On the number of vertices belonging to all maximum stable sets of a graph

Authors:
Endre Boros;Martin C. Golumbic;Vadim E. Levit
Affiliations:
RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway, NJ;Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel;Department of Computer Science, Holon Academic Institute of Technology, 52 Golomb Str., P.O. B 305, Holon 58102, Israel
Venue:
Discrete Applied Mathematics - Workshop on discrete optimization DO'99, contributions to discrete optimization
Year:
2002

Citing 2
Cited 8

The structure and maximum number of maximum independent sets in trees

Journal of Graph Theory
Graphs whose vertex independence number is unaffected by single edge addition or deletion

Discrete Applied Mathematics

Crown reductions for the Minimum Weighted Vertex Cover problem

Discrete Applied Mathematics
The union of minimal hitting sets: Parameterized combinatorial bounds and counting

Journal of Discrete Algorithms
Loss optimal monotone relabeling of noisy multi-criteria data sets

Information Sciences: an International Journal
The union of minimal hitting sets: parameterized combinatorial bounds and counting

STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
Minimum d-blockers and d-transversals in graphs

Journal of Combinatorial Optimization
Vertices Belonging to All Critical Sets of a Graph

SIAM Journal on Discrete Mathematics
Note: On maximum matchings in König-Egerváry graphs

Discrete Applied Mathematics
Note: On the intersection of all critical sets of a unicyclic graph

Discrete Applied Mathematics

Quantified Score

Hi-index	0.00

Visualization

Abstract

Let us denote by α(G) the size of a maximum stable set, and by µ(G) the size of a maximum matching of a graph G, and let ξ(G) be the number of vertices which belong to all maximum stable sets. We shall show that ξ(G) ≥ 1+α(G)-µ(G) holds for any connected graph, whenever α(G) µ(G). This inequality improves on related results by Hammer et al. (SIAM J. Algebraic Discrete Methods 3 (1982) 511) and by Levit and Mandrescu [(prE-print math. CO/9912047 (1999) 13pp.)].We also prove that on one hand, ξ(G) 0 can be recognized in polynomial time whenever µ(G) V(G)|/3, and on the other hand determining whether ξ(G) k is, in general, NP-complete for any fixed k ≥ 0.