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 ($kle 4$) admit $mathsf{L}$-embeddings.
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