Induced Path¶

A graph path whose selected vertices have no host-graph edges except the consecutive edges of the path.

Version: v1 · 2026-10-04 · History
Domain-specific #: 13738
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Graph Theory → Mathematics
Aliases: Chordless path, Monophonic path

Core Idea¶

An induced path is a path in a host graph whose selected vertices have exactly the path's consecutive edges and no other edges among them. Write the ordered distinct vertices as v₁,…,vₜ. Every vᵢvᵢ₊₁ must be an edge, and no vᵢvⱼ may be an edge when |i−j|>1. Equivalently, the subgraph induced by those vertices is the path graph Pₜ. An ordinary simple path can traverse the same vertices even if the host contains a shortcut edge; that shortcut is precisely what the induced path forbids.^[1]^[2]

Induced Cycle, hole, antihole, chordal/perfect-graph facts and longest-path terminology are related but distinct objects or applications. An induced cycle is closed, not an alias of this open path, and deserves separate treatment. Maximum induced-path length should not be called the “detour number” without qualification, because conventional detour terminology often refers to a longest path without the induced restriction.^[3]^[4]

Structural Signature¶

Sig role-phrases:

Host graph — Supplies the full adjacency relation. Inducedness is checked against all edges of this graph among the chosen vertices, not just the edges the path chooses to walk.^[1]
Ordered distinct vertices — v₁,…,vₜ specifies one simple candidate route. Repeating a vertex would describe a walk rather than the path object at issue.
Consecutive adjacency — Every neighboring pair in the sequence is joined by a host edge, so the sequence is a path.^[2]
No nonconsecutive chord — Every non-neighboring pair in the sequence lacks a host edge. Add one such edge and the sequence remains a simple path but is no longer induced.^[1]
Induced-subgraph equality — G[{v₁,…,vₜ}] ≅ Pₜ summarizes both the required and forbidden edges in one test.^[2]

Condensed: host graph + ordered simple path + all consecutive edges + zero nonconsecutive selected-pair edges.

What It Is Not¶

Not every simple path. Distinct vertices and consecutive adjacency are necessary but do not rule out a host-graph shortcut.^[1]
Not a shortest path by definition. A shortest path between fixed endpoints is induced in an unweighted simple graph, but an induced path can be longer than the shortest one.
Not an induced cycle. Closing the endpoints changes the graph object; it yields an induced cycle only if the resulting vertex-induced subgraph has exactly the cycle edges.^[3]
Not an antihole. That is a complement-graph construction on a hole, not an open path.
Not the detour number. The longest induced path is not safely synonymous with a longest unrestricted path; detour conventions vary, and a standard detour number can concern the latter.^[4]

Scope of Application¶

The definition applies to finite simple undirected graphs whenever one can inspect all edges among a selected vertex set. In forbidden-induced-subgraph theory, excluding Pₜ specifies a graph class. In a hypercube graph, a long induced path is a snake: successive binary strings differ in one bit, but no nonconsecutive strings in the selected sequence may be adjacent in the cube. A corresponding long induced Cycle is a coil, underscoring the identity split.^[2]^[3]

The host graph may represent many things, but the graph-theoretic criterion stays exact. A physical or social route called “shortcut-free” is only an analogy unless its vertices and full adjacency relation are specified and the induced-subgraph test holds. Length, extremal search difficulty and recognition algorithms are questions about induced paths, not additional constitutive roles.^[1]

Clarity¶

The key distinction is between edges selected for traversal and edges present among selected vertices. A path description lists only the former; inducedness inspects the latter as well. If v₁−v₂−v₃−v₄ is traversable but the host also contains v₁v₃, the route is simple yet not induced. One must test all nonconsecutive pairs, not just whether the displayed drawing has a visible shortcut.^[1]

This also clarifies why induced-path and induced-cycle questions should not be merged. Adding an edge between the endpoints of a four-vertex induced path removes the open induced-path status on that vertex set and may create an induced four-cycle; it does not turn every induced path into a hole regardless of other edges.^[3]

Manages Complexity¶

For a fixed selected vertex sequence, the whole condition reduces to two checks: every consecutive pair is adjacent, and every nonconsecutive pair is not. This converts a visually ambiguous “no shortcut” claim into a precise induced-subgraph equality. It lets algorithms enumerate or reject candidate chordless paths based on local adjacency facts.^[1]

In larger graph-class reasoning, Pₜ serves as a compact forbidden motif. “No induced P₅” is a checkable structural condition rather than a prose description of all routes. The compression does not make extremal search easy in every graph; it only makes the object being sought exact.^[2]

Abstract Reasoning¶

Choose a host graph and an ordered list of distinct vertices. Check the required t−1 consecutive edges. Then inspect every other selected pair for a host edge. If none exists, the induced subgraph is exactly Pₜ. This permits reasoning about long chordless sequences, forbidden Pₜ graph classes or hypercube snakes without confusing a merely traversable route with an induced one.^[1]^[3]

A counterfactual edge addition is decisive. Add v₁v₃ to the host while holding selected vertices fixed: the displayed consecutive route still exists, but the induced-path claim fails. Delete a required consecutive edge: it is no longer even a path. Add an edge from a selected vertex to a vertex outside the selected set: inducedness of this particular selected path is unchanged, because the induced subgraph is taken only on its chosen vertices.

