Graph Theory: Connectivity & Planarity
Graph theory’s concepts of connectivity and planarity explore how vertices in a graph are linked and whether a graph can be drawn on a plane without edges crossing. Connectivity measures the minimum number of elements—nodes or edges—that must be removed to disconnect the remaining nodes. Planarity determines if a graph can be represented without overlapping edges, which is crucial in circuit design and network visualization. Both concepts are fundamental in understanding graph structure and applications.
Graph Theory: Connectivity & Planarity
Graph theory’s concepts of connectivity and planarity explore how vertices in a graph are linked and whether a graph can be drawn on a plane without edges crossing. Connectivity measures the minimum number of elements—nodes or edges—that must be removed to disconnect the remaining nodes. Planarity determines if a graph can be represented without overlapping edges, which is crucial in circuit design and network visualization. Both concepts are fundamental in understanding graph structure and applications.
💡 Key Takeaways
- Understand vertex-connectivity and edge-connectivity and how removing nodes or edges can disconnect a graph.
- Learn what planarity means and identify graphs that can be drawn without edge crossings, including classic non-planar examples like K5 and K3,3.
- Apply core planar-graph results such as Euler's formula for connected planar graphs and the bound E ≤ 3V − 6 for simple planar graphs.
- Relate connectivity to network resilience and planarity to graph drawing and algorithm design.
❓ Frequently Asked Questions
What is graph connectivity?
Connectivity measures how hard it is to separate a graph. It includes vertex connectivity κ(G) (minimum vertices to remove to disconnect the graph) and edge connectivity λ(G) (minimum edges to remove to disconnect the graph).
What is planarity?
A graph is planar if it can be drawn on the plane without any edge crossings; edges may only meet at shared endpoints in a planar embedding.
How can you test planarity quickly?
A quick check: for a connected simple planar graph with n ≥ 3, the number of edges m satisfies m ≤ 3n − 6. If m > 3n − 6, the graph is non-planar. A definitive test uses Kuratowski’s theorem: non-planar graphs contain a subdivision of K5 or K3,3.
What is a planar embedding?
A planar embedding is a specific drawing of a planar graph on the plane with no edge crossings, which also reveals the faces (regions) of the drawing.
What are classic examples of non-planar graphs?
K5 (the complete graph on five vertices) and K3,3 (the complete bipartite graph with partitions of size 3) are famous non-planar graphs.