modular-width

tags: module

Definition: Modular-width is the minimum number $k$ such that there is a partition into at most $k$ modules where each module either contains a single vertex or the graph induced by the module has modular-width $k$.

Relations

Other		Relation from	Relation to
acyclic chromatic number	blue■	exclusion	exclusion
admissibility	blue■	exclusion	exclusion
arboricity	blue■	exclusion	exclusion
average degree	blue■	exclusion	exclusion
average distance	red■	exclusion	upper bound
bandwidth	cyan■	unknown to HOPS	exclusion
bipartite	blue■	unbounded	exclusion
bipartite number	blue■	exclusion	exclusion
bisection bandwidth	blue■	exclusion	exclusion
block	blue■	unbounded	exclusion
book thickness	blue■	exclusion	exclusion
boolean width	red■	exclusion	upper bound
bounded components	cyan■	unknown to HOPS	exclusion
bounded expansion	blue■	exclusion	avoids
boxicity	blue■	exclusion	exclusion
branch width	blue■	exclusion	exclusion
c-closure	blue■	exclusion	exclusion
carving-width	blue■	exclusion	exclusion
chi-bounded	red■	exclusion	upper bound
chordal	blue■	unbounded	exclusion
chordality	blue■	exclusion	exclusion
chromatic number	blue■	exclusion	exclusion
clique cover number	blue■	exclusion	exclusion
clique-tree-width	magenta■	exclusion	unknown to HOPS
clique-width	red■	exclusion	upper bound
cluster	green■	upper bound	exclusion
co-cluster	cyan■	unknown to HOPS	exclusion
cograph	cyan■	unknown to HOPS	exclusion
complete	green■	upper bound	exclusion
connected	blue■	exclusion	avoids
contraction complexity	blue■	exclusion	exclusion
cutwidth	blue■	exclusion	exclusion
cycle	cyan■	unknown to HOPS	exclusion
cycles	cyan■	unknown to HOPS	exclusion
d-admissibility	magenta■	exclusion	unknown to HOPS
d-path-free	cyan■	unknown to HOPS	exclusion
degeneracy	blue■	exclusion	exclusion
degree treewidth	blue■	exclusion	exclusion
diameter	red■	exclusion	upper bound
diameter+max degree	cyan■	unknown to HOPS	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	green■	upper bound	exclusion
distance to edgeless	green■	upper bound	exclusion
distance to forest	blue■	exclusion	exclusion
distance to interval	blue■	exclusion	exclusion
distance to linear forest	blue■	exclusion	exclusion
distance to maximum degree	blue■	exclusion	exclusion
distance to outerplanar	blue■	exclusion	exclusion
distance to perfect	blue■	exclusion	exclusion
distance to planar	blue■	exclusion	exclusion
distance to stars	cyan■	unknown to HOPS	exclusion
domatic number	blue■	exclusion	exclusion
domination number	blue■	exclusion	exclusion
domino treewidth	blue■	exclusion	exclusion
edge clique cover number	green■	upper bound	exclusion
edge connectivity	blue■	exclusion	exclusion
edge-cut width	blue■	exclusion	exclusion
edge-treewidth	blue■	exclusion	exclusion
edgeless	green■	upper bound	avoids
excluded minor	blue■	exclusion	avoids
excluded planar minor	cyan■	unknown to HOPS	avoids
excluded top-minor	blue■	exclusion	avoids
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	blue■	exclusion	exclusion
interval	blue■	unbounded	exclusion
iterated type partitions	lime■	upper bound	unknown to HOPS
linear clique-width	magenta■	exclusion	unknown to HOPS
linear forest	blue■	unbounded	exclusion
linear NLC-width	magenta■	exclusion	unknown to HOPS
linear rank-width	magenta■	exclusion	unknown to HOPS
maximum clique	blue■	exclusion	exclusion
maximum degree	blue■	exclusion	exclusion
maximum independent set	blue■	exclusion	exclusion
maximum induced matching	magenta■	exclusion	unknown to HOPS
maximum leaf number	cyan■	unknown to HOPS	exclusion
maximum matching	green■	upper bound	exclusion
maximum matching on bipartite graphs	green■	upper bound	exclusion
merge-width	red■	exclusion	upper bound
mim-width	red■	exclusion	upper bound
minimum degree	blue■	exclusion	exclusion
mm-width	blue■	exclusion	exclusion
modular-width	yellow■	equal	equal
module-width	red■	exclusion	upper bound
monadically dependent	red■	exclusion	upper bound
monadically stable	magenta■	exclusion	unknown to HOPS
neighborhood diversity	green■	upper bound	exclusion
NLC-width	red■	exclusion	upper bound
NLCT-width	magenta■	exclusion	unknown to HOPS
nowhere dense	magenta■	exclusion	unknown to HOPS
odd cycle transversal	blue■	exclusion	exclusion
outerplanar	cyan■	unknown to HOPS	exclusion
overlap treewidth	blue■	exclusion	exclusion
path	blue■	unbounded	exclusion
pathwidth	blue■	exclusion	exclusion
pathwidth+maxdegree	blue■	exclusion	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	blue■	exclusion	exclusion
sim-width	red■	exclusion	upper bound
size	green■	upper bound	exclusion
slim tree-cut width	blue■	exclusion	exclusion
sparse twin-width	blue■	exclusion	exclusion
star	green■	upper bound	exclusion
stars	cyan■	unknown to HOPS	exclusion
strong coloring number	blue■	exclusion	exclusion
strong d-coloring number	magenta■	exclusion	unknown to HOPS
strong inf-coloring number	blue■	exclusion	exclusion
topological bandwidth	cyan■	unknown to HOPS	exclusion
tree	blue■	unbounded	exclusion
tree-cut width	blue■	exclusion	exclusion
tree-independence number	magenta■	exclusion	unknown to HOPS
tree-partition-width	blue■	exclusion	exclusion
treebandwidth	blue■	exclusion	exclusion
treedepth	cyan■	unknown to HOPS	exclusion
treelength	red■	exclusion	upper bound
treespan	blue■	exclusion	exclusion
treewidth	blue■	exclusion	exclusion
twin-cover number	green■	upper bound	exclusion
twin-width	red■	exclusion	upper bound
vertex connectivity	gray■	unknown to HOPS	unknown to HOPS
vertex cover	green■	upper bound	exclusion
vertex integrity	cyan■	unknown to HOPS	exclusion
weak coloring number	blue■	exclusion	exclusion
weak d-coloring number	magenta■	exclusion	unknown to HOPS
weak inf-coloring number	cyan■	unknown to HOPS	exclusion
weakly sparse	magenta■	exclusion	unknown to HOPS
weakly sparse and merge width	blue■	exclusion	exclusion

