Abstract
We consider the problem of recognizing whether a simple undirected graph is a P 4-comparability graph. This problem has been considered by Hoàng and Reed who described an O(n4)-time algorithm for its solution, where n is the number of vertices of the given graph. Faster algorithms have recently been presented by Raschle and Simon and by Nikolopoulos and Palios; the time complexity of both algorithms is O(n + m 2), where m is the number of edges of the graph. In this paper, we describe an O(nm)-time, O(n+m)-space algorithm for the recognition of P4-comparability graphs. The algorithm computes the P 4s of the input graph G by means of the BFS-trees of the complement of G rooted at each of its vertices, without however explicitly computing the complement of G. Our algorithm is simple, uses simple data structures, and leads to an O(n m)-time algorithm for computing an acyclic P 4- transitive orientation of a P 4-comparability graph.
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References
A.V. Aho, J.E. Hopcroft, and J.D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974.
S.R. Arikati and U.N. Peled, A polynomial algorithm for the parity path problem on perfectly orderable graphs, Discrete Appl. Math. 65, 5–20, 1996.
A. Brandstädt, V.B. Lê, and J.P. Spinrad, Graph Classes: A Survey, Monographs on Discrete Mathematics and Applications 3, SIAM, 1999.
V. Chvátal, Perfectly ordered graphs, Annals of Discrete Math. 21, 63–65, 1984.
E. Dahlhaus, J. Gustedt, and R.M. McConnell, Efficient and practical modular decomposition, Proc. 8th ACM-SIAM Symp. on Discrete Algorithms (SODA’97), 26–35, 1997.
C.M.H. de Figueiredo, J. Gimbel, C.P. Mello, and J.L. Szwarcfiter, Even and odd pairs in comparability and in P 4-comparability graphs, Discrete Appl. Math. 91, 293–297, 1999.
P.C. Gilmore and A.J. Hoffman, A characterization of comparability graphs and of interval graphs, Canad. J. Math. 16, 539–548, 1964.
M.C. Golumbic, Algorithmic graph theory and perfect graphs, Academic Press, Inc., New York, 1980.
M.C. Golumbic, D. Rotem, and J. Urrutia, Comparability graphs and intersection graphs, Discrete Math. 43, 37–46, 1983.
C.T. Hoàng, Efficient algorithms for minimum weighted colouring of some classes of perfect graphs, Discrete Appl. Math. 55, 133–143, 1994.
C.T. Hoàng and B.A. Reed, Some classes of perfectly orderable graphs, J. Graph Theory 13, 445–463, 1989.
C.T. Hoàng and B.A. Reed, P 4-comparability graphs, Discrete Math. 74, 173–200, 1989.
H. Ito and M. Yokoyama, Linear time algorithms for graph search and connectivity determination on complement graphs, Inform. Process. Letters 66, 209–213, 1998.
R.M. McConnell and J. Spinrad, Linear-time transitive orientation, Proc. 8th ACM-SIAM Symp. on Discrete Algorithms (SODA’97), 19–25, 1997.
M. Middendorf and F. Pfeiffer, On the complexity of recognizing perfectly orderable graphs, Discrete Math. 80, 327–333, 1990.
S.D. Nikolopoulos and L. Palios, Recognition and orientation algorithms for P 4-comparability graphs, Proc. 12th Symp. on Algorithms and Computation (ISAAC’01), 320–331, 2001.
T. Raschle and K. Simon, On the P 4-components of graphs, Discrete Appl. Math. 100, 215–235, 2000.
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© 2002 Springer-Verlag Berlin Heidelberg
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Nikolopoulos, S.D., Palios, L. (2002). On the Recognition of P 4 -Comparability Graphs. In: Goos, G., Hartmanis, J., van Leeuwen, J., Kučera, L. (eds) Graph-Theoretic Concepts in Computer Science. WG 2002. Lecture Notes in Computer Science, vol 2573. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36379-3_31
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DOI: https://doi.org/10.1007/3-540-36379-3_31
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