neighborhood diversity

abbr: nd

tags: module

Definition: Vertices can be partitioned into $\mathrm{nd}$ parts, each consisting of only false or true twins.

Relations

Results

• 2024 Parameterized complexity for iterated type partitions and modular-width by Cordasco, Gargano, Rescigno
  • page 3 : neighborhood diversity upper bounds iterated type partitions by a linear function – … $itp(G) \le nd(G)$. Actually $itp(G)$ can be arbitrarily smaller than $nd(G)$.
  • page 3 : bounded iterated type partitions does not imply bounded neighborhood diversity – … $itp(G) \le nd(G)$. Actually $itp(G)$ can be arbitrarily smaller than $nd(G)$.
• 2022 Expanding the Graph Parameter Hierarchy by Tran
  • page 14 : neighborhood diversity – The neighborhood diversity $nd(G)$ of a graph $G$ is the smallest number $k$ such that there is a $k$-partition $(V_1,\dots,V_k)$ of $G$, where each subset $V_i$, $i \in [k]$ is a module and is either a complete set or an independent set.
  • 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 : vertex cover upper bounds neighborhood diversity by an exponential function – Proposition 4.3. Vertex Cover Number strictly upper bounds Neighborhood Diversity.
  • page 23 : bounded neighborhood diversity does not imply bounded vertex cover – Proposition 4.3. Vertex Cover Number strictly upper bounds Neighborhood Diversity.
  • page 26 : graph class path has unbounded neighborhood diversity – The Neighborhood Diversity of a Path $P$ with length $n > 3$ is $n$.
  • page 26 : neighborhood diversity upper bounds boxicity by a polynomial function – Note that given a path of length $n > 3$. The boxicity is 1 while … neighborhood diversity is $n$. … a graph … with neighborhood diversity of $k$, its boxicity is at most $k+k^2$.
  • page 26 : bounded boxicity does not imply bounded neighborhood diversity – Note that given a path of length $n > 3$. The boxicity is 1 while … neighborhood diversity is $n$. … a graph … with neighborhood diversity of $k$, its boxicity is at most $k+k^2$.
  • page 28 : bounded neighborhood diversity does not imply bounded twin-cover number – Proposition 4.12. Modular-width is incomparable to Distance to Twin Cover Number.
  • page 28 : bounded twin-cover number does not imply bounded neighborhood diversity – Proposition 4.12. Modular-width is incomparable to Distance to Twin Cover Number.
• 2019 Shrub-depth: Capturing Height of Dense Graphs by Ganian, Hliněný, Nešetřil, Obdržálek, Mendez
  • neighborhood diversity upper bounds shrub-depth by a constant – $\mathcal{TM}_m(1)$ is exactly the class of graphs of neighborhood diversity at most $m$.
• 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 : bounded modular-width does not imply 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, …
• 2012 Algorithmic Meta-theorems for Restrictions of Treewidth by Lampis
  • neighborhood diversity – We will say that two vertices $v, v’$ of a graph $G(V, E)$ have the same type iff they have the same colors and $N(v) \setminus {v}=N(v’) \setminus {v}$, where $N(v)$ denotes the set of neighbors of $v$. … A colored graph $G(V, E)$ has neighborhood diversity at most $w$, if there exists a partition of $V$ into at most $w$ sets, such that all the vertices in each set have the same type.
• assumed
  • neighborhood diversity is equivalent to neighborhood diversity – assumed
• https://www.fi.muni.cz/~hlineny/papers/shrubdepth-warw18-slipp.pdf
  • page 9 : neighborhood diversity upper bounds shrub-depth by a constant – Shrub-depth 1: e.g., cliques, stars, \dots, gen. BND – bounded neighborhood diversity.
• 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.
  • vertex cover upper bounds neighborhood diversity by an exponential function
  • bounded twin-cover number does not imply bounded neighborhood diversity
  • bounded vertex integrity does not imply bounded neighborhood diversity