  In an $\mathsf{L}$-embedding of a graph, each vertex is represented by an
$\mathsf{L}$-segment, and two segments intersect each other if and only if the
corresponding vertices are adjacent in the graph. If the corner of each
$\mathsf{L}$-segment in an $\mathsf{L}$-embedding lies on a straight line, we
call it a monotone $\mathsf{L}$-embedding. In this paper we give a full
characterization of monotone $\mathsf{L}$-embeddings by introducing a new class
of graphs which we call "non-jumping" graphs. We show that a graph admits a
monotone $\mathsf{L}$-embedding if and only if the graph is a non-jumping
graph. Further, we show that outerplanar graphs, convex bipartite graphs,
interval graphs, 3-leaf power graphs, and complete graphs are subclasses of
non-jumping graphs. Finally, we show that distance-hereditary graphs and
$k$-leaf power graphs ($k\le 4$) admit $\mathsf{L}$-embeddings.
