On the Critical Difference of Almost Bipartite Graphs
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A set S⊆ V is independent in a graph G=( V,E) if no two vertices from S are adjacent. The independence number α(G) is the cardinality of a maximum independent set, while μ(G) is the size of a maximum matching in G. If α(G)+μ(G) equals the order of G, then G is called a König-Egerváry graph dem,ster. The number d( G) ={ A - N( A) :A⊆ V} is called the critical difference of G Zhang (where N( A) ={ v:v∈ V,N( v) ∩ A≠∅} ). It is known that α(G)-μ(G)≤ d( G) holds for every graph Levman2011a,Lorentzen1966,Schrijver2003. In LevMan5 it was shown that d(G)=α(G)-μ(G) is true for every König-Egerváry graph. A graph G is (i)unicyclic if it has a unique cycle, (ii)almost bipartite if it has only one odd cycle. It was conjectured in LevMan2012a,LevMan2013a and validated in Bhattacharya2018 that d(G)=α(G)-μ(G) holds for every unicyclic non-König-Egerváry graph G. In this paper we prove that if G is an almost bipartite graph of order n( G) , then α(G)+μ(G)∈{ n( G) -1,n( G) } . Moreover, for each of these two values, we characterize the corresponding graphs. Further, using these findings, we show that the critical difference of an almost bipartite graph G satisfies d(G)=α(G)-μ(G)=core(G) - N(core(G)) , where by core( G) we mean the intersection of all maximum independent sets.
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