On the stable degree of graphs

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. © 2012 Springer-Verlag Berlin Heidelberg.