Path cover

From HandWiki

Given a directed graph G = (VE), a path cover is a set of directed paths such that every vertex v ∈ V belongs to at least one path. Note that a path cover may include paths of length 0 (a single vertex).[1] A path cover may also refer to a vertex-disjoint path cover, i.e., a set of paths such that every vertex v ∈ V belongs to exactly one path.[2]

Properties

A theorem by Gallai and Milgram shows that the number of paths in a smallest path cover cannot be larger than the number of vertices in the largest independent set.[3] In particular, for any graph G, there is a path cover P and an independent set I such that I contains exactly one vertex from each path in P. Dilworth's theorem follows as a corollary of this result.

Computational complexity

Given a directed graph G, the minimum path cover problem consists of finding a path cover for G having the fewest paths.

A minimum path cover consists of one path if and only if there is a Hamiltonian path in G. The Hamiltonian path problem is NP-complete, and hence the minimum path cover problem is NP-hard. However, if the graph is acyclic, the problem is in complexity class P and can therefore be solved in polynomial time by transforming it into a matching problem, see https://walkccc.me/CLRS/Chap26/Problems/26-2/.

Applications

The applications of minimum path covers include software testing.[4] For example, if the graph G represents all possible execution sequences of a computer program, then a path cover is a set of test runs that covers each program statement at least once.

See also

  • Covering (disambiguation)#Mathematics

Notes

  1. (Diestel 2005), Section 2.5.
  2. (Franzblau Raychaudhuri).
  3. (Diestel 2005), Theorem 2.5.1.
  4. (Ntafos Hakimi)

References