Thomassen characterized some 1-plane embedding as the forbidden configuration such that
a given 1-plane embedding of a graph is drawable in straight lines
if and only if it does not contain the configuration [C. Thomassen, Rectilinear drawings of graphs,
J. Graph Theory, 10(3), 335-341, 1988].In this paper, we characterize
some 1-plane embedding as the forbidden configuration such that
a given 1-plane embedding of a graph can be re-embedded into
a straight-line drawable 1-plane embedding of the same graph
if and only if it does not contain the configuration.
Re-embedding of a 1-plane embedding preserves the same set of pairs of
crossing edges.
We give a linear-time algorithm for finding a
straight-line drawable 1-plane re-embedding or
the forbidden configuration.