More on spanning 2-connected subgraphs in truncated rectangular grid graphs

A grid graph is a finite induced subgraph of the infinite 2-dimensional grid defined by $Z \times Z$ and all edges between pairs of vertices from $Z \times Z$ at Euclidean distance precisely 1. An $m\times n$-rectangular grid graph is induced by all vertices with coordinates $1$ to $m$ and $1$ to $n$, respectively. A natural drawing of a (rectangular) grid graph $G$ is obtained by drawing its vertices in $\mathbb{R}^2$ according to their coordinates. We consider a subclass of the rectangular grid graphs obtained by deleting some vertices from the corners. Apart from the outer face, all (inner) faces of these graphs have area one (bounded by a 4-cycle) in a natural drawing of these graphs. We determine which of these graphs contain a Hamilton cycle, i.e. a cycle containing all vertices, and solve the problem of determining a spanning 2-connected subgraph with as few edges as possible for all these graphs.