Alternating Whitney sums and matchings in trees, part 1
Discrete Mathematics
Finding critical independent sets and critical vertex subsets are polynomial problems
SIAM Journal on Discrete Mathematics
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
On the structure of &agr;-stable graphs
Discrete Mathematics
On the number of vertices belonging to all maximum stable sets of a graph
Discrete Applied Mathematics - Workshop on discrete optimization DO'99, contributions to discrete optimization
Crown reductions for the Minimum Weighted Vertex Cover problem
Discrete Applied Mathematics
The critical independence number and an independence decomposition
European Journal of Combinatorics
Critical Independent Sets and König–Egerváry Graphs
Graphs and Combinatorics
Note: On the intersection of all critical sets of a unicyclic graph
Discrete Applied Mathematics
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Let $G=(V,E)$ be a graph. A set $S\subseteq V$ is independent if no two vertices from $S$ are adjacent, while $\mathrm{core}(G)$ is the intersection of all maximum independent sets [V. E. Levit and E. Mandrescu, Discrete Appl. Math., 117 (2002), pp. 149-161]. The independence number $\alpha(G)$ is the cardinality of a largest independent set, and $\mu(G)$ is the size of a maximum matching of $G$. The neighborhood of $A\subseteq V$ is $\mathcal{N}(A)=\{v\in V:\mathcal{N}(v)\cap A\neq\emptyset\}$. The number $d_{c}(G)=\max\{\vert X\vert -\vert \mathcal{N}(X)\vert :X\subseteq V\}$ is called the critical difference of $G$, and $A$ is critical if $\vert A\vert -\vert \mathcal{N}% (A)\vert =d_{c}(G)$ [C. Q. Zhang, SIAM J. Discrete Math., 3 (1990), pp. 431-438]. We define $\mathrm{\ker}(G)$ as the intersection of all critical sets. In this paper we prove that if $d_{c}(G)\geq1$, then $\mathrm{\ker}(G)\subseteq\mathrm{core}(G)$ and $\vert \mathrm{\ker}(G)\vert d_{c}(G) \geq\alpha(G) -\mu(G)$.