c-closure
equivalent to: c-closure
Relations
| Other | Relation from | Relation to |
|---|---|---|
| acyclic chromatic number | exclusion | exclusion |
| arboricity | exclusion | exclusion |
| average degree | exclusion | exclusion |
| average distance | exclusion | exclusion |
| bandwidth | upper bound | exclusion |
| bipartite | unknown to HOPS | exclusion |
| bipartite number | exclusion | unknown to HOPS |
| bisection bandwidth | exclusion | exclusion |
| block | unknown to HOPS | exclusion |
| book thickness | exclusion | exclusion |
| boolean width | exclusion | exclusion |
| bounded components | upper bound | exclusion |
| boxicity | exclusion | exclusion |
| branch width | exclusion | exclusion |
| carving-width | upper bound | exclusion |
| chordal | unknown to HOPS | exclusion |
| chordality | exclusion | unknown to HOPS |
| chromatic number | exclusion | exclusion |
| clique cover number | exclusion | exclusion |
| clique-tree-width | exclusion | exclusion |
| clique-width | exclusion | exclusion |
| cluster | unknown to HOPS | exclusion |
| co-cluster | unknown to HOPS | exclusion |
| cograph | unknown to HOPS | exclusion |
| complete | unknown to HOPS | exclusion |
| connected | unknown to HOPS | unknown to HOPS |
| cutwidth | upper bound | exclusion |
| cycle | constant | exclusion |
| cycles | constant | exclusion |
| d-path-free | exclusion | exclusion |
| degeneracy | exclusion | exclusion |
| degree treewidth | upper bound | exclusion |
| diameter | exclusion | exclusion |
| diameter+max degree | upper bound | exclusion |
| disjoint cycles | unknown to HOPS | exclusion |
| distance to bipartite | exclusion | exclusion |
| distance to block | exclusion | exclusion |
| distance to bounded components | exclusion | exclusion |
| distance to chordal | exclusion | exclusion |
| distance to cluster | exclusion | exclusion |
| distance to co-cluster | exclusion | exclusion |
| distance to cograph | exclusion | exclusion |
| distance to complete | exclusion | exclusion |
| distance to edgeless | exclusion | exclusion |
| distance to forest | exclusion | exclusion |
| distance to interval | exclusion | exclusion |
| distance to linear forest | exclusion | exclusion |
| distance to maximum degree | exclusion | exclusion |
| distance to outerplanar | exclusion | exclusion |
| distance to perfect | exclusion | exclusion |
| distance to planar | exclusion | exclusion |
| distance to stars | exclusion | exclusion |
| domatic number | exclusion | exclusion |
| domination number | exclusion | exclusion |
| edge clique cover number | exclusion | exclusion |
| edge connectivity | exclusion | unknown to HOPS |
| edgeless | constant | exclusion |
| feedback edge set | upper bound | exclusion |
| feedback vertex set | exclusion | exclusion |
| forest | constant | exclusion |
| genus | exclusion | exclusion |
| girth | exclusion | exclusion |
| grid | constant | exclusion |
| h-index | exclusion | exclusion |
| inf-flip-width | exclusion | exclusion |
| interval | unknown to HOPS | exclusion |
| iterated type partitions | exclusion | exclusion |
| linear clique-width | exclusion | exclusion |
| linear forest | constant | exclusion |
| linear NLC-width | exclusion | exclusion |
| linear rank-width | exclusion | exclusion |
| maximum clique | exclusion | exclusion |
| maximum degree | upper bound | exclusion |
| maximum independent set | exclusion | exclusion |
| maximum induced matching | exclusion | exclusion |
| maximum leaf number | upper bound | exclusion |
| maximum matching | exclusion | exclusion |
| maximum matching on bipartite graphs | unknown to HOPS | exclusion |
| mim-width | exclusion | unknown to HOPS |
| minimum degree | exclusion | exclusion |
| mm-width | exclusion | exclusion |
| modular-width | exclusion | exclusion |
| module-width | exclusion | exclusion |
| neighborhood diversity | exclusion | exclusion |
| NLC-width | exclusion | exclusion |
| NLCT-width | exclusion | exclusion |
| odd cycle transversal | exclusion | exclusion |
| outerplanar | unknown to HOPS | exclusion |
| path | constant | exclusion |
| pathwidth | exclusion | exclusion |
| pathwidth+maxdegree | upper bound | exclusion |