Results

2024 Parameterized complexity for iterated type partitions and modular-width by Cordasco, Gargano, Rescigno
- page 3 : iterated type partitions upper bounds modular-width by a linear function – By definition
2022 Expanding the Graph Parameter Hierarchy by Tran
- page 14 : modular-width – The modular-width $mw(G)$ of a graph $G$ is the smallest number $h$ such that a $k$-partition $(V_1,\dots,V_k)$ of $G$ exists, where $k \le h$ and each subset $V_i$, $i \in [k]$ is a module and either contains a single vertex or for which the modular-subgraph $G[V_i]$ has a modular-width of $h$.
- page 24 : graph class path is not constant modular-width – The Modular-width of a path $P$ with length $n > 3$ is $n$.
- page 25 : modular-width upper bounds clique-width by a computable function – Proposition 4.6. Modular-width strictly upper bounds Clique-width.
- page 25 : graph classes with bounded clique-width are not bounded modular-width – Proposition 4.6. Modular-width strictly upper bounds Clique-width.
- page 25 : modular-width upper bounds diameter by a computable function – Theorem 4.7. Modular-width strictly upper bounds Max Diameter of Components.
- page 25 : graph classes with bounded diameter are not bounded modular-width – Theorem 4.7. Modular-width strictly upper bounds Max Diameter of Components.
- page 28 : graph classes with bounded modular-width are not bounded distance to cluster – Proposition 4.10. Modular-width is incomparable to Distance to Cluster.
- page 28 : graph classes with bounded distance to cluster are not bounded modular-width – Proposition 4.10. Modular-width is incomparable to Distance to Cluster.
- page 28 : graph classes with bounded modular-width are not bounded distance to co-cluster – Proposition 4.11. Modular-width is incomparable to Distance to Co-Cluster.
- page 28 : graph classes with bounded distance to co-cluster are not bounded modular-width – Proposition 4.11. Modular-width is incomparable to Distance to Co-Cluster.
- page 30 : graph classes with bounded modular-width are not bounded chordality – Theorem 4.16. Modular-width is incomparable to Chordality.
- page 30 : graph classes with bounded chordality are not bounded modular-width – Theorem 4.16. Modular-width is incomparable to Chordality.
2013 Parameterized Algorithms for Modular-Width by Gajarský, Lampis, Ordyniak
- page 6 : neighborhood diversity upper bounds modular-width by a linear function – Theorem 3. Let $G$ be a graph. Then $\mathrm{mw}(G) \le \mathrm{nd}(G)$ and $\mathrm{mw}(G) \le 2\mathrm{tc}(G) + \mathrm{tc}(G)$. Furthermore, both inequalities are strict, …
- page 6 : graph classes with bounded modular-width are not bounded neighborhood diversity – Theorem 3. Let $G$ be a graph. Then $\mathrm{mw}(G) \le \mathrm{nd}(G)$ and $\mathrm{mw}(G) \le 2\mathrm{tc}(G) + \mathrm{tc}(G)$. Furthermore, both inequalities are strict, …
- page 6 : twin-cover number upper bounds modular-width by an exponential function – Theorem 3. Let $G$ be a graph. Then $\mathrm{mw}(G) \le \mathrm{nd}(G)$ and $\mathrm{mw}(G) \le 2\mathrm{tc}(G) + \mathrm{tc}(G)$. Furthermore, both inequalities are strict, …
- page 6 : graph classes with bounded modular-width are not bounded twin-cover number – Theorem 3. Let $G$ be a graph. Then $\mathrm{mw}(G) \le \mathrm{nd}(G)$ and $\mathrm{mw}(G) \le 2\mathrm{tc}(G) + \mathrm{tc}(G)$. Furthermore, both inequalities are strict, …
- page 8 : graph classes with bounded modular-width are not bounded shrub-depth – Theorem 4. There are classes of graphs with unbounded modular-width and bounded shrub-depth and vice versa.
- page 8 : graph classes with bounded shrub-depth are not bounded modular-width – Theorem 4. There are classes of graphs with unbounded modular-width and bounded shrub-depth and vice versa.
unknown source
- modular-width upper bounds clique-width by a computable function
- modular-width upper bounds diameter by a computable function
assumed
- modular-width is equivalent to modular-width – assumed