Greedoids on Vertex Sets of Unicycle Graphs

欧拉分支贪婪拟阵的贪婪拟阵多项式根无关性的几何证明

A geometric proof for the root-independence of the greedoid polynomial of Eulerian branching greedoids

该研究通过定义正则有向拟阵的根多胞体，证明欧拉分支贪婪拟阵的贪婪拟阵多项式等价于对应图拟阵对偶根多胞体的$h^*$-多项式，从而给出其根无关性的几何证明，还得到边反转不改变该多项式的结论。

Local Maximum Stable Sets Greedoids Stemmed from Very Well-Covered Graphs

A maximum stable set in a graph G is a stable set of maximum cardinality. S is called a local maximum stable set of G if S is a maximum stable set of the subgraph induced by the closed neighborhood of S. A greedoid (V,F) is called a local maximum stable set greedoid if there exists a graph G=(V,E) such that its family of local maximum stable sets coinsides with (V,F). It has been shown that the family local maximum stable sets of a forest T forms a greedoid on its vertex set. In this paper we demonstrate that if G is a very well-covered graph, then its family of local maximum stable sets is a …

Interval greedoids and families of local maximum stable sets of 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. Nemhauser and Trotter Jr. proved in 1975 that any local maximum stable set is a subset of a maximum stable set of G. In 2002 we showed that the family of all local maximum stable sets of a forest forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, well-covered, while the family of all local maximum stable sets is a greedoid, were analyzed in 2004, 2007, and 2008, respective…

Graph Operations that are Good for Greedoids

S is a local maximum stable set of a graph G, if the set S is a maximum stable set of the subgraph induced by its closed neighborhood. In (Levit, Mandrescu, 2002) we have proved that the family of all local maximum stable sets is a greedoid for every forest. The cases of bipartite graphs and triangle-free graphs were analyzed in (Levit, Mandrescu, 2004) and (Levit, Mandrescu, 2007), respectively. In this paper we give necessary and sufficient conditions for the family of all local maximum stable sets of a graph G to form a greedoid, where G is: (a) the disjoint union of a family of graphs; (b)…

Very Well-Covered Graphs of Girth at least Four and Local Maximum Stable Set Greedoids

A \textit{maximum stable set} in a graph $G$ is a stable set of maximum cardinality. $S$ is a \textit{local maximum stable set} of $G$, and we write $S\inΨ(G)$, if $S$ is a maximum stable set of the subgraph induced by $S\cup N(S)$, where $N(S)$ is the neighborhood of $S$. Nemhauser and Trotter Jr. (1975), proved that any $S\inΨ(G)$ is a subset of a maximum stable set of $G$. In (Levit & Mandrescu, 2002) we have shown that the family $Ψ(T)$ of a forest $T$ forms a greedoid on its vertex set. The cases where $G$ is bipartite, triangle-free, well-covered, while $Ψ(G)$ is a greedoid, were analyze…

Bipartite graphs with uniquely restricted maximum matchings and their corresponding greedoids

A maximum stable set in a graph G is a stable set of maximum size. S is a local maximum stable set if it is a maximum stable set of the subgraph of G spanned by the union of S and N(S), where N(S) is the neighborhood of S. A matching M is uniquely restricted if its saturated vertices induce a subgraph which has a unique perfect matching, namely M itself. One theorem of Nemhauser and Trotter Jr., working as a useful sufficient local optimality condition for the weighted maximum stable set problem, ensures that any local maximum stable set of G can be enlarged to a maximum stable set of G. In on…