maximum leaf number

providers: ISGCI

Definition: Largest number of tree leaves in any of graph’s spanning trees.

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	red■	exclusion	upper bound
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	magenta■	exclusion	unknown to HOPS
bounded expansion	red■	exclusion	upper bound
boxicity	red■	exclusion	upper bound
branch width	red■	exclusion	upper bound
c-closure	red■	exclusion	upper bound
carving-width	red■	exclusion	upper bound
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	red■	exclusion	upper bound
cutwidth	red■	exclusion	upper bound
cycle	green■	upper bound	exclusion
cycles	magenta■	unbounded	unknown to HOPS
d-admissibility	red■	exclusion	upper bound
d-path-free	magenta■	exclusion	unknown to HOPS
degeneracy	red■	exclusion	upper bound
degree treewidth	red■	exclusion	upper bound
diameter	magenta■	exclusion	unknown to HOPS
diameter+max degree	magenta■	exclusion	unknown to HOPS
distance to bipartite	red■	exclusion	upper bound
distance to block	red■	exclusion	upper bound
distance to bounded components	magenta■	exclusion	unknown to HOPS
distance to chordal	red■	exclusion	upper bound
distance to cluster	magenta■	exclusion	unknown to HOPS
distance to co-cluster	magenta■	exclusion	unknown to HOPS
distance to cograph	magenta■	exclusion	unknown to HOPS
distance to complete	magenta■	exclusion	unknown to HOPS
distance to edgeless	magenta■	exclusion	unknown to HOPS
distance to forest	red■	exclusion	upper bound
distance to interval	red■	exclusion	upper bound
distance to linear forest	red■	exclusion	upper bound
distance to maximum degree	red■	exclusion	upper bound
distance to outerplanar	red■	exclusion	upper bound
distance to perfect	red■	exclusion	upper bound
distance to planar	red■	exclusion	upper bound
distance to stars	magenta■	exclusion	unknown to HOPS
domatic number	red■	exclusion	upper bound
domination number	magenta■	exclusion	unknown to HOPS
domino treewidth	red■	exclusion	upper bound
edge clique cover number	magenta■	exclusion	unknown to HOPS
edge connectivity	red■	exclusion	upper bound
edge-cut width	red■	exclusion	upper bound
edge-treewidth	red■	exclusion	upper bound
edgeless	cyan■	unknown to HOPS	avoids
excluded minor	red■	exclusion	upper bound
excluded planar minor	gray■	unknown to HOPS	unknown to HOPS
excluded top-minor	red■	exclusion	upper bound
feedback edge set	red■	exclusion	upper bound
feedback vertex set	red■	exclusion	upper bound
flip-width	red■	exclusion	upper bound
forest	blue■	unbounded	exclusion
genus	red■	exclusion	upper bound
grid	blue■	unbounded	exclusion
h-index	red■	exclusion	upper bound
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	red■	exclusion	upper bound
maximum independent set	magenta■	exclusion	unknown to HOPS
maximum induced matching	magenta■	exclusion	unknown to HOPS
maximum leaf number	yellow■	equal	equal
maximum matching	magenta■	exclusion	unknown to HOPS
maximum matching on bipartite graphs	blue■	exclusion	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	red■	exclusion	upper bound
outerplanar	cyan■	unknown to HOPS	exclusion
overlap treewidth	red■	exclusion	upper bound
path	cyan■	unknown to HOPS	exclusion
pathwidth	red■	exclusion	upper bound
pathwidth+maxdegree	red■	exclusion	upper bound
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	lime■	upper bound	unknown to HOPS
slim tree-cut width	red■	exclusion	upper bound
sparse twin-width	red■	exclusion	upper bound
star	blue■	unbounded	exclusion
stars	blue■	unbounded	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	red■	exclusion	upper bound
tree	blue■	unbounded	exclusion
tree-cut width	red■	exclusion	upper bound
tree-independence number	red■	exclusion	upper bound
tree-partition-width	red■	exclusion	upper bound
treebandwidth	red■	exclusion	upper bound
treedepth	magenta■	exclusion	unknown to HOPS
treelength	magenta■	exclusion	unknown to HOPS
treespan	red■	exclusion	upper bound
treewidth	red■	exclusion	upper bound
twin-cover number	magenta■	exclusion	unknown to HOPS
twin-width	red■	exclusion	upper bound
vertex connectivity	magenta■	exclusion	unknown to HOPS
vertex cover	magenta■	exclusion	unknown to HOPS
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

2019 The Graph Parameter Hierarchy by Sorge
- page 8 : maximum leaf number upper bounds distance to linear forest by a computable function – Lemma 4.10 ([14]). The max-leaf number $\mathrm{ml}$ upper bounds the distance to disjoint paths $d$. We have $d \le \mathrm{ml}-1$.
- page 10 : maximum leaf number upper bounds bandwidth by a linear function – Lemma 4.25. The max leaf number $\mathrm{ml}$ strictly upper bounds the bandwidth $\mathrm{bw}$.
unknown source
- maximum leaf number upper bounds distance to linear forest by a computable function
- maximum leaf number upper bounds feedback edge set by a polynomial function
- cycle upper bounds maximum leaf number by a constant
- maximum leaf number upper bounds feedback edge set by a polynomial function – M. Bentert (personal communication)
assumed
- size upper bounds maximum leaf number by a linear function – By definition
- maximum leaf number is equivalent to maximum leaf number – assumed