Greedoids on Vertex Sets of Unicycle Graphs

A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set of G, if S is a maximum stable set of the subgraph induced by its closed neighborhood. It is known that the family of all local maximum stable sets of a forest forms a greedoid on its vertex set. Bipartite, triangle-free, and well-covered graphs whose families of local maximum stable sets form greedoids have been analyzed as well. A unicycle graph owns only one cycle. In this paper we characterize the unicycle graphs whose families of local maximum stable sets form greedoids.

💡 Research Summary

The paper investigates the combinatorial structure formed by all local maximum stable sets (LMSS) of a graph G, focusing on the class of unicycle graphs—connected graphs that contain exactly one cycle. A stable set is a set of pairwise non‑adjacent vertices; a local maximum stable set S is a stable set that is maximum within the subgraph induced by its closed neighbourhood N