Distance matching in punctured planar triangulations

Distance matching extension with prescribed and proscribed edges in planar triangulations has been previously studied. In the present work, matching extension behavior is investigated when the graph families are slightly more general than triangulations. More particularly, we replace the triangulation hypothesis with the weaker hypotheses that (a) the graph is locally connected and (b) the graph has at most two non-triangular faces. We investigate which distance matching properties enjoyed by triangulations are retained and which are lost.