Overview
Hopcroft-Tarjan Planarity Test is a planarity testing method for planarity detection. It works by using depth-first search lowpoint structure to detect planarity obstructions.
SpecializedSource checked
Hopcroft-Tarjan Planarity Test
Hopcroft-Tarjan Planarity Test addresses planarity detection by using depth-first search lowpoint structure to detect planarity obstructions.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Graph Traversal
- Difficulty
- Advanced
- Family
- Planarity Testing
- Tier
- Specialized
- Status
- Verified
- Review
- Source Checked
- Sources
- 2
- Formulas
- 0
- Relationships
- 4
- Code example
- Not ready
- Visualization
- Ready
Core Concepts
planarity detectionplanarity testingdepth-first searchlowlink valuesvisibility orderinggraph traversal
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(V + E)
Average
O((V + E) log V)
Worst
O(V^2 + E)
Space
O(V + E)
When to Use
- Use Hopcroft-Tarjan Planarity Test when solving planarity detection and you want to benefit from using depth-first search lowpoint structure to detect planarity obstructions.
- Choose it when O((V + E) log V) average/worst-case time and O(V + E) memory match your workload profile.
- Prefer it over Boyer-Myrvold Planarity Test or Planar Embedding when the problem structure favors planarity detection.
When Not to Use
- Avoid when the graph model does not match the method's direction, weight, or connectivity assumptions.
- Avoid when a simpler traversal gives the same information with less bookkeeping.
What It Does
- Computes planarity detection within Planarity Testing workloads (O((V + E) log V) time, O(V + E) space).
- Works by using depth-first search lowpoint structure to detect planarity obstructions.
- Maintains key invariants across planarity detection, planarity testing, depth-first search.
Use Cases
- Production and research workloads requiring planarity detection via using depth-first search lowpoint structure to detect planarity obstructions.
- Planarity Testing pipelines where O((V + E) log V) time and O(V + E) space fit system constraints.
- Engineering comparisons against Boyer-Myrvold Planarity Test and Planar Embedding when tuning planarity detection.
Inputs
- Undirected graph G = (V, E)
Outputs
- Boolean planarity verdict
- Combinatorial planar embedding (cyclic edge ordering) or Kuratowski subgraph
Pseudocode
- 1.Construct the graph representation G = (V, E) and initialize auxiliary structures for planarity detection (planarity testing).
- 2.Seed the initial frontier, partition, or residual capacities according to Hopcroft-Tarjan Planarity Test's invariants.
- 3.Iteratively process vertices or edges by using depth-first search lowpoint structure to detect planarity obstructions.
- 4.Maintain structural invariants (planarity detection, planarity testing, depth-first search) and prune or contract resolved subproblems.
- 5.Extract and return the final planarity detection structure from the converged graph state.
Canonical Example
- Input
- Undirected graph G = (V, E) configured for a representative planarity detection test case
- Output
- Boolean planarity verdict satisfying Hopcroft-Tarjan Planarity Test invariants
- Explanation
- Hopcroft-Tarjan Planarity Test processes the input by using depth-first search lowpoint structure to detect planarity obstructions, tracking planarity detection and planarity testing to produce the verified planarity detection result.
Implementation Notes
- Represent vertices and edges explicitly before optimizing storage.
- Keep graph direction, weights, and multiedges clear in tests.
- Record tie-breaking when multiple valid outputs exist.
Common Pitfalls
- Using the method on the wrong graph type, such as directed versus undirected or weighted versus unweighted.
- Forgetting disconnected components or isolated vertices.
- Letting tie-breaking hide nondeterministic output in tests.
Interactive Mechanism Walkthrough
Graph Frontier & Edge RelaxationShow using depth-first search lowpoint structure to detect planarity obstructions on a small planarity detection example.
Step 1 of 5: Construct the graph representation G = (V, E) and initialize auxiliary structures for planarity detection (planarity testing).
graph-traversal-mechanismNotes
Hopcroft-Tarjan Planarity Test operates in O((V + E) log V) time and O(V + E) space within the Planarity Testing family. Often compared with Boyer-Myrvold Planarity Test and Planar Embedding.
Sources (2)
What each reference supports for this entry.
Computational Geometry: Algorithms and Applications
BookTextbook support for Hopcroft-Tarjan Planarity Test's computational geometry problem family, predicates, structures, or geometric construction.
Open sourceThe Algorithm Design Manual
BookAlgorithm catalog support for Hopcroft-Tarjan Planarity Test in the graph, flow, matching, and combinatorial optimization benchmark area.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Relationships
View full relationship pageConceptual neighborBoyer-Myrvold Planarity Test
Hopcroft-Tarjan Planarity Test and Boyer-Myrvold Planarity Test share a concrete planarity testing mechanism for planarity detection.
Conceptual neighborPlanar Embedding
Hopcroft-Tarjan Planarity Test and Planar Embedding share a concrete planarity testing mechanism for planarity detection.
Related Algorithms
Tags
GraphTraversal