@techreport{TR-IC-25-03,
   number = {IC-25-03},
   author = {A. A. {Pereira} and C. N. {Campos}},
   title = {{Extending unlocked matchings in graphs}},
   month = {October},
   year = {2025},
   institution = {Institute of Computing, University of Campinas},
   note = {In English, 09 pages.
    \par\selectlanguage{english}\textbf{Abstract}
       Let  $M$ be a matching of a graph $G$ and let $S(M)$ and $U(M)$
       be  the  sets  of $M$-saturated and $M$-unsaturated vertices of
       $G$,  respectively.  A vertex $v \in U(M)$ is unlocked if there
       exists  a  vertex  $u  \in N(v) \cap U(M)$. The matching $M$ is
       unlocked  if  every vertex in $U(M)$ is unlocked. We say that a
       graph  $G$  is  $k$-fully-extendable if every unlocked matching
       $M$ in $G$ of size $k$ can be extended to a perfect matching of
       $G$.  In  this  work,  we  study $k$-fully-extendable graphs of
       order  $2n$,  characterizing the cases $k = n-1$ and $k = n-2$.
       For  the  case  $k  =  n-2$,  we  provide  additional necessary
       conditions,  based  on  degree  bounds,  as  well as sufficient
       conditions, based on classes of graphs.
  }
}