Functionality of box intersection graphs

TL;DR

This work investigates the boundedness of the graph-functionality parameter and the symmetric-difference measure for box intersection graphs across dimensions. It proves that functionality is bounded for interval graphs, with an explicit upper bound of $8$, while unbounded for box intersection graphs in $\mathbb{R}^3$, establishing a sharp dimension threshold. Additionally, it shows that symmetric difference is unbounded for interval graphs and for unit box intersection graphs in $\mathbb{R}^2$, via constructive ABC-graph embeddings, which also imply unbounded twin-width and clique-width in those classes. Overall, the results map the landscape of these graph-structure parameters for geometric intersection graphs and pose open problems for unit box graphs in $\mathbb{R}^2$ and higher dimensions.

Abstract

Functionality is a graph complexity measure that extends a variety of parameters, such as vertex degree, degeneracy, clique-width, or twin-width. In the present paper, we show that functionality is bounded for box intersection graphs in $\mathbb{R}^1$, i.e. for interval graphs, and unbounded for box intersection graphs in $\mathbb{R}^3$. We also study a parameter known as symmetric difference, which is intermediate between twin-width and functionality, and show that this parameter is unbounded both for interval graphs and for unit box intersection graphs in $\mathbb{R}^2$.

Figures (2)

• Figure 1: A schematic representation of the graph $G_k$, for $k = 4$, as defined in \ref{['ABC-unbounded-sd']}. The bottom-most vertices belong to the clique $A$; the right-most vertices belong to the clique $C$; vertices on the grid belong to the clique $B$. For each $i \in [t]$, vertex $a_i$ (on column $i$) is adjacent to all the vertices in $B$ on a column $j > i$. Similarly, for each $i \in [t]$, vertex $c_i$ (on row $i$) is adjacent to all the vertices in $B$ on a row $j < i$.
• Figure 2: On the left, an ABC-graph with 15 vertices represented by intervals. On the right, the same graph, this time represented as a unit box intersection graph in $\mathbb{R}^2$.

Theorems & Definitions (19)

• Definition 2.1
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Lemma 4.1
• proof
• Corollary 4.2
• proof
• Lemma 4.3