topological bandwidth

Definition: Minimum over grpah subdivisions, bandwidth.

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	magenta■	exclusion	unknown to HOPS
bandwidth	lime■	upper bound	unknown to HOPS
bipartite	blue■	unbounded	exclusion
bipartite number	magenta■	exclusion	unknown to HOPS
bisection bandwidth	red■	exclusion	upper bound
block	blue■	unbounded	exclusion
book thickness	red■	exclusion	upper bound
boolean width	red■	exclusion	upper bound
bounded components	gray■	unknown to HOPS	unknown to HOPS
bounded expansion	red■	exclusion	upper bound
boxicity	red■	exclusion	upper bound
branch width	red■	exclusion	upper bound
c-closure	magenta■	exclusion	unknown to HOPS
carving-width	gray■	unknown to HOPS	unknown to HOPS
chi-bounded	red■	exclusion	upper bound
chordal	blue■	unbounded	exclusion
chordality	red■	exclusion	upper bound
chromatic number	red■	exclusion	upper bound
clique cover number	magenta■	exclusion	unknown to HOPS
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	gray■	unknown to HOPS	unknown to HOPS
cutwidth	gray■	unknown to HOPS	unknown to HOPS
cycle	green■	upper bound	exclusion
cycles	gray■	unknown to HOPS	unknown to HOPS
d-admissibility	red■	exclusion	upper bound
d-path-free	magenta■	exclusion	unknown to HOPS
degeneracy	red■	exclusion	upper bound
degree treewidth	gray■	unknown to HOPS	unknown to HOPS
diameter	magenta■	exclusion	unknown to HOPS
diameter+max degree	gray■	unknown to HOPS	unknown to HOPS
distance to bipartite	blue■	exclusion	exclusion
distance to block	blue■	exclusion	exclusion
distance to bounded components	magenta■	exclusion	unknown to HOPS
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	blue■	exclusion	exclusion
distance to forest	blue■	exclusion	exclusion
distance to interval	blue■	exclusion	exclusion
distance to linear forest	blue■	exclusion	exclusion
distance to maximum degree	magenta■	exclusion	unknown to HOPS
distance to outerplanar	blue■	exclusion	exclusion
distance to perfect	blue■	exclusion	exclusion
distance to planar	blue■	exclusion	exclusion
distance to stars	blue■	exclusion	exclusion
domatic number	red■	exclusion	upper bound
domination number	magenta■	exclusion	unknown to HOPS
domino treewidth	gray■	unknown to HOPS	unknown to HOPS
edge clique cover number	magenta■	exclusion	unknown to HOPS
edge connectivity	red■	exclusion	upper bound
edge-cut width	magenta■	exclusion	unknown to HOPS
edge-treewidth	magenta■	exclusion	unknown to HOPS
edgeless	cyan■	unknown to HOPS	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	blue■	exclusion	exclusion
feedback vertex set	blue■	exclusion	exclusion
flip-width	red■	exclusion	upper bound
forest	blue■	unbounded	exclusion
genus	blue■	exclusion	exclusion
grid	blue■	unbounded	exclusion
h-index	magenta■	exclusion	unknown to HOPS
interval	blue■	unbounded	exclusion
iterated type partitions	magenta■	exclusion	unknown to HOPS
linear clique-width	red■	exclusion	upper bound
linear forest	cyan■	unknown to HOPS	exclusion
linear NLC-width	red■	exclusion	upper bound
linear rank-width	red■	exclusion	upper bound
maximum clique	red■	exclusion	upper bound
maximum degree	magenta■	exclusion	unknown to HOPS
maximum independent set	magenta■	exclusion	unknown to HOPS
maximum induced matching	magenta■	exclusion	unknown to HOPS
maximum leaf number	green■	upper bound	exclusion
maximum matching	blue■	exclusion	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	red■	exclusion	upper bound
modular-width	magenta■	exclusion	unknown to HOPS
module-width	red■	exclusion	upper bound
monadically dependent	red■	exclusion	upper bound
monadically stable	red■	exclusion	upper bound
neighborhood diversity	magenta■	exclusion	unknown to HOPS
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	magenta■	exclusion	unknown to HOPS
path	cyan■	unknown to HOPS	exclusion
pathwidth	red■	exclusion	upper bound
pathwidth+maxdegree	gray■	unknown to HOPS	unknown to HOPS
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	magenta■	exclusion	unknown to HOPS
sparse twin-width	red■	exclusion	upper bound
star	cyan■	unknown to HOPS	exclusion
stars	cyan■	unknown to HOPS	exclusion
strong coloring number	red■	exclusion	upper bound
strong d-coloring number	red■	exclusion	upper bound
strong inf-coloring number	red■	exclusion	upper bound
topological bandwidth	yellow■	equal	equal
tree	blue■	unbounded	exclusion
tree-cut width	magenta■	exclusion	unknown to HOPS
tree-independence number	red■	exclusion	upper bound
tree-partition-width	magenta■	exclusion	unknown to HOPS
treebandwidth	magenta■	exclusion	unknown to HOPS
treedepth	magenta■	exclusion	unknown to HOPS
treelength	magenta■	exclusion	unknown to HOPS
treespan	gray■	unknown to HOPS	unknown to HOPS
treewidth	red■	exclusion	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	blue■	exclusion	exclusion
vertex integrity	magenta■	exclusion	unknown to HOPS
weak coloring number	red■	exclusion	upper bound
weak d-coloring number	red■	exclusion	upper bound
weak inf-coloring number	magenta■	exclusion	unknown to HOPS
weakly sparse	red■	exclusion	upper bound
weakly sparse and merge width	red■	exclusion	upper bound

Results

2013 The Power of Data Reduction: Kernels for Fundamental Graph Problems by Jansen
- topological bandwidth – The \emph{topological bandwidth} of a graph $G$ is the minimum bandwidth over all subdivisions of $G$
1998 A partial $k$-arboretum of graphs with bounded treewidth by Bodlaender
- page 22 : topological bandwidth – The \emph{topological bandwidth} of a graph $G$ is the minimum bandwidth over all graphs $G’$ which are obtained by addition of an arbitrary number of vertices along edges of $G$.
- page 23 : topological bandwidth upper bounds pathwidth by a linear function – Theorem 45. For every graph $G$, the pathwidth of $G$ is at most the topological band-width of $G$.
assumed
- bandwidth upper bounds topological bandwidth by a linear function – By definition
- topological bandwidth is equivalent to topological bandwidth – assumed
unknown source
- topological bandwidth upper bounds bisection bandwidth by a linear function – Order vertices by their bandwidth integer. We split the graph in the middle of this ordering. There are at most roughly $k^2/2$ edges over this split.