Contact graphs of boxes with unidirectional contacts

TL;DR

The paper introduces and studies $d$-CBU graphs—contact graphs of axis-aligned boxes with unidirectional contacts—relating them to boxicity and cover graphs. It provides a structural and labeling-theoretic foundation via homogeneous arc labeling, proves NP-hardness of recognizing membership for $d\ge3$, and establishes that CBU is a proper subset of cover graphs. The work explores planar and outerplanar cases, giving containment results (e.g., $2$-CBU for many triangle-free outerplanar graphs, $3$-CBU for bipartite planar graphs) and counterexamples, and it analyzes the chromatic and optimization problem complexity on these classes. Collectively, the results yield a cohesive picture of how geometric box-contact representations constrain graph structure and algorithmic tractability, while highlighting several open questions about tight bounds and recognition.

Abstract

This paper is devoted to the study of particular geometrically defined intersection classes of graphs. Those were previously studied by Magnant and Martin, who proved that these graphs have arbitrary large chromatic number, while being triangle-free. We give several structural properties of these graphs, and we raise several questions.

Figures (11)

• Figure 1: $(i)$ Example of a good orientation of a $C_5$ with some valid labels, $(ii)$ example of bad orientation. Once a label $x$ is fixed for one arc, this label is propagated to all the arcs leading to the conclusion that $x < x$. $(iii)$ the two valid orientations of a $C_4$
• Figure 2: The top boxes with respect to $\ell_1$, $\ell_2$, and $\ell_3$ are $b$, $d$, and $a$, respectively. The top sequence of this 2-CBU representation is $a,b,d,e,a,g,h,g,a$.
• Figure 3: Left part : Adding a degree one vertex $v_i$ in the representation. This is done in the vertical stripe where the neighbor of $v_i$, $v_{i-1}=v_{i+1}$, is the top box. Right part : Adding $v_{i+1},\ldots,v_{j-1}$. This is done in the vertical stripe where $v_i$ and $v_{j}$ are the top box, successively.
• Figure 4: An example of $2$-CBU graph and its associated acyclic orientation
• Figure 5: The series-parallel graph $G$ of Theorem \ref{['thm:series-parallel']}.

Theorems & Definitions (34)

• Claim 1
• Claim 2
• Claim 3
• Corollary 4
• Remark 5
• Theorem 6
• Theorem 8
• Corollary 9
• Corollary 10
• Theorem 11