boxicity

tags: topology

equivalent to: boxicity

Relations

Results

• 2022 Expanding the Graph Parameter Hierarchy by Tran
  • page 16 : boxicity – The boxicity of a graph $G$ is the minimum amount of interval graphs required, such that their intersection (ed: fixed typo) results in $G$.
  • 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 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.
  • page 42 : bounded degeneracy does not imply bounded boxicity – Proposition 7.1. Degeneracy is incomparable to Boxicity.
  • page 42 : bounded boxicity does not imply bounded degeneracy – Proposition 7.1. Degeneracy is incomparable to Boxicity.
• 2019 The Graph Parameter Hierarchy by Sorge
  • page 8 : acyclic chromatic number upper bounds boxicity by a polynomial function – Lemma 4.9. The boxicity $b$ is upper bounded by the acyclic chromatic number $\chi_a$ (for every graph with $\chi_a>1$). We have $b \le \chi_a(\chi_a-1)$.
  • page 9 : boxicity upper bounds chordality by a linear function – Lemma 4.15 ([8,11]). The boxicity $b$ upper-bounds the chordality $c$. We have $c \le b$.
  • page 9 : distance to interval upper bounds boxicity by a linear function – Lemma 4.16. The distance $i$ to an interval graph upper bounds the boxicity $b$. We have $b \le i+1$.
• 2011 Chordal Bipartite Graphs with High Boxicity by Chandran, Francis, Mathew
  • page 9 : bounded distance to bipartite does not imply bounded boxicity – Theorem 2 For any $b \in \mathbb N^+$, there exists a chordal bipartite graph $G$ (ed: i.e. bipartite graph with no induced cycle on more than 4 vertices) with $\mathrm{box}(G) > b$.
• https://en.wikipedia.org/wiki/Boxicity
  • boxicity – The boxicity of a graph is the minimum dimension in which a given graph can be represented as an intersection graph of axis-parallel boxes.
• Comparing Graph Parameters by Schröder
  • page 14 : bounded distance to co-cluster does not imply bounded boxicity – Proposition 3.4
  • page 27 : bounded distance to chordal does not imply bounded boxicity – Proposition 3.25
• assumed
  • distance to interval upper bounds boxicity by a linear function – By definition
• unknown source
  • acyclic chromatic number upper bounds boxicity by a computable function