treebandwidth

Definition: A \emph{tree-layout} of $G=(V,E)$ is a rooted tree $T$ whose nodes are the vertices of $V$, and such that, for every edge $xy \in E$, $x$ is an ancestor of $y$ or vice-versa. The bandwidth of $T$ is then the maximum distance in $T$ between pairs of neighbors in $G$. We call \emph{treebandwidth} of $G$, the minimum bandwidth over tree-layouts of $G$, and denote it by ${\rm tbw}(G)$.

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	orange■	unknown to HOPS	upper bound
c-closure	magenta■	exclusion	unknown to HOPS
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	cyan■	unknown to HOPS	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	cyan■	unknown to HOPS	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	cyan■	unknown to HOPS	exclusion
distance to forest	cyan■	unknown to HOPS	exclusion
distance to interval	blue■	exclusion	exclusion
distance to linear forest	cyan■	unknown to HOPS	exclusion
distance to maximum degree	blue■	exclusion	exclusion
distance to outerplanar	cyan■	unknown to HOPS	exclusion
distance to perfect	blue■	exclusion	exclusion
distance to planar	blue■	exclusion	exclusion
distance to stars	cyan■	unknown to HOPS	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	gray■	unknown to HOPS	unknown to HOPS
excluded top-minor	red■	exclusion	upper bound
feedback edge set	green■	upper bound	exclusion
feedback vertex set	cyan■	unknown to HOPS	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	cyan■	unknown to HOPS	exclusion
maximum matching on bipartite graphs	cyan■	unknown to HOPS	exclusion
merge-width	red■	exclusion	upper bound
mim-width	red■	exclusion	upper bound
minimum degree	red■	exclusion	upper bound
mm-width	orange■	unknown to HOPS	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	cyan■	unknown to HOPS	exclusion
overlap treewidth	gray■	unknown to HOPS	unknown to HOPS
path	green■	upper bound	exclusion
pathwidth	cyan■	unknown to HOPS	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	orange■	unknown to HOPS	upper bound
topological bandwidth	cyan■	unknown to HOPS	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	yellow■	equal	equal
treedepth	cyan■	unknown to HOPS	exclusion
treelength	magenta■	exclusion	unknown to HOPS
treespan	green■	upper bound	exclusion
treewidth	orange■	unknown to HOPS	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	cyan■	unknown to HOPS	exclusion
vertex integrity	cyan■	unknown to HOPS	exclusion
weak coloring number	red■	exclusion	upper bound
weak d-coloring number	red■	exclusion	upper bound
weak inf-coloring number	cyan■	unknown to HOPS	exclusion
weakly sparse	red■	exclusion	upper bound
weakly sparse and merge width	red■	exclusion	upper bound

Results

2025 On a tree-based variant of bandwidth and forbidding simple topological minors by Jacob, Lochet, Paul
- page 1 : treebandwidth – A \emph{tree-layout} of $G=(V,E)$ is a rooted tree $T$ whose nodes are the vertices of $V$, and such that, for every edge $xy \in E$, $x$ is an ancestor of $y$ or vice-versa. The bandwidth of $T$ is then the maximum distance in $T$ between pairs of neighbors in $G$. We call \emph{treebandwidth} of $G$, the minimum bandwidth over tree-layouts of $G$, and denote it by ${\rm tbw}(G)$.
- page 7 : tree-partition-width upper bounds treebandwidth by a linear function – By rooting the tree-partition arbitrarily and replacing each bag by an arbitrary linear ordering of its vertices one derives ${\rm tbw}(G) \le 2 \cdot {\rm tpw}(G)$. However, some graphs of treebandwidth 2 have unbounded tree-partition-width: …
- page 7 : graph classes with bounded treebandwidth are not bounded tree-partition-width – By rooting the tree-partition arbitrarily and replacing each bag by an arbitrary linear ordering of its vertices one derives ${\rm tbw}(G) \le 2 \cdot {\rm tpw}(G)$. However, some graphs of treebandwidth 2 have unbounded tree-partition-width: …
- page 38 : treebandwidth upper bounds treewidth by a computable function – [implied through obstructions]
assumed
- treebandwidth is equivalent to treebandwidth – assumed