On well-covered triangulations: Part III

A graph GG is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is a triangle is called a triangulation. It was proved in an earlier paper Finbow et al. (2004) [3] that there are no 5-connected planar well-covered triangulations, and in Finbow et al. (submitted for publication) [4] that there are exactly four 4-connected well-covered triangulations containing two adjacent vertices of degree 4. It is the aim of the present paper to complete the characterization of 4-connected well-covered triangulations by showing that each such graph contains two adjacent vertices of degree 4.