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 $n\ge 5$, 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 $\mathcal{NP}$-hard: for the existence and optimization of interval-order subgraphs of line-graphs, or of interval-completions of their complement.

翻译：图的箱性是指能够将图表示为若干$d$维轴平行盒子的相交图所需的最小维度$d$。我们提出了一种通用的简单方法，通过研究图的“区间序子图”来确定其箱性。首先，该方法在若干先前难以处理的著名图类上展示了其有效性：彼得森图的箱性为$3$，更一般地，当$n\ge 5$时，Kneser图$K(n,2)$的箱性为$n-2$，从而证实了Caoduro与Lichev的猜想[Discrete Mathematics, Vol. 346, 5, 2023]。由于每个线图都是$K(n,2)$补图的诱导子图，所发展的工具进一步表明：线图仅具有多项式数量的边极大区间序子图。这为某些通常属于$\mathcal{NP}$-困难的问题（如线图中区间序子图的存在性与优化问题，或线图补图的区间完备化问题）开辟了多项式时间算法的可能性。