insight - Algorithms and Data Structures - # Contact Graphs of Axis-Parallel Boxes with Unidirectional Contacts

Core Concepts

The paper studies a new class of geometric intersection graphs, called Contact Graphs of Boxes with Unidirectional contacts (CBU), where vertices correspond to interior-disjoint axis-parallel boxes in Rd, and two boxes intersect only on a (d-1)-dimensional box orthogonal to one specific dimension. The authors provide several structural properties of these graphs and analyze the computational complexity of related problems.

Abstract

The paper introduces and studies a new class of geometric intersection graphs called Contact Graphs of Boxes with Unidirectional contacts (CBU). In this model, the vertices correspond to interior-disjoint axis-parallel boxes in Rd, and two boxes are adjacent if they intersect only on a (d-1)-dimensional box orthogonal to one specific dimension.
The key findings are:
CBU graphs are triangle-free and can be characterized by the existence of a homogeneous arc labeling of their orientation.
The class of CBU graphs is strictly contained in the class of cover graphs, and there are planar graphs that are not CBU.
Bipartite graphs of boxicity b belong to (b+1)-CBU, and 1-subdivisions of graphs with proper box representations belong to (b+1)-CBU.
Recognizing whether a graph belongs to d-CBU is NP-complete for d≥3, and there is no polynomial-time constant-factor approximation algorithm for the minimum d such that a graph belongs to d-CBU.
CBU graphs have bounded fractional chromatic number and large independent sets, but 3-CBU graphs can have arbitrarily large chromatic number.
Several optimization problems remain NP-hard on 2-CBU or 3-CBU graphs.

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Deeper Inquiries

The recognition problem for CBU graphs can be solved in polynomial time for specific graph classes. For planar graphs, the problem can be tackled efficiently. By characterizing the orientations that allow for homogeneous labeling, we can determine if a planar graph belongs to CBU. Additionally, for graphs with large girth, the recognition problem becomes more complex. While some graphs with arbitrary large girth are not in CBU, others, like shift graphs, have bounded fractional chromatic numbers and linear size independent sets. Therefore, for specific graph classes like planar graphs, the recognition problem for CBU graphs can be solved in polynomial time.

The largest value of c such that every graph with a circular chromatic number at most c belongs to CBU is 5/2. This is supported by Theorem 35, which states that every graph with a circular chromatic number of 5/2 or less belongs to CBU. Therefore, the largest value of c for which this property holds is 5/2.

For graphs of girth at least g, there is no known function f(g) that guarantees every graph of girth at least g belongs to f(g)-CBU. The recognition of graphs based on girth is a challenging problem, as demonstrated by the existence of graphs with large girth that do not belong to CBU. While there are constructions like the Jones graphs that exhibit specific properties in terms of girth and independence number, a general function f(g) that encompasses all graphs of girth g and their membership in f(g)-CBU remains an open question.

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