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

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A graph is $P_3$ -free if and only if every connected component is complete

Statement

A finite graph is $P_3$ -free if and only if each of its connected components is complete. Such graphs are called cluster graphs.

Facts & Assumptions

Given: A finite graph $G$ .

[F1]

$P_3$ -free means having no induced copy of the three-vertex path ( $H$ -free and $\mathcal F$ -free graphs under the induced-subgraph convention, Induced embeddings and induced copies of a graph).

[L1]

Connected components partition the vertices and contain every path between their vertices (The connected components of a graph partition its vertex set and are its maximal connected subgraphs).

[F2]

$P_3$ has two consecutive edges and no edge between its endpoints (Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices).

Verification

technique · direct

1.1

If every component is complete, three vertices in one component induce all three edges, while vertices meeting more than one component induce edges only within their component parts and therefore at most one edge. Neither pattern is $P_3$ , so $G$ is $P_3$ -free.

L1F1F2

1.2

Conversely, suppose a component contains nonadjacent vertices $x,y$ . Choose a shortest $x$ - $y$ path $x=v_0,v_1,\ldots,v_k=y$ , where $k\ge2$ .

chooseL1

2.1

Minimality of the path gives $v_0v_2\notin E(G)$ , so $G[\{v_0,v_1,v_2\}]\cong P_3$ .

step 1.2F2

3.1

Therefore a $P_3$ -free graph cannot have a noncomplete component. Combining both directions proves the equivalence.

step 1.1step 2.1F1∎

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: 22 results over 10 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

ISGCI, Cluster graphs (standard reference, not scraped)

A graph is P3P_3P3​-free if and only if every connected component is complete