منابع مشابه
On Duality between Local Maximum Stable Sets of a Graph and Its Line-Graph
G is a König-Egerváry graph provided α(G) +μ(G) = |V (G)|, where μ(G) is the size of a maximum matching and α(G) is the cardinality of a maximum stable set, [3], [22]. S is a local maximum stable set of G, and we write S ∈ Ψ(G), if S is a maximum stable set of the subgraph induced by S ∪N(S), where N(S) is the neighborhood of S, [12]. Nemhauser and Trotter Jr. proved that any S ∈ Ψ(G) is a subs...
متن کاملCombinatorial properties of the family of maximum stable sets of a graph
The stability number α(G) of a graph G is the size of a maximum stable set of G, core(G) = ∩{S : S is a maximum stable in G}, and ξ(G) = |core(G)|. In this paper we prove that for a graph G without isolated vertices, the following assertions are true: (i) if ξ(G) ≤ 1, then G is quasi-regularizable; (ii) if G is of order n and α(G) > (n + k − 1)/2, for some k ≥ 1, then ξ(G) ≥ k + 1, and ξ(G) ≥ k...
متن کاملOn the number of vertices belonging to all maximum stable sets of a graph
Let us denote by (G) the size of a maximum stable set, and by (G) the size of a maximum matching of a graph G, and let (G) be the number of vertices which belong to all maximum stable sets. We shall show that (G)¿ 1+ (G)− (G) holds for any connected graph, whenever (G)¿ (G). This inequality improves on related results by Hammer et al. (SIAM J. Algebraic Discrete Methods 3 (1982) 511) and by Lev...
متن کاملDifferent-Distance Sets in a Graph
A set of vertices $S$ in a connected graph $G$ is a different-distance set if, for any vertex $w$ outside $S$, no two vertices in $S$ have the same distance to $w$.The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.We prove that a different-distance set induces either a special type of path or an independent...
متن کاملCritical and Maximum Independent Sets of a Graph
Let G be a simple graph with vertex set V (G). A set S ⊆ V (G) is independent if no two vertices from S are adjacent. By Ind(G) we mean the family of all independent sets of G, while core (G) and corona (G) denote the intersection and the union of all maximum independent sets, respectively. The number d (X) = |X| − |N(X)| is the difference of X ⊆ V (G), and a set A ∈ Ind(G) is critical if d(A) ...
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ژورنال
عنوان ژورنال: Journal of Mathematical Analysis and Applications
سال: 1969
ISSN: 0022-247X
DOI: 10.1016/0022-247x(69)90233-9