Abstract
We consider a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or kPC for short, on interval graphs. Given a graph G and a subset \(\mathcal{T}\) of k vertices of V(G), a k-fixed-endpoint path cover of G with respect to \(\mathcal{T}\) is a set of vertex-disjoint paths ℘ that covers the vertices of G such that the k vertices of \(\mathcal{T}\) are all endpoints of the paths in ℘. The kPC problem is to find a k-fixed-endpoint path cover of G of minimum cardinality; note that, if \(\mathcal{T}\) is empty the stated problem coincides with the classical path cover problem. In this paper, we study the 1-fixed-endpoint path cover problem on interval graphs, or 1PC for short, generalizing the 1HP problem which has been proved to be NP-complete even for small classes of graphs. Motivated by a work of Damaschke (Discrete Math. 112:49–64, 1993), where he left both 1HP and 2HP problems open for the class of interval graphs, we show that the 1PC problem can be solved in polynomial time on the class of interval graphs. We propose a polynomial-time algorithm for the problem, which also enables us to solve the 1HP problem on interval graphs within the same time and space complexity.
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Adhar, G.S., Peng, S.: Parallel algorithm for path covering, Hamiltonian path, and Hamiltonian cycle in cographs. In: International Conference on Parallel Processing, vol. III: Algorithms and Architecture, pp. 364–365. Pennsylvania State University Press, Pennsylvania (1990)
Arikati, S.R., Rangan, C.P.: Linear algorithm for optimal path cover problem on interval graphs. Inf. Process. Lett. 35, 149–153 (1990)
Asdre, K., Nikolopoulos, S.D.: A linear-time algorithm for the k-fixed-endpoint path cover problem on cographs. Networks 50, 231–240 (2007)
Asdre, K., Nikolopoulos, S.D.: A polynomial solution for the k-fixed-endpoint path cover problem on proper interval graphs. In: Proc. of the 18th International Conference on Combinatorial Algorithms (IWOCA’07), Lake Macquarie, Newcastle, Australia (2007)
Bertossi, A.A.: Finding Hamiltonian circuits in proper interval graphs. Inf. Process. Lett. 17, 97–101 (1983)
Bertossi, A.A., Bonuccelli, M.A.: Finding Hamiltonian circuits in interval graph generalizations. Inf. Process. Lett. 23, 195–200 (1986)
Brandstädt, A., Le, V.B., Spinrad, J.: Graph Classes—A Survey. SIAM Monographs in Discrete Mathematics and Applications. SIAM, Philadelphia (1999)
Damaschke, P.: Paths in interval graphs and circular arc graphs. Discrete Math. 112, 49–64 (1993)
Fortune, S., Hopcroft, J.E., Wyllie, J.: The directed subgraph homeomorphism problem. J. Theor. Comput. Sci. 10, 111–121 (1980)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)
Garey, M.R., Johnson, D.S., Tarjan, R.E.: The planar Hamiltonian circuit problem is NP-complete. SIAM J. Comput. 5, 704–714 (1976)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980) (2nd edn., in Ann. Discrete Math. 57, Elsevier, 2004)
Hsieh, S.Y.: An efficient parallel strategy for the two-fixed-endpoint Hamiltonian path problem on distance-hereditary graphs. J. Parallel Distributed Comput. 64, 662–685 (2004)
Hsieh, S.Y., Ho, C.W., Hsu, T.S., Ko, M.T.: The Hamiltonian problem on distance-hereditary graphs. Discrete Appl. Math. 154, 508–524 (2006)
Hung, R.W., Chang, M.S.: Linear-time algorithms for the Hamiltonian problems on distance-hereditary graphs. Theor. Comput. Sci. 341, 411–440 (2005)
Hung, R.W., Chang, M.S.: Solving the path cover problem on circular-arc graphs by using an approximation algorithm. Discrete Appl. Math. 154, 76–105 (2006)
Karp, R.M.: On the complexity of combinatorial problems. Networks 5, 45–68 (1975)
Keil, J.M.: Finding Hamiltonian circuits in interval graphs. Inf. Process. Lett. 20, 201–206 (1985)
Lin, R., Olariu, S., Pruesse, G.: An optimal path cover algorithm for cographs. Comput. Math. Appl. 30, 75–83 (1995)
Müller, H.: Hamiltonian circuits in chordal bipartite graphs. Discrete Math. 156, 291–298 (1996)
Nakano, K., Olariu, S., Zomaya, A.Y.: A time-optimal solution for the path cover problem on cographs. Theor. Comput. Sci. 290, 1541–1556 (2003)
Nikolopoulos, S.D.: Parallel algorithms for Hamiltonian problems on quasi-threshold graphs. J. Parallel Distributed Comput. 64, 48–67 (2004)
Park, J.H.: One-to-many disjoint path covers in a graph with faulty elements. In: Proc. of the 10th International Computing and Combinatorics Conference (COCOON’04), pp. 392–401 (2004)
Ramalingam, G., Rangan, C.P.: A unified approach to domination problems on interval graphs. Inf. Process. Lett. 27, 271–274 (1988)
Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory Ser. B 63, 65–110 (1995)
Seymour, P.D.: Disjoint paths in graphs. Discrete Math. 29, 293–309 (1980)
Shiloach, Y.: A polynomial solution to the undirected two paths problem. J. Assoc. Comput. Mach. 27, 445–456 (1980)
Srikant, R., Sundaram, R., Singh, K.S., Rangan, C.P.: Optimal path cover problem on block graphs and bipartite permutation graphs. Theor. Comput. Sci. 115, 351–357 (1993)
Suzuki, Y., Kaneko, K., Nakamori, M.: Node-disjoint paths algorithm in a transposition graph. IEICE Trans. Inf. Syst. E89-D, 2600–2605 (2006)
Thomassen, C.: 2-linked graphs. Eur. J. Comb. 1, 371–378 (1980)
Uehara, R., Uno, Y.: On computing longest paths in small graph classes. Int. J. Found. Comput. Sci. 18, 911–930 (2007)
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Asdre, K., Nikolopoulos, S.D. The 1-Fixed-Endpoint Path Cover Problem is Polynomial on Interval Graphs. Algorithmica 58, 679–710 (2010). https://doi.org/10.1007/s00453-009-9292-5
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DOI: https://doi.org/10.1007/s00453-009-9292-5