False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01

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Every hereditary graph class has a finite forbidden induced-subgraph basis

False Statement

Every hereditary graph class has a finite minimal forbidden induced-subgraph basis.

Facts & Assumptions

Given: The hereditary class $\mathcal B$ of finite bipartite graphs.

[L1]

A graph is bipartite if and only if it contains no odd cycle (A bipartite graph and a proper two-colouring of its vertices, A finite graph is bipartite if and only if it has no odd cycle).

[F1]

For every $k\ge1$ , $C_{2k+1}$ is an odd cycle, and distinct lengths give nonisomorphic graphs (Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices).

[F2]

A minimal forbidden induced subgraph is outside the class while all proper induced subgraphs are inside (Minimal forbidden induced subgraphs and forbidden bases).

[L2]

The family of all such minimal graphs is the unique minimal basis (Every hereditary graph class is determined by its unique minimal forbidden induced subgraphs).

Refutation

technique · contradiction

1.1

Suppose the minimal forbidden basis of $\mathcal B$ is finite.

assume-contra

1.2

For every $k\ge1$ , the graph $C_{2k+1}$ is not bipartite. Every proper induced subgraph of this chordless cycle is a disjoint union of paths, hence is bipartite. Thus $C_{2k+1}$ is minimally forbidden.

L1F1F2

2.1

The minimal basis therefore contains the pairwise nonisomorphic graphs $C_3,C_5,C_7,\ldots$ .

step 1.2L2

3.1

This is an infinite family, contradicting step 1.1. Hence a hereditary class need not have a finite minimal forbidden basis.

step 1.1step 2.1discharge-contradiction∎

Remarks

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 43 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Valerio Boncompagni, On hereditary graph classes defined by forbidding Truemper configurations (PhD thesis, 2018) (standard reference, not scraped)
Reinhard Diestel, Graph Theory, Preview Chapter 1 (standard reference, not scraped)