shrub-depth
equivalent to: shrub-depth
Relations
Other | Relation from | Relation to |
---|---|---|
acyclic chromatic number | exclusion | exclusion |
arboricity | exclusion | exclusion |
average degree | exclusion | exclusion |
average distance | exclusion | unknown to HOPS |
bandwidth | unknown to HOPS | exclusion |
bipartite | unbounded | exclusion |
bipartite number | exclusion | unknown to HOPS |
bisection bandwidth | exclusion | exclusion |
block | unknown to HOPS | exclusion |
book thickness | exclusion | exclusion |
boolean width | exclusion | upper bound |
bounded components | upper bound | exclusion |
boxicity | exclusion | unknown to HOPS |
branch width | unknown to HOPS | exclusion |
c-closure | exclusion | exclusion |
carving-width | unknown to HOPS | exclusion |
chordal | unknown to HOPS | exclusion |
chordality | exclusion | unknown to HOPS |
chromatic number | exclusion | exclusion |
clique cover number | exclusion | exclusion |
clique-tree-width | unknown to HOPS | upper bound |
clique-width | exclusion | upper bound |
cluster | constant | exclusion |
co-cluster | unknown to HOPS | exclusion |
cograph | unknown to HOPS | exclusion |
complete | constant | exclusion |
connected | unbounded | unknown to HOPS |
cutwidth | unknown to HOPS | exclusion |
cycle | unknown to HOPS | exclusion |
cycles | unknown to HOPS | exclusion |
d-path-free | upper bound | exclusion |
degeneracy | exclusion | exclusion |
degree treewidth | unknown to HOPS | exclusion |
diameter | exclusion | unknown to HOPS |
diameter+max degree | upper bound | exclusion |
disjoint cycles | unknown to HOPS | exclusion |
distance to bipartite | exclusion | exclusion |
distance to block | unknown to HOPS | exclusion |
distance to bounded components | upper bound | exclusion |
distance to chordal | exclusion | exclusion |
distance to cluster | unknown to HOPS | exclusion |
distance to co-cluster | unknown to HOPS | exclusion |
distance to cograph | unknown to HOPS | exclusion |
distance to complete | upper bound | exclusion |
distance to edgeless | upper bound | exclusion |
distance to forest | unknown to HOPS | exclusion |
distance to interval | exclusion | exclusion |
distance to linear forest | unknown to HOPS | exclusion |
distance to maximum degree | exclusion | exclusion |
distance to outerplanar | unknown to HOPS | exclusion |
distance to perfect | exclusion | exclusion |
distance to planar | exclusion | exclusion |
distance to stars | upper bound | exclusion |
domatic number | exclusion | exclusion |
domination number | exclusion | exclusion |
edge clique cover number | upper bound | exclusion |
edge connectivity | exclusion | exclusion |
edgeless | constant | exclusion |
feedback edge set | unknown to HOPS | exclusion |
feedback vertex set | unknown to HOPS | exclusion |
forest | unknown to HOPS | exclusion |
genus | exclusion | exclusion |
girth | exclusion | unknown to HOPS |
grid | unbounded | exclusion |
h-index | exclusion | exclusion |
inf-flip-width | exclusion | upper bound |
interval | unknown to HOPS | exclusion |
iterated type partitions | unknown to HOPS | exclusion |
linear clique-width | unknown to HOPS | upper bound |
linear forest | unknown to HOPS | exclusion |
linear NLC-width | unknown to HOPS | upper bound |
linear rank-width | unknown to HOPS | upper bound |
maximum clique | exclusion | exclusion |
maximum degree | exclusion | exclusion |
maximum independent set | exclusion | exclusion |
maximum induced matching | exclusion | unknown to HOPS |
maximum leaf number | unknown to HOPS | exclusion |
maximum matching | unknown to HOPS | exclusion |
maximum matching on bipartite graphs | upper bound | exclusion |
mim-width | exclusion | upper bound |
minimum degree | exclusion | exclusion |
mm-width | unknown to HOPS | exclusion |
modular-width | exclusion | exclusion |
module-width | exclusion | upper bound |
neighborhood diversity | upper bound | exclusion |
NLC-width | exclusion | upper bound |
NLCT-width | unknown to HOPS | upper bound |
odd cycle transversal | exclusion | exclusion |
outerplanar | unknown to HOPS | exclusion |
path | unknown to HOPS | exclusion |
pathwidth | unknown to HOPS | exclusion |
pathwidth+maxdegree | unknown to HOPS | exclusion |
perfect | unbounded | exclusion |
planar | unbounded | exclusion |
radius-r flip-width | exclusion | upper bound |
rank-width | exclusion | upper bound |
sim-width | exclusion | upper bound |
star | constant | exclusion |
stars | constant | exclusion |
topological bandwidth | unknown to HOPS | exclusion |
tree | unknown to HOPS | exclusion |
tree-independence number | unknown to HOPS | unknown to HOPS |
treedepth | upper bound | exclusion |
treelength | exclusion | unknown to HOPS |
treewidth | unknown to HOPS | exclusion |
twin-cover number | upper bound | exclusion |
twin-width | exclusion | upper bound |
vertex connectivity | unknown to HOPS | exclusion |
vertex cover | upper bound | exclusion |
vertex integrity | upper bound | exclusion |
Results
- 2019 Shrub-depth: Capturing Height of Dense Graphs by Ganian, Hliněný, Nešetřil, Obdržálek, Mendez
- shrub-depth upper bounds linear clique-width by a linear function – Proposition 3.4. Let $\mathcal G$ be a graph class and $d$ an integer. Then: … b) If $\mathcal G$ is of bounded shrub-depth, then $\mathcal G$ is of bounded linear clique-width.
- neighborhood diversity upper bounds shrub-depth by a constant – $\mathcal{TM}_m(1)$ is exactly the class of graphs of neighborhood diversity at most $m$.
- treedepth upper bounds shrub-depth by a linear function – Proposition 3.2. If $G$ is of tree-depth $d$, then $G \in \mathcal{TM}_{2^d}(d)$. …
- 2015 Improving Vertex Cover as a Graph Parameter by Ganian
- page 20 : twin-cover number upper bounds shrub-depth by a constant – Let $\mathcal H_k$ be the class of graphs of twin-cover $k$. Then $\mathcal H_k \subseteq \mathcal{TM}_{2^k+k}(2)$ and a tree-model of any $G \in \mathcal H_k$ may be constructed in single-exponential FPT time.
- 2013 Parameterized Algorithms for Modular-Width by Gajarský, Lampis, Ordyniak
- page 8 : bounded modular-width does not imply bounded shrub-depth – Theorem 4. There are classes of graphs with unbounded modular-width and bounded shrub-depth and vice versa.
- page 8 : bounded shrub-depth does not imply bounded modular-width – Theorem 4. There are classes of graphs with unbounded modular-width and bounded shrub-depth and vice versa.
- https://www.fi.muni.cz/~hlineny/papers/shrubdepth-warw18-slipp.pdf
- page 7 : shrub-depth – Tree-model of $m$ colors and depth $d$: a rooted tree $T$ of height $d$, leaves are the vertices of $G$, each leaf has one of $m$ colors, an associated signature determining the edge set of $G$ as follows: for $i=1,2,\dots,d$, let $u$ and $v$ be leaves with the least common ancestor at height $i$ in $T$, then $uv \in E(G)$ iff the color pair of $u,v$ is in the signature at height $i$.
- page 9 : neighborhood diversity upper bounds shrub-depth by a constant – Shrub-depth 1: e.g., cliques, stars, \dots, gen. BND – bounded neighborhood diversity.