Abstract
The compaction problem is to partition the vertices of an input graph G onto the vertices of a fixed target graph H, such that adjacent vertices of G remain adjacent in H, and every vertex and non-loop edge of H is covered by some vertex and edge of G respectively, i.e., the partition is a homomorphism of G onto H (except the loop edges). Various computational complexity results, including both NP-completeness and polynomial time solvability, have been presented earlier for this problem for various class of target graphs H. In this paper, we pay attention to the input graphs G, and present polynomial time algorithms for the problem for some class of input graphs, keeping the target graph H general as any reflexive or irreflexive graph. Our algorithms also give insight as for which instances of the input graphs, the problem could possibly be NP-complete for certain target graphs. With the help of our results, we are able to further refine the structure of the input graph that would be necessary for the problem to be possibly NP-complete, when the target graph is a cycle. Thus, when the target graph is a cycle, we enhance the class of input graphs for which the problem is polynomial time solvable.
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
Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. The MIT Press, McGraw-Hill Book Company, Cambridge, Massachusetts, New York (1990)
Feder, T., Hell, P., Klein, S., Motwani, R.: Complexity of Graph Partition Problems. In: Proceedings of the 31st Annual ACM Symposium on Theory of Computing (STOC), Atlanta, Georgia (1999)
Feder, T., Hell, P., Klein, S., Motwani, R.: List Partitions. SIAM Journal on Discrete Mathematics 16, 449–478 (2003)
Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)
Hell, P., Nesetril, J.: On the Complexity of H-colouring. Journal of Combinatorial Theory, Series B 48, 92–110 (1990)
Vikas, N.: Computational Complexity of Compaction to Cycles. In: Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Baltimore, Maryland (1999)
Vikas, N.: Connected and Loosely Connected List Homomorphisms. In: Kučera, L. (ed.) WG 2002. LNCS, vol. 2573, pp. 399–412. Springer, Heidelberg (2002)
Vikas, N.: Computational Complexity of Compaction to Reflexive Cycles. SIAM Journal on Computing 32, 253–280 (2003)
Vikas, N.: Computational Complexity of Compaction to Irreflexive Cycles. Journal of Computer and System Sciences 68, 473–496 (2004a)
Vikas, N.: Compaction, Retraction, and Constraint Satisfaction. SIAM Journal on Computing 33, 761–782 (2004b)
Vikas, N.: Computational Complexity Classification of Partition under Compaction and Retraction. In: Chwa, K.-Y., Munro, J.I.J. (eds.) COCOON 2004. LNCS, vol. 3106, pp. 380–391. Springer, Heidelberg (2004c)
Vikas, N.: A Complete and Equal Computational Complexity Classification of Compaction and Retraction to All Graphs with at most Four Vertices. Journal of Computer and System Sciences 71, 406–439 (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Vikas, N. (2011). Algorithms for Partition of Some Class of Graphs under Compaction. In: Fu, B., Du, DZ. (eds) Computing and Combinatorics. COCOON 2011. Lecture Notes in Computer Science, vol 6842. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22685-4_29
Download citation
DOI: https://doi.org/10.1007/978-3-642-22685-4_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22684-7
Online ISBN: 978-3-642-22685-4
eBook Packages: Computer ScienceComputer Science (R0)