edge clique cover number
tags: covering edges
equivalent to: edge clique cover number
Relations
| Other | Relation from | Relation to |
|---|---|---|
| acyclic chromatic number | exclusion | exclusion |
| arboricity | exclusion | exclusion |
| average degree | exclusion | exclusion |
| average distance | exclusion | upper bound |
| bandwidth | exclusion | exclusion |
| bipartite | unbounded | exclusion |
| bipartite number | exclusion | upper bound |
| bisection bandwidth | exclusion | exclusion |
| block | unbounded | exclusion |
| book thickness | exclusion | exclusion |
| boolean width | exclusion | upper bound |
| bounded components | unknown to HOPS | exclusion |
| boxicity | exclusion | upper bound |
| branch width | exclusion | exclusion |
| c-closure | exclusion | exclusion |
| carving-width | exclusion | exclusion |
| chordal | unbounded | exclusion |
| chordality | exclusion | upper bound |
| chromatic number | exclusion | exclusion |
| clique cover number | exclusion | exclusion |
| clique-tree-width | exclusion | upper bound |
| clique-width | exclusion | upper bound |
| cluster | unknown to HOPS | exclusion |
| co-cluster | unknown to HOPS | exclusion |
| cograph | unknown to HOPS | exclusion |
| complete | constant | exclusion |
| connected | unbounded | unknown to HOPS |
| cutwidth | exclusion | exclusion |
| cycle | unbounded | exclusion |
| cycles | unbounded | exclusion |
| d-path-free | exclusion | exclusion |
| degeneracy | exclusion | exclusion |
| degree treewidth | exclusion | exclusion |
| diameter | exclusion | upper bound |
| diameter+max degree | unknown to HOPS | exclusion |
| disjoint cycles | unbounded | 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 | upper bound | 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 | unknown to HOPS |
| distance to stars | exclusion | exclusion |
| domatic number | exclusion | exclusion |
| domination number | exclusion | exclusion |
| edge connectivity | exclusion | exclusion |
| edgeless | unknown to HOPS | exclusion |
| feedback edge set | exclusion | exclusion |
| feedback vertex set | exclusion | exclusion |
| forest | unbounded | exclusion |
| genus | exclusion | exclusion |
| girth | exclusion | upper bound |
| grid | unbounded | exclusion |
| h-index | exclusion | exclusion |
| inf-flip-width | exclusion | upper bound |
| interval | unbounded | exclusion |
| iterated type partitions | exclusion | upper bound |
| linear clique-width | exclusion | upper bound |
| linear forest | unbounded | exclusion |
| linear NLC-width | exclusion | upper bound |
| linear rank-width | exclusion | 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 | exclusion | exclusion |
| maximum matching | exclusion | exclusion |
| maximum matching on bipartite graphs | unknown to HOPS | exclusion |
| mim-width | exclusion | upper bound |
| minimum degree | exclusion | exclusion |
| mm-width | exclusion | exclusion |
| modular-width | exclusion | upper bound |
| module-width | exclusion | upper bound |
| neighborhood diversity | exclusion | upper bound |
| NLC-width | exclusion | upper bound |
| NLCT-width | exclusion | upper bound |
| odd cycle transversal | exclusion | exclusion |
| outerplanar | unbounded | exclusion |
| path | unbounded | exclusion |
| pathwidth | exclusion | exclusion |
| pathwidth+maxdegree | exclusion | exclusion |
| perfect | unbounded | exclusion |
| planar | unbounded | exclusion |
| radius-r flip-width | exclusion | upper bound |
| rank-width | exclusion | upper bound |
| shrub-depth | exclusion | upper bound |
| sim-width | exclusion | upper bound |
| star | unknown to HOPS | exclusion |
| stars | unknown to HOPS | exclusion |
| topological bandwidth | exclusion | exclusion |
| tree | unbounded | exclusion |
| tree-independence number | exclusion | unknown to HOPS |
| treedepth | exclusion | exclusion |
| treelength | exclusion | upper bound |
| treewidth | exclusion | exclusion |
| twin-cover number | exclusion | exclusion |
| twin-width | exclusion | upper bound |
| vertex connectivity | unknown to HOPS | exclusion |
| vertex cover | exclusion | exclusion |
| vertex integrity | exclusion | exclusion |
Results
- 2022 Expanding the Graph Parameter Hierarchy by Tran
- page 14 : edge clique cover number – The edge clique cover number $eccn(G)$ of a graph $G$ is the minimum number of complete subgraphs required such that each edge is contained in at least one of them.
- page 22 : edge clique cover number upper bounds neighborhood diversity by an exponential function – Theorem 4.1. Edge Clique Cover Number strictly upper bounds Neighborhood Diversity.
- page 22 : bounded neighborhood diversity does not imply bounded edge clique cover number – Theorem 4.1. Edge Clique Cover Number strictly upper bounds Neighborhood Diversity.
- page 23 : distance to complete upper bounds edge clique cover number by a polynomial function – Proposition 4.2. Disatnce to Clique strictly upper bounds Edge Clique Cover Number.
- page 23 : bounded edge clique cover number does not imply bounded distance to complete – Proposition 4.2. Disatnce to Clique strictly upper bounds Edge Clique Cover Number.
- page 29 : bounded edge clique cover number does not imply bounded vertex cover – Proposition 4.13. Edge Clique Cover Number is incomparable to Vertex Cover Number.
- page 29 : bounded vertex cover does not imply bounded edge clique cover number – Proposition 4.13. Edge Clique Cover Number is incomparable to Vertex Cover Number.
- page 29 : bounded edge clique cover number does not imply bounded domination number – Proposition 4.14. Edge Clique Cover Number is incomparable to Domination Number.
- page 29 : bounded domination number does not imply bounded edge clique cover number – Proposition 4.14. Edge Clique Cover Number is incomparable to Domination Number.
- page 29 : bounded edge clique cover number does not imply bounded distance to perfect – Proposition 4.15. Edge Clique Cover Number is incomparable to Distance to Perfect.
- page 29 : bounded distance to perfect does not imply bounded edge clique cover number – Proposition 4.15. Edge Clique Cover Number is incomparable to Distance to Perfect.
- unknown source
- edge clique cover number upper bounds neighborhood diversity by an exponential function – Label vertices by the cliques they are contained in, each label is its own group in the neighborhood diversity, connect accordingly.
- distance to complete upper bounds edge clique cover number by a polynomial function – Cover the remaining clique, cover each modulator vertex and its neighborhood outside of it with another clique, cover each edge within the modulator by its own edge.