Edge cover
a set of edges that touches every vertex of a graph
In graph theory, an edge cover of a graph is a set of edges such that every vertex of the graph is an endpoint of at least one edge of the set.
In computer science, the minimum edge cover problem is the problem of finding an edge cover of minimum size. It is an optimization problem that belongs to the class of covering problems and can be solved in polynomial time.
Definition
Formally, an edge cover of a graph G is a set of edges C such that each vertex in G is incident with at least one edge in C. The set C is said to cover the vertices of G. The following figure shows examples of edge coverings in two graphs (the set C is marked with red).
A minimum edge covering is an edge covering of smallest possible size. The edge covering number ρ(G) is the size of a minimum edge covering. The following figure shows examples of minimum edge coverings (again, the set C is marked with red).
Note that the figure on the right is not only an edge cover but also a matching. In particular, it is a perfect matching: a matching M in which every vertex is incident with exactly one edge in M. A perfect matching (if it exists) is always a minimum edge covering.
Examples
The set of all edges is an edge cover, assuming that there are no degree-0 vertices.
Begin with the source’s own compact description: “Edge cover” is a set of edges that touches every vertex of a graph. The dossier treats that line as a proposition to test through Edge, cover and edges, not as a finished interpretation.
Why this record matters
The phrase “a set of edges that touches every vertex of a graph” supplies a clear boundary for inquiry. It also exposes the unanswered questions: who defined that boundary, when it became stable and which sources sit outside it.
Vocabulary and entity names are the principal evidence signals here, because they determine the precision of every later search. The source revision retrieved here is dated Jun 15, 2025. The linked authority identifier is Q594001.
A concise general-reference account can conceal disagreements about scope, terminology or the weight assigned to individual sources. The lead is largely declarative, so disagreement and counter-evidence require a deliberate search beyond the opening account. Authority statements aid reconciliation but still require their own references, qualifiers and ranks to be checked.
How to read it
Use the entry as an orientation point, then follow its citations and revision history. Names, dates and institutional relationships should be checked against the original record.
- Subject orientation
- Search vocabulary
- Locating named sources
The closest primary source, responsible institution and strongest cited specialist reference.
Three-step research path
- Establish the record: confirm the title “Edge cover”, its source revision and the description used here.
- Expand the search: follow Edge cover primary sources, Edge cover archive and Edge research across catalogues and specialist indexes.
- Test the account: compare the strongest cited source with the responsible institution’s current record and note any disagreement.
Questions for further research
- Which source most directly establishes the central claim about “Edge cover”?
- What terminology or title could unlock a more precise catalogue search?
- Which cited source is closest to the event, object or claim?
Search terms from this dossier
This entry incorporates text from “Edge cover” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.