Knowledge Transfer¶

The no-chord test transfers literally from ordinary graphs to hypercubes and forbidden-subgraph arguments. The application goal changes: one setting may maximize path length, another may classify graphs by whether a fixed Pₜ occurs. Neither goal belongs to the definition of an individual induced path.^[3]^[2]

Path covers the ordered traversable edge sequence. Induced Path adds a graph-specific host-induced-subgraph constraint, so every instance is a stricter kind of Path.

Examples¶

Four selected vertices with no shortcut¶

In a graph containing v₁v₂, v₂v₃ and v₃v₄, suppose no other edge joins any pair among those four vertices. Other edges may connect them to outside vertices. The selected four induce P₄, so their ordered sequence is an induced path.^[1]

Mapped back: host = full graph including outside vertices; sequence = v₁,v₂,v₃,v₄; consecutive adjacency = three displayed edges; no chord = v₁v₃, v₁v₄, v₂v₄ absent; equality = selected subgraph is P₄.

Snake in a hypercube¶

In the three-dimensional cube Q₃, take the explicit sequence 000 → 001 → 011 → 111 → 110. Its consecutive Hamming distances are 1,1,1,1, so all four route edges exist. The six nonconsecutive pairs have distances d(000,011)=2, d(000,111)=3, d(000,110)=2, d(001,111)=2, d(001,110)=3, and d(011,110)=2. Since an edge of Q₃ requires distance one, none of those pairs is a chord. This is an elementary constructed snake candidate, not a sequence quoted from the source; the cited snake-in-the-box study supplies the hypercube induced-path setting, not these five particular bit strings.^[3] Maximizing length is a further problem, not part of recognizing this candidate.

Mapped back: host = Q₃; sequence = the five displayed binary strings; consecutive adjacency = four distance-one pairs; no chord = all six nonconsecutive distances are two or three; equality = the selected vertices induce precisely P₅.

Near miss: endpoint shortcut¶

Add an edge from v₁ to v₄ in the first example. The ordered route still follows three distinct consecutive edges, but its four selected vertices now contain an extra edge. It is not an induced path on that vertex set; if no other extra edges exist, they instead induce a four-cycle.

Structural Tensions¶

An induced path is a formal graph object and has no intrinsic two-sided design tradeoff. Adding an edge between nonconsecutive selected vertices does not buy an in-instance benefit at a cost; it simply makes that selected route fail the induced-subgraph definition. Likewise, maximizing a hypercube snake or forbidding Pₜ is a separate problem about paths, not a tension within the identity. Diagnostic: check every nonconsecutive selected pair against the host adjacency relation before applying a length or exclusion claim.^[1]^[3]^[2]

Structural–Framed Character¶

Induced Path is highly structural but graph-theory-framed. Its no-chord vocabulary travels unchanged among host graphs and hypercubes; applying the name to a general workflow imports an explicit graph adjacency model rather than recognizing the relation automatically. It has no evaluative weight beyond mathematical properties and depends on no institution's policy. Human practice selects a graph representation and which pairs count as edges, while the induced-subgraph test itself is formal once that representation is fixed. Its broadly portable part is Path; the additional host-edge exclusion is graph-specific. Its character: formally structural inside graph theory, yet domain-specific as a named induced-subgraph object.

Structural Core vs. Domain Accent¶

The skeleton is an ordered traversable route. The domain accent is that the entire host-induced subgraph on its vertices has exactly the route edges. Remove that inducedness constraint and the object is a plain Path; remove required consecutive adjacency and it is not a path. The named entry does not clear the prime bar simply because paths travel broadly: inducedness refers to a formal host graph and its complete selected-pair adjacency relation. Broad reach belongs to Path, the strict parent.

This entry is a kind of Path.

Strict parent: Path. Every induced path is a path with an additional host-graph no-chord condition; ordinary paths may have chords. Network supplies a host relation but does not capture this identity. Induced Cycle, hole and antihole remain distinct graph objects, and the longest induced-path statistic is not relabeled detour number by name.

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 (53^rd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Graph Structures & Combinatorial Objects (44 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Do not confuse a displayed simple path with an induced path unless all nonconsecutive host edges have been checked. An induced cycle is closed and must satisfy its own chordless condition. A longest path may have chords, so unqualified detour-number terminology is not used for longest induced-path length here. A hypercube snake is an induced path in a particular host graph, not the definition of the general object.^[4]^[3]

References¶

[1] “An Efficient Algorithm for Enumerating Chordless Cycles and Chordless Paths”, original graph-algorithms paper. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j

[2] “On Graphs with No Induced Five-Vertex Path or Paraglider”, original Pₜ-free graph-class study. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g

[3] David Allison and Daniel Paulusma, “New Bounds for the Snake-in-the-Box Problem”, original induced-path/induced-cycle hypercube study. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i

[4] “On Detours in Graphs”, original abstract defining maximum-length path terminology. registry ↩a ↩b ↩c