On the critical difference of almost bipartite graphs

A set S subset of V is independent in a graph G = (V, E) if no two vertices from S are adjacent. The independence number alpha(G) is the cardinality of a maximum independent set, while mu(G) is the size of a maximum matching in G. If alpha(G) + mu(G) equals the order of G, then G is called a Konig-Egervary graph (Deming in Discrete Math 27:23-33, 1979; Sterboul in J Combin Theory Ser B 27:228-229, 1979). The number d (G) = max{vertical bar A vertical bar - vertical bar N (A)vertical bar : A subset of V} is called the critical difference of G (Zhang in SIAM J Discrete Math 3:431-438, 1990) (where N (A) = {v : v is an element of V, N (v) boolean AND A not equal theta}). It is known that alpha(G) - mu(G) <= d (G) holds for every graph (Levit and Mandrescu in SIAM J Discrete Math 26:399-403, 2012; Lorentzen in Notes on covering of arcs by nodes in an undirected graph, Technical report ORC 66-16. University of California, Berkeley, CA, Operations Research Center, 1966; Schrijver in Combinatorial optimization. Springer, Berlin, 2003). In Levit and Mandrescu (Graphs Combin 28:243-250, 2012), it was shown that d(G) = alpha(G) - mu(G) is true for every Konig-Egervary graph. A graph G is (i) unicyclic if it has a unique cycle and (ii) almost bipartite if it has only one odd cycle. It was conjectured in Levit and Mandrescu (in: Abstracts of the SIAM conference on discrete mathematics, Halifax, Canada, p 40, abstract MS21, 2012, 3rd international conference on discrete mathematics, June 10-14, Karnatak University. Dharwad, India, 2013) and validated in Bhattacharya et al. (Discrete Math 341:1561-1572, 2018) that d(G) = alpha(G)-mu(G) holds for every unicyclic non-Konig-Egervary graph G. In this paper, we prove that if G is an almost bipartite graph of order n (G), then alpha(G) + mu(G) is an element of {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) = alpha(G) - mu(G) = vertical bar core(G)vertical bar - vertical bar N(core(G))vertical bar, where by core(G) we mean the intersection of all maximum independent sets.