## Abstract A graph is well covered if every maximal independent set has the same cardinality. A vertex __x__, in a wellβcovered graph __G__, is called extendable if __G β {x}__ is well covered and Ξ²(__G__) = Ξ²(__G β {x}__). If __G__ is a connected, wellβcovered graph containing no 4β nor 5βcycles
The structure of well-covered graphs with no cycles of length 4
β Scribed by J.I. Brown; R.J. Nowakowski; I.E. Zverovich
- Book ID
- 108113750
- Publisher
- Elsevier Science
- Year
- 2007
- Tongue
- English
- Weight
- 187 KB
- Volume
- 307
- Category
- Article
- ISSN
- 0012-365X
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The main theorem of that paper is the following: let G be a graph of order n, of size at least (nZ -3n + 6 ) / 2 . For any integers k, n,, n2,. . . , nk such that n = n, + n2 + ... + nk and n, 2 3, there exists a covering of the vertices of G by disjoint cycles (C,),=,..,k with ICjl = n,, except whe
A graph is well-covered if all its maximal independent sets are of the same cardinality. Deciding whether a given graph is well-covered is known to be NP-hard in general, and solvable in polynomial time, if the input is restricted to certain families of graphs. We present here a simple structural ch