The critical independence number and an independence decomposition

Abstract

An independent set Ic is a critical independent set if |Ic|−|N (Ic)| ≥ |J| − |N (J)|, for any independent set J. The critical independence number of a graph is the cardinality of a maximum critical independent set. This number is a lower bound for the independence number and can be computed in polynomial-time. Any graph can be decomposed into two subgraphs where the independence number of one subgraph equals its critical independence number, where the critical independence number of the other subgraph is zero, and where the sum of the independence numbers of the subgraphs is the independence number of the graph. A proof of a conjecture of Graffiti.pc yields a new characterization of König-Egervary graphs: these are exactly the graphs whose independence and critical independence numbers are equal.

On Some Conjectures Concerning Critical Independent Sets of a Graph

“…Note that for some graphs the empty set is the only critical independent set, for example odd cycles or complete graphs. See [2,7,8,16] for more background and properties of critical independent sets.…”

A parallel algorithm for computing the critical independence number and related sets

“…The first author's Graffiti.pc program conjectured (number 329 in [5]) a characterization in terms of the critical independence number: a graph G is a König-Egerváry graph if, and only if, α(G) = α (G). The conjecture was first proven by Larson in [12]. In [14] Levit & Mandrescu extended the statement of this result as follows.…”

Computing unique maximum matchings in $$O(m)$$ O ( m ) time for König–Egerváry graphs and unicyclic graphs

“…As a well-known example, every bipartite graph is a König-Egerváry graph [6,13]. Several properties of König-Egerváry graphs are presented in [12,15,24,25,28]. Theorem 1 may fail for non-bipartite König-Egerváry graphs; e.g., the graphs G 1 and G 2 from Figure 1 have core(G 1 ) = {a}, and core(G 2 ) = {u}.…”