclusion	upper bound
boxicity	red■	exclusion	upper bound
branch width	yellow■	equal	equal
c-closure	blue■	exclusion	exclusion
carving-width	green■	upper bound	exclusion
chi-bounded	red■	exclusion	upper bound
chordal	blue■	unbounded	exclusion
chordality	red■	exclusion	upper bound
chromatic number	red■	exclusion	upper bound
clique cover number	blue■	exclusion	exclusion
clique-tree-width	red■	exclusion	upper bound
clique-width	red■	exclusion	upper bound
cluster	blue■	unbounded	exclusion
co-cluster	blue■	unbounded	exclusion
cograph	blue■	unbounded	exclusion
complete	blue■	unbounded	exclusion
connected	blue■	exclusion	avoids
contraction complexity	green■	upper bound	exclusion
cutwidth	green■	upper bound	exclusion
cycle	green■	upper bound	exclusion
cycles	green■	upper bound	exclusion
d-admissibility	red■	exclusion	upper bound
d-path-free	green■	upper bound	exclusion
degeneracy	red■	exclusion	upper bound
degree treewidth	green■	upper bound	exclusion
diameter	blue■	exclusion	exclusion
diameter+max degree	green■	upper bound	exclusion
distance to bipartite	blue■	exclusion	exclusion
distance to block	blue■	exclusion	exclusion
distance to bounded components	green■	upper bound	exclusion
distance to chordal	blue■	exclusion	exclusion
distance to cluster	blue■	exclusion	exclusion
distance to co-cluster	blue■	exclusion	exclusion
distance to cograph	blue■	exclusion	exclusion
distance to complete	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	red■	exclusion	upper bound
domination number	blue■	exclusion	exclusion
domino treewidth	green■	upper bound	exclusion
edge clique cover number	blue■	exclusion	exclusion
edge connectivity	red■	exclusion	upper bound
edge-cut width	green■	upper bound	exclusion
edge-treewidth	green■	upper bound	exclusion
edgeless	green■	upper bound	avoids
excluded minor	magenta■	exclusion	unknown to HOPS
excluded planar minor	lime■	upper bound	unknown to HOPS
excluded top-minor	red■	exclusion	upper bound
feedback edge set	green■	upper bound	exclusion
feedback vertex set	green■	upper bound	exclusion
flip-width	red■	exclusion	upper bound
forest	green■	upper bound	exclusion
genus	blue■	exclusion	exclusion
grid	blue■	unbounded	exclusion
h-index	blue■	exclusion	exclusion
interval	blue■	unbounded	exclusion
iterated type partitions	blue■	exclusion	exclusion
linear clique-width	magenta■	exclusion	unknown to HOPS
linear forest	green■	upper bound	exclusion
linear NLC-width	magenta■	exclusion	unknown to HOPS
linear rank-width	magenta■	exclusion	unknown to HOPS
maximum clique	red■	exclusion	upper bound
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
merge-width	red■	exclusion	upper bound
mim-width	red■	exclusion	upper bound
minimum degree	red■	exclusion	upper bound
mm-width	yellow■	upper bound	upper bound
modular-width	blue■	exclusion	exclusion
module-width	red■	exclusion	upper bound
monadically dependent	red■	exclusion	upper bound
monadically stable	red■	exclusion	upper bound
neighborhood diversity	blue■	exclusion	exclusion
NLC-width	red■	exclusion	upper bound
NLCT-width	red■	exclusion	upper bound
nowhere dense	red■	exclusion	upper bound
odd cycle transversal	blue■	exclusion	exclusion
outerplanar	green■	upper bound	exclusion
overlap treewidth	lime■	upper bound	unknown to HOPS
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-inf flip-width	red■	exclusion	upper bound
radius-r flip-width	red■	exclusion	upper bound
rank-width	red■	exclusion	upper bound
