Outerplanar graph: planar embedding where all vertices lie on the outer face boundary. Equivalent: no K_4 or K_{2,3} minor. Strictly contained in planar graphs. O(n) recognition. Treewidth ≤ 2. Book thickness = 1. Many NP-hard problems become linear.
How to Use
Select graph:
- Test: Outerplanar?
- Minor: K₄, K_{2,3}
- tw: Treewidth ≤ 2
Characterization
G is outerplanar iff (1) no K_4 minor and no K_{2,3} minor, (2) every biconnected component is a single edge or has a Hamiltonian face, (3) treewidth ≤ 2. Recognizable in O(n) time.
Algorithms
Because tw ≤ 2: many NP-hard problems become O(n). Maximum independent set, minimum dominating set, Hamiltonian cycle, vertex coloring — all linear on outerplanar graphs via tree decomposition DP.
Step-by-Step Instructions
- 1Select graph.
- 2Check outerplanar.
- 3Find minor obstruction.
- 4Compute treewidth.
- 5Apply DP.