Boxicity and Interval-Orders: Petersen and the Complements of Line Graphs
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The boxicity of a graph is the smallest dimension d allowing a representation of it as the intersection graph of a set of d-dimensional axis-parallel boxes. We present a simple general approach to determining the boxicity of a graph based on studying its “interval-order subgraphs”. The power of the method is first tested on the boxicity of some popular graphs that have resisted previous attempts: the boxicity of the Petersen graph is 3, and more generally, that of the Kneser-graphs K(n,2) is n-2 if n≥ 5, confirming a conjecture of Caoduro and Lichev [Discrete Mathematics, Vol. 346, 5, 2023]. Since every line graph is an induced subgraph of the complement of K(n,2), the developed tools show furthermore that line graphs have only a polynomial number of edge-maximal interval-order subgraphs. This opens the way to polynomial-time algorithms for problems that are in general 𝒩𝒫-hard: for the existence and optimization of interval-order subgraphs of line-graphs, or of interval-completions of their complement.
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