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.
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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
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DOI: https://doi.org/10.1007/978-3-642-34611-8_17
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