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We study the problem of finding a minimum $k$-critical-bipartite graph oforder $(n,m)$: a bipartite graph $G=(U,V;E)$, with $|U|=n$, $|V|=m$, and$n>m>1$, which is $k$-critical-bipartite, and the tuple $(|E|, \Delta_U,\Delta_V)$, where $\Delta_U$ and $\Delta_V$ denote the maximum degree in $U$and $V$, respectively, is lexicographically minimum over all such graphs. $G$is $k$-critical-bipartite if deleting at most $k=n-m$ vertices from $U$ yields$G'$ that has a complete matching, i.e., a matching of size $m$. Cichacz andSuchan solved the problem for biregular bipartite graphs. Here, we extend theirresults to bipartite graphs that are not biregular. We also prove tight lowerbounds on the connectivity of $k$-critical-bipartite graphs.

Minimum $k$-critical-bipartite graphs: the irregular Case