L-Graphs and Monotone L-Graphs

An L-segment consists of a horizontal and a vertical straight line which form an L. In an L-embedding of a graph, each vertex is represented by an Lsegment, and two segments intersect each other if and only if the corresponding vertices are adjacent in the graph. If the corner of each L-segment in an Lembedding lies on a straight line, we call it a monotone L-embedding. In this paper we give a full characterization of monotone L-embeddings by introducing a new class of graphs which we call “non-jumping" graphs. We show that a graph admits a monotone 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 ≤ 4) admit L-embeddings.