  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.
