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A graph is well-covered if every independent set can be extended to a maximum independent set. We show that it is co-NP-complete to determine whether an arbitrary graph is well-covered, even when restricted to the family of circulant graphs. Despite the intractability of characterizing the complete set of well-covered circulant graphs, we apply the theory of independence polynomials to show that several...
If sk denotes the number of independent sets of cardinality k and α(G) is the size of a maximum independent set in graph G, then I(G;x)=s0+s1x+⋯+sα(G)xα(G) is the independence polynomial of G (Gutman and Harary, 1983) [8].In this paper we provide an elementary proof of the inequality|I(G;−1)|≤2φ(G) (Engström, 2009) [7], where φ(G) is the decycling number of G (Beineke and Vandell, 1997) [3], namely,...
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