Abstract
In this paper, we design the first linear-time algorithm for computing the prime decomposition of a digraph G with regard to the cartesian product. A remarkable feature of our solution is that it computes the decomposition of G from the decomposition of its underlying undirected graph, for which there exists a linear-time algorithm. First, this allows our algorithm to remain conceptually very simple and in addition, it provides new insight into the connexions between the directed and undirected versions of cartesian product of graphs.
This work was partially supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aurenhammer, F., Hagauer, J., Imrich, W.: Cartesian graph factorization at logarithmic cost per edge. Computational Complexity 2, 331–349 (1992)
Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Annals of Mathematics 164(1), 51–229 (2006)
Feder, T.: Product graph representations. Journal of Graph Theory 16, 467–488 (1992)
Feigenbaum, J.: Directed cartesian-product graphs have unique factorizations that can be computed in polynomial time. Discr. Appl. Math. 15, 105–110 (1986)
Feigenbaum, J., Hershberger, J., Schäffer, A.A.: A polynomial time algorithm for finding the prime factors of cartesian-product graphs. Discrete Applied Mathematics 12, 123–138 (1985)
Hammack, R., Imrich, W., Klavzar, S.: Handbook of Product Graphs. CRC Press (2011)
Imrich, W., Peterin, I.: Recognizing cartesian products in linear time. Discrete Mathematics 307(3-5), 472–483 (2007)
Krebs, M., Schmid, J.: Ordering the order of a distributive lattice by itself. Journal of Logic and Algebraic Programming 76, 198–208 (2008)
Sabidussi, G.: Graph multiplication. Mathematische Zeitschrift 72(1), 446–457 (1960)
Spinrad, J.P.: Efficient graph representations. Fields Institute Monographs, vol. 19. American Mathematical Society (2003)
Vizing, V.G.: The cartesian product of graphs. Vyčisl. Sistemy 9, 30–43 (1963)
Walker, J.W.: Strict refinement for graphs and digraphs. Journal of Combinatorial Theory Series B 43(2), 140–150 (1987)
Winkler, P.M.: Factoring a graph in polynomial time. European Journal on Combinatorics 8, 209–212 (1987)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Crespelle, C., Thierry, E., Lambert, T. (2013). A Linear-Time Algorithm for Computing the Prime Decomposition of a Directed Graph with Regard to the Cartesian Product. In: Du, DZ., Zhang, G. (eds) Computing and Combinatorics. COCOON 2013. Lecture Notes in Computer Science, vol 7936. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38768-5_42
Download citation
DOI: https://doi.org/10.1007/978-3-642-38768-5_42
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38767-8
Online ISBN: 978-3-642-38768-5
eBook Packages: Computer ScienceComputer Science (R0)