Well-covered graph: every maximal independent set has the same size. No 'bad' greedy choices! K_n: well-covered (all MIS have size 1). Cycles: C_{2k} well-covered (size k), C_{2k+1} NOT (sizes k and k+1). Recognition is co-NP-complete.
How to Use
Select graph:
- Test: Well-covered?
- MIS: Sizes equal?
- α: Independence #
Theory
Plummer (1970): introduced well-covered graphs. Every very well-covered graph (well-covered + n=2α) has nice structure. Chvátal-Slater: polynomial for K_{1,3}-free graphs. General recognition: co-NP-complete (Sankaranarayana-Stewart).
Examples
Complete graphs K_n: well-covered (MIS size 1). Even cycles C_{2k}: well-covered (size k). Rook's graphs: well-covered. Complete bipartite K_{a,b} (a≠b): NOT well-covered. Pentagon C_5: NOT well-covered.
Step-by-Step Instructions
- 1Select graph.
- 2Find all MIS.
- 3Compare sizes.
- 4Determine well-covered.
- 5Apply properties.