Edge Cover

An edge cover is a subset of edges defined similarly to the vertex cover (Skiena 1990, p. 219), namely a collection of graph edges such that the union of edge endpoints corresponds to the entire vertex set of the graph. Therefore, only graphs with no isolated points have an edge cover.

A graph can be tested in the Wolfram Language to see if it is an edge cover of a given graph using EdgeCoverQ[g]. Precomputed edge covers for many named graphs can be looked up using GraphData[graph, "EdgeCovers"].

An edge cover having the smallest possible number of edges for a given graph is known as a minimum edge cover. A minimum edge cover of a graph can be found in the Wolfram Language using FindEdgeCover[g]. An edge cover that does not contain any other edge cover as a proper subset is known as a minimal edge cover.

REFERENCES

Pemmaraju, S. and Skiena, S. Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Cambridge, England: Cambridge University Press, p. 318, 2003.

Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 178, 1990.