Abstract
We define the stable degree s(G) of a graph G by s(G) = min U max v ∈ U d G (v), where the minimum is taken over all maximal independent sets U of G. For this new parameter we prove the following. Deciding whether a graph has stable degree at most k is NP-complete for every fixed k ≥ 3; and the stable degree is hard to approximate. For asteroidal triple-free graphs and graphs of bounded asteroidal number the stable degree can be computed in polynomial time. For graphs in these classes the treewidth is bounded from below and above in terms of the stable degree.
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
Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM Journal on Algebraic and Discrete Methods 8, 277–284 (1987)
Bodlaender, H.L., Koster, A.M.C.A.: Treewidth computations II. Lower bounds. Information & Computation 209, 1103–1119 (2011)
Bouchitté, V., Todinca, I.: Approximating the treewidth of AT-free graphs. Discrete Applied Mathematics 131, 11–37 (2003)
Bouchitté, V., Kratsch, D., Müller, H., Todinca, I.: On treewidth approximations. Discrete Applied Mathematics 136, 183–196 (2004)
Broersma, H.-J., Kloks, T., Kratsch, D., Müller, H.: Independent sets in asteroidal triple-free graphs. SIAM Journal on Discrete Mathematics 12, 276–287 (1999)
Broersma, H.-J., Kloks, T., Kratsch, D., Müller, H.: A generalization of AT-free graphs and a generic algorithm for solving triangulation problems. Algorithmica 32, 594–610 (2002)
Corneil, D.G., Olariu, S., Stewart, L.: Asteroidal triple-free graphs. SIAM Journal on Discrete Mathematics 10, 399–430 (1997)
Courcelle, B.: The monadic second-order logic of graphs. III: Tree-decompositions, minors and complexity issues. RAIRO. Informatique Théorique et Applications 26, 257–286 (1992)
Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems 33, 125–150 (2000)
Kloks, T., Kratsch, D., Spinrad, J.: On treewidth and minimum fill-in of asteroidal triple-free graphs. Theoretical Computer Science 175, 309–335 (1997)
Kloks, T., Kratsch, D., Müller, H.: On the structure of graphs with bounded asteroidal number. Graphs and Combinatorics 17, 295–306 (2001)
Kratsch, D.: Domination and total domination on asteroidal triple-free graphs. Discrete Applied Mathematics 99, 111–123 (2000)
Lekkerkerker, C.G., Boland, J.C.: Representation of a finite graph by a set of intervals on the real line. Fundamenta Mathematicae 51, 45–64 (1962)
Möhring, R.H.: Triangulating graphs without asteroidal triples. Discrete Applied Mathematics 64, 281–287 (1996)
Ramachandramurthi, S.: A Lower Bound for Treewidth and its Consequences. In: Mayr, E.W., Schmidt, G., Tinhofer, G. (eds.) WG 1994. LNCS, vol. 903, pp. 14–25. Springer, Heidelberg (1995)
Tovey, C.A.: A simplified NP-complete satisfiability problem. Discrete Applied Mathematics 8, 85–89 (1984)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Müller, H. (2012). On the Stable Degree of Graphs. In: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2012. Lecture Notes in Computer Science, vol 7551. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34611-8_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-34611-8_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34610-1
Online ISBN: 978-3-642-34611-8
eBook Packages: Computer ScienceComputer Science (R0)