# Nontrivial path covers of graphs: existence, minimization and maximization

- 14 Downloads

## Abstract

Let *G* be a graph and \({{{\mathcal {P}}}}\) be a set of pairwise vertex-disjoint paths in *G*. We say that \({{\mathcal {P}}}\) is a *path cover* if every vertex of *G* belongs to a path in \({{\mathcal {P}}}\). In the *minimum path cover problem*, one wishes to find a path cover of minimum cardinality. In this problem, known to be \({\textsc {NP}}\)-hard, the set \({{\mathcal {P}}}\) may contain trivial (single-vertex) paths. We study the problem of finding a path cover composed only of nontrivial paths. First, we show that the corresponding existence problem can be reduced to a matching problem. This reduction gives, in polynomial time, a certificate for both the yes-answer and the no-answer. When trivial paths are forbidden, for the feasible instances, one may consider either minimizing or maximizing the number of paths in the cover. We show that, the minimization problem on feasible instances is computationally equivalent to the minimum path cover problem: their optimum values coincide and they have the same approximation threshold. We show that the maximization problem can be solved in polynomial time. We also consider a weighted version of the path cover problem, in which we seek a path cover with minimum or maximum total weight (the number of paths do not matter), and we show that while the first is polynomial, the second is NP-hard, but admits a constant-factor approximation algorithm. We also describe a linear-time algorithm on (weighted) trees, and mention results for graphs with bounded treewidth.

## Keywords

Covering Min path cover Max path cover [1, 2]-factor Bounded treewidth## Notes

### Acknowledgements

We thank the referees for the valuable suggestions.

## References

- Anstee R (1985) An algorithmic proof of Tutte’s \(f\)-factor theorem. J Algorithms 6(1):112–131MathSciNetzbMATHCrossRefGoogle Scholar
- Arikati SR, Pandu Rangan C (1990) Linear algorithm for optimal path cover problem on interval graphs. Inf Process Lett 35(3):149–153MathSciNetzbMATHCrossRefGoogle Scholar
- Corneil DG, Dalton B, Habib M (2013) LDFS-based certifying algorithm for the minimum path cover problem on cocomparability graphs. SIAM J Comput 42(3):792–807MathSciNetzbMATHCrossRefGoogle Scholar
- Courcelle B (1990) The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf Comput 85(1):12–75. https://doi.org/10.1016/0890-5401(90)90043-H MathSciNetzbMATHCrossRefGoogle Scholar
- Courcelle B, Engelfriet J (2012) Graph structure and monadic second-order logic-a language-theoretic approach, encyclopedia of mathematics and its applications, vol 138. Cambridge University Press, CambridgezbMATHGoogle Scholar
- Dulmage A, Mendelsohn N (1958) Coverings of bipartite graphs. Can J Math 10:517–534MathSciNetzbMATHCrossRefGoogle Scholar
- Franzblau DS, Raychaudhuri A (2002) Optimal Hamiltonian completions and path covers for trees, and a reduction to maximum flow. ANZIAM J 44(2):193–204MathSciNetzbMATHCrossRefGoogle Scholar
- Garey M, Johnson D, Tarjan R (1976) The planar Hamiltonian circuit problem is NP-complete. SIAM J Comput 5(4):704–714MathSciNetzbMATHCrossRefGoogle Scholar
- Georges J, Mauro D, Whittlesey M (1994) Relating path coverings to vertex labellings with a condition at distance two. Discret Math 135(1):103–111MathSciNetzbMATHCrossRefGoogle Scholar
- Gómez R, Wakabayashi Y (2018) Covering a graph with nontrivial vertex-disjoint paths: existence and optimization. In: Graph-theoretic concepts in computer science—44th international workshop, WG 2018, Cottbus, Germany, June 27–29, 2018. Lecture Notes in Computer Science, vol 11159, pp 228–238. https://doi.org/10.1007/978-3-030-00256-5_19 zbMATHCrossRefGoogle Scholar
- Heinrich K, Hell P, Kirkpatrick D, Liu G (1990) A simple existence criterion for \((g<f)\)-factors. Discret Math 85(3):313–317MathSciNetzbMATHCrossRefGoogle Scholar
- Henning M, Wash K (2017) Matchings, path covers and domination. Discret Math 340(1):3207–3216MathSciNetzbMATHCrossRefGoogle Scholar
- Kano M, Saito A (1983) \([a,\, b]\)-factors of graphs. Discret Math 47(1):113–116zbMATHCrossRefGoogle Scholar
- Las Vergnas M (1978) An extension of Tutte’s 1-factor theorem. Discret Math 23(3):241–255MathSciNetzbMATHCrossRefGoogle Scholar
- Li Y, Mao-cheng C (1998) A degree condition for a graph to have \([a, b]\)-factors. J Graph Theory 27(1):1–6MathSciNetzbMATHCrossRefGoogle Scholar
- Lovász L (1970) Subgraphs with prescribed valencies. J Comb Theory 8:391–416MathSciNetzbMATHCrossRefGoogle Scholar
- Lovász L, Plummer M (1986) Matching theory, North-Holland mathematics studies, vol 121. North-Holland, AmsterdamGoogle Scholar
- Magnant C, Martin D (2009) A note on the path cover number of regular graphs. Australas J Comb 43:211–217MathSciNetzbMATHGoogle Scholar
- Moran S, Wolfstahl Y (1991) Optimal covering of cacti by vertex-disjoint paths. Theor Comput Sci 84(2):179–197MathSciNetzbMATHCrossRefGoogle Scholar
- Müller H (1996) Hamiltonian circuits in chordal bipartite graphs. Discret Math 156(1–3):291–298MathSciNetzbMATHCrossRefGoogle Scholar
- Papadimitriou CH, Yannakakis M (1993) The traveling salesman problem with distances one and two. Math Oper Res 18(1):1–11MathSciNetzbMATHCrossRefGoogle Scholar
- Rao M (2007) MSOL partitioning problems on graphs of bounded treewidth and clique-width. Theor Comput Sci 377(1–3):260–267. https://doi.org/10.1016/j.tcs.2007.03.043 MathSciNetzbMATHCrossRefGoogle Scholar
- Reed B (1996) Paths, stars and the number three. Comb Probab Comput 5(3):277–295MathSciNetzbMATHCrossRefGoogle Scholar
- Schrijver A (2003) Combinatorial optimization. Polyhedra and efficiency. Vol A, algorithms and combinatorics, vol 24. Springer, Berlin (Paths, flows, matchings, Chapters 1–38)Google Scholar
- Vishwanathan S (1992) An approximation algorithm for the asymmetric travelling salesman problem with distances one and two. Inf Process Lett 44(6):297–302MathSciNetzbMATHCrossRefGoogle Scholar
- Yu G (2017) Covering 2-connected 3-regular graphs with disjoint paths. J Graph Theory 88(3):385–401MathSciNetzbMATHCrossRefGoogle Scholar