  The notion of a 12-representable graph was introduced by Jones et al.. This
notion generalizes the notions of the much studied permutation graphs and
co-interval graphs. It is known that any 12-representable graph is a
comparability graph, and also that a tree is 12-representable if and only if it
is a double caterpillar. Moreover, Jones et al.\ initiated the study of
12-representability of induced subgraphs of a grid graph, and asked whether it
is possible to characterize such graphs. This question in is meant to be about
induced subgraphs of a grid graph that consist of squares, which we call square
grid graphs. However, an induced subgraph in a grid graph does not have to
contain entire squares, and we call such graphs line grid graphs.
  In this paper we answer the question of Jones et al.\ by providing a complete
characterization of $12$-representable square grid graphs in terms of forbidden
induced subgraphs. Moreover, we conjecture such a characterization for the line
grid graphs and give a number of results towards solving this challenging
conjecture. Our results are a major step in the direction of characterization
of all 12-representable graphs since beyond our characterization, we also
discuss relations between graph labelings and 12-representability, one of the
key open questions in the area.
