DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-02

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Vertex covers, edge covers and the covering numbers $τ (G)$ and $ρ (G)$

Definition

For a finite simple graph $G = (V, E)$ , a vertex cover is a set $C \subseteq V$ such that every edge has an endpoint in $C$ ; its least possible size is $τ (G) : = \min {∣ C ∣ : C is a vertex cover of G} .$ An edge cover is a set $F \subseteq E$ such that every vertex is incident with an edge of $F$ ; if $G$ has no isolated vertices, its least possible size is $ρ (G) : = \min {∣ F ∣ : F is an edge cover of G} .$ Both extrema exist whenever their displayed families are nonempty, since $V$ and $E$ are finite. An isolated vertex prevents every edge cover.

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. Goemans, Lecture notes on bipartite matching (standard reference, not scraped)

Vertex covers, edge covers and the covering numbers τ(G)τ(G) and ρ(G)ρ(G)