series-parallel	gray■	unknown to HOPS	unknown to HOPS
shrub-depth	magenta■	exclusion	unknown to HOPS
sim-width	red■	exclusion	upper bound
size	green■	upper bound	exclusion
slim tree-cut width	green■	upper bound	exclusion
sparse twin-width	red■	exclusion	upper bound
star	green■	upper bound	exclusion
stars	green■	upper bound	exclusion
strong coloring number	red■	exclusion	upper bound
strong d-coloring number	red■	exclusion	upper bound
strong inf-coloring number	yellow■	upper bound	upper bound
topological bandwidth	green■	upper bound	exclusion
tree	green■	upper bound	exclusion
tree-cut width	green■	upper bound	exclusion
tree-independence number	red■	exclusion	upper bound
tree-partition-width	green■	upper bound	exclusion
treebandwidth	lime■	upper bound	unknown to HOPS
treedepth	green■	upper bound	exclusion
treelength	magenta■	exclusion	unknown to HOPS
treespan	green■	upper bound	exclusion
treewidth	yellow■	upper bound	upper bound
twin-cover number	blue■	exclusion	exclusion
twin-width	red■	exclusion	upper bound
vertex connectivity	gray■	unknown to HOPS	unknown to HOPS
vertex cover	green■	upper bound	exclusion
vertex integrity	green■	upper bound	exclusion
weak coloring number	red■	exclusion	upper bound
weak d-coloring number	red■	exclusion	upper bound
weak inf-coloring number	green■	upper bound	exclusion
weakly sparse	red■	exclusion	upper bound
weakly sparse and merge width	red■	exclusion	upper bound

Results

1998 A partial $k$-arboretum of graphs with bounded treewidth by Bodlaender
- page 5 : branch width – A \emph{branch decomposition} of a graph $G=(V,E)$ is a pair $(T=(I,F),\sigma)$, where $T$ is a tree with every node in $T$ of degree one of three, and $\sigma$ is a bijection from $E$ to the set of leaves in $T$. The \emph{order} of an edge $f \in F$ is the number of vertices $v \in V$, for which there exist adjacent edges $(v,w),(v,x) \in E$, such that the path in $T$ from $\sigma(v,w)$ to $\sigma(v,x)$ uses $f$. The \emph{width} of branch decomposition $(T=(I,F),\sigma)$, is the maximum order over all edges $f \in F$. The \emph{branchwidth} of $G$ is the minimum width over all branch decompositions of $G$.
1991 Graph minors. X. Obstructions to tree-decomposition by Robertson, Seymour
- page 12 : branch width – A \emph{branch-width} of a hypergraph $G$ is a pair $(T,\tau)$, where $T$ is a ternary tree and $\tau$ is a bijection from the set of leaves of $T$ to $E(G)$. The \emph{order} of an edge $e$ of $T$ is the number of vertices $v$ of $G$ such that there are leaves $t_1,t_2$ of $T$ in different components of $T \setminus e$, with $\tau(t_1),\tau(t_2)$ both incident with $v$. The \emph{width} of $(T,\tau)$ is the maximum order of the edges of $T$, and the \emph{branch-width} $\beta(G)$ of $G$ is the minimum width of all branch-decompositions of $G$ (or 0 if $|E(G)| \le 1$, when $G$ has no branch-decompositions).
- page 16 : treewidth upper bounds branch width by a linear function – (5.1) For any hypergraph $G$, $\max(\beta(G), \gamma(G)) \le \omega(G) + 1 \le \max(\lfloor(3/2)\beta(G)\rfloor, \gamma(G), 1)$.
- page 16 : branch width upper bounds treewidth by a linear function – (5.1) For any hypergraph $G$, $\max(\beta(G), \gamma(G)) \le \omega(G) + 1 \le \max(\lfloor(3/2)\beta(G)\rfloor, \gamma(G), 1)$.
unknown source
- branch width upper bounds boolean width by a linear function
- branch width upper bounds rank-width by a linear function
assumed
- branch width is equivalent to branch width – assumed