Path cover

Given a directed graph G = (V, E), 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 least number of 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 in a matching problem.

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 (2002).
  3. Diestel (2005), Theorem 2.5.1.
  4. Ntafos & Hakimi (1979)

References
  • Bang-Jensen, Jørgen; Gutin, Gregory (2006), Digraphs: Theory, Algorithms and Applications (1st ed.), Springer.
  • Diestel, Reinhard (2005), Graph Theory (3rd ed.), Springer.
  • Franzblau, D. S.; Raychaudhuri, A. (2002), "Optimal Hamiltonian completions and path covers for trees, and a reduction to maximum flow", ANZIAM Journal, 44 (2): 193–204, doi:10.1017/S1446181100013894.
  • Ntafos, S. C.; Hakimi, S. Louis. (1979), "On path cover problems in digraphs and applications to program testing", IEEE Transactions on Software Engineering, 5 (5): 520–529, doi:10.1109/TSE.1979.234213.
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