Planar Embeddings

Graph Theory · Axiom Academy

Discover how the same graph can be drawn in many different ways on a plane. This is a simple planar graph - a graph that can be drawn on a plane without any edges crossing. Click and drag any vertex to move it around! Try moving the vertices around. Notice how the edges stretch and bend but never cross! When a planar graph is drawn without crossings, it divides the plane into regions called faces . Try dragging the vertices to create different shapes! Click the button to highlight all the faces in different colors Step 3: Same Graph, Different Embeddings Here are two completely different drawings of the exact same graph. They look different, but they're the same graph! Watch as one embedding smoothly transforms into the other Step 4: The Face Count is Invariant No matter how you rearrange the vertices, certain properties stay the same. Let's count the faces in different embeddings! A planar embedding is a specific way of drawing a planar graph on the plane without edge crossings. The same graph can have many different embeddings - they all represent the same abstract structure but look visually different. Every planar embedding divides the plane into faces (regions). This includes the special outer face that extends to infinity. Different embeddings might make different faces the outer face, but the total number of faces stays constant.

This is the written version of the interactive lesson above. See the full Graph Theory course.