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

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 $\mathrm{bw}$.
assumed
- size upper bounds maximum leaf number by a linear function – By definition
- maximum leaf number is equivalent to maximum leaf number – assumed
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)
Comparing Graph Parameters by Schröder
- page 21 : bounded maximum leaf number does not imply bounded girth – Proposition 3.14