Simultaneous embedding

Simultaneous embedding is a technique in <a href="/facts/Graph_drawing/uhh73BVo">graph drawing</a> and <a href="/facts/Information_visualization/DSHteV85">information visualization</a> for visualizing two or more different <a href="/facts/Graph_theory/VVPCd4TX">graphs</a> on the same or overlapping sets of labeled <a href="/facts/Vertex_(graph_theory)/WhlWmL3o">vertices</a>, while avoiding <a href="/facts/Crossing_number_(graph_theory)/x3Yz6UhF">crossings</a> within both graphs. Crossings between an <a href="/facts/Edge_(geometry)/3lFN6sEz">edge</a> of one graph and an edge of the other graph are allowed.
If edges are allowed to be drawn as <a href="/facts/Polyline/GRQ6BMQS">polylines</a> or <a href="/facts/Curve/xBQ0pTBW">curves</a>, then any <a href="/facts/Planar_graph/plREagr9">planar graph</a> may be drawn without crossing with its vertices in arbitrary positions in the plane, where the same vertex placement provides a simultaneous embedding.
There are two restricted models: simultaneous geometric embedding, where each graph must be drawn planarly with line segments representing its edges rather than more complex curves, restricting the two given graphs to subclasses of the planar graphs, and simultaneous embedding with fixed edges, where curves or bends are allowed in the edges, but any edge in both graphs must be represented by the same curve in both drawings.
In the unrestricted model, any two planar graphs can have a simultaneous embedding.

Simultaneous embedding open-in-new

Simultaneous embedding