Induced Path¶
A graph path whose selected vertices have no host-graph edges except the consecutive edges of the path.
Core Idea¶
An induced path is a sequence of distinct graph vertices with edges between every consecutive pair and no host-graph edge between any nonconsecutive selected pair. Equivalently, the subgraph induced by its vertices is exactly a path graph. A displayed simple route can still have a shortcut edge in the host; that makes it non-induced.[^ref-6f4a2c140642]
Scope of Application¶
The definition applies in graph theory wherever all selected-pair adjacencies are known. In a hypercube, a long induced path is a snake-in-the-box candidate; excluding an induced Pₜ can define a graph class. Induced Cycle, hole and antihole are distinct related identities, not aliases.[ref-d4f2f005f6f4][ref-16b2042ed22d]
Clarity¶
Check edges the route does not use. If a four-vertex route has an edge between its first and third vertices, it remains a simple path but is not induced. An edge to an unselected outside vertex does not change inducedness of this selected path.
For a concrete hypercube case, 000→001→011→111→110 has four consecutive one-bit changes in Q₃; its six nonconsecutive Hamming distances are 2,3,2,2,3,2, so none is an extra cube edge. The five vertices induce P₅. This sequence is an elementary constructed illustration, not a sourced record-length snake.[^ref-d4f2f005f6f4]
Manages Complexity¶
For selected vertices, one exact test replaces an informal “no shortcut” judgment: consecutive pairs must be adjacent, every other selected pair nonadjacent. That supports enumeration and forbidden-motif reasoning.[^ref-6f4a2c140642]
Abstract Reasoning¶
Specify a host graph and ordered distinct vertices. Test required path edges, then test all nonconsecutive pairs for forbidden edges. Adding a shortcut destroys inducedness while preserving ordinary path traversal; deleting a consecutive edge destroys the path itself.
Knowledge Transfer¶
The no-chord test transfers between ordinary graphs and hypercubes, but maximal-length searches or graph-class exclusions are applications, not part of the definition. Live prime Path is the independently challenged staged strict parent; induced cycles remain different objects. The seed's unqualified “detour number” synonym for longest induced-path length has been removed.[^ref-a173b6082379]
[^ref-6f4a2c140642]: Original chordless-path enumeration study.
[^ref-d4f2f005f6f4]: Original snake-in-the-box study.
[^ref-16b2042ed22d]: Original forbidden-Pₜ graph study.
[^ref-a173b6082379]: “On Detours in Graphs”, longest-path terminology.
Relationships to Other Abstractions¶
Current abstraction Induced Path Domain-specific
Parents (1) — more general patterns this builds on
-
Induced Path is a kind of Path Prime
Induced paths are chordless graph paths.
Hierarchy path (1) — routes to 1 parentless root
- Induced Path → Path → Network → Reservoir-Flux Network → Conservation Laws → Invariance
Neighborhood in Abstraction Space¶
Induced Path sits in a moderately populated region (53rd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Graph Structures & Combinatorial Objects (44 abstractions)
Nearest neighbors
- Connected Dominating Set — 0.88
- Hadwiger number — 0.86
- Even-hole-free graph — 0.86
- Twin-width — 0.86
- Comparability Graph — 0.86
Computed from structural-signature embeddings · 2026-10-08