Resumen de On maximum independent sets of almost bipartite graphs

Vadim E. Levit, Eugen Mandrescu

• The independence number α(G) is the cardinality of a maximum independent set, whileµ(G) is the size of a maximum matching in the graph G = (V, E). If α(G) + µ(G) equals theorder of G, then G is a K¨onig-Egerv´ary graph [4, 18]. The number d (G) = max{|A|−|N (A)| :A ⊆ V } is called the critical difference of G [19] (where N (A) = {v : v ∈ V, N (v) ∩ A ̸= ∅}).A set X ⊆ V is critical if |X| − |N (X)| = d(G).A graph G is unicyclic if it has a unique cycle and almost bipartite if it has only one oddcycle.Let ker(G) = T{S : S is a critical independent set}, core(G) be the intersection of allmaximum independent sets, and corona(G) be the union of all maximum independent sets.It is known that ker(G) ⊆ core(G) is true for every graph [14], while the equality holds forbipartite graphs [15], and for unicyclic non-K¨onig-Egerv´ary graphs [16].In this paper, we prove that if G is an almost bipartite non-K¨onig-Egerv´ary graph, then:(i) core(G) = ker(G), like for bipartite graphs;(ii) corona(G) ∪ N(core(G)) = V , like for K¨onig-Egerv´ary graphs.