| perfect | unknown to HOPS | exclusion |
| planar | unknown to HOPS | exclusion |
| radius-r flip-width | exclusion | unknown to HOPS |
| rank-width | exclusion | exclusion |
| shrub-depth | exclusion | exclusion |
| sim-width | exclusion | unknown to HOPS |
| star | constant | exclusion |
| stars | constant | exclusion |
| topological bandwidth | unknown to HOPS | exclusion |
| tree | constant | exclusion |
| tree-independence number | exclusion | unknown to HOPS |
| treedepth | exclusion | exclusion |
| treelength | exclusion | unknown to HOPS |
| treewidth | exclusion | exclusion |
| twin-cover number | exclusion | exclusion |
| twin-width | exclusion | exclusion |
| vertex connectivity | exclusion | exclusion |
| vertex cover | exclusion | exclusion |
| vertex integrity | exclusion | exclusion |
Results
- 2022 Expanding the Graph Parameter Hierarchy by Tran
- page 14 : c-closure – The c-closure $\mathrm{cc}(G)$ of a graph $G$ is the smallest number $c$ such that any pair of vertices $v,w \in V(G)$ with $|N_G(v) \cap N_G(w)| \ge c$ is adjacent. …
- page 32 : maximum degree upper bounds c-closure by a linear function – Proposition 5.1. Maximum Degree strictly upper bounds $c$-Closure.
- page 32 : bounded c-closure does not imply bounded maximum degree – Proposition 5.1. Maximum Degree strictly upper bounds $c$-Closure.
- page 32 : feedback edge set upper bounds c-closure by a linear function – Theorem 5.2. Feedback Edge Number strictly upper bounds $c$-Closure.
- page 32 : bounded c-closure does not imply bounded feedback edge set – Theorem 5.2. Feedback Edge Number strictly upper bounds $c$-Closure.
- page 34 : bounded c-closure does not imply bounded vertex cover – Proposition 5.3. $c$-Closure is incomparable to Vertex Cover Number.
- page 34 : bounded vertex cover does not imply bounded c-closure – Proposition 5.3. $c$-Closure is incomparable to Vertex Cover Number.
- page 34 : bounded c-closure does not imply bounded distance to complete – Proposition 5.4. $c$-Closure is incomparable to Distance to Clique.
- page 34 : bounded distance to complete does not imply bounded c-closure – Proposition 5.4. $c$-Closure is incomparable to Distance to Clique.
- page 34 : bounded c-closure does not imply bounded bisection bandwidth – Proposition 5.5. $c$-Closure is incomparable to Bisection Width.
- page 34 : bounded bisection bandwidth does not imply bounded c-closure – Proposition 5.5. $c$-Closure is incomparable to Bisection Width.
- page 34 : bounded c-closure does not imply bounded genus – Proposition 5.6. $c$-Closure is incomparable to Genus.
- page 34 : bounded genus does not imply bounded c-closure – Proposition 5.6. $c$-Closure is incomparable to Genus.
- page 34 : bounded c-closure does not imply bounded vertex connectivity – Observation 5.7. $c$-Closure is incomparable to Distance to Disconnected, Domatic Number and Maximum Clique.
- page 34 : bounded vertex connectivity does not imply bounded c-closure – Observation 5.7. $c$-Closure is incomparable to Distance to Disconnected, Domatic Number and Maximum Clique.
- page 34 : bounded c-closure does not imply bounded domatic number – Observation 5.7. $c$-Closure is incomparable to Distance to Disconnected, Domatic Number and Maximum Clique.
- page 34 : bounded domatic number does not imply bounded c-closure – Observation 5.7. $c$-Closure is incomparable to Distance to Disconnected, Domatic Number and Maximum Clique.
- page 34 : bounded c-closure does not imply bounded maximum clique – Observation 5.7. $c$-Closure is incomparable to Distance to Disconnected, Domatic Number and Maximum Clique.
- page 34 : bounded maximum clique does not imply bounded c-closure – Observation 5.7. $c$-Closure is incomparable to Distance to Disconnected, Domatic Number and Maximum Clique.
- page 35 : bounded c-closure does not imply bounded boxicity – Proposition 5.8. $c$-Closure is incomparable to Boxicity.
- page 35 : bounded boxicity does not imply bounded c-closure – Proposition 5.8. $c$-Closure is incomparable to Boxicity.
- unknown source
- maximum degree upper bounds c-closure by a computable function
- feedback edge set upper bounds c-closure by a computable function