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A grid graph is a finite induced subgraph of the infinite 2-dimensional grid defined by \( \mathbb{Z} \times \mathbb{Z} \) and all edges between pairs of vertices from \( \mathbb{Z} \times \mathbb{Z} \) at Euclidean distance precisely \( 1 \). An \( m \times n \)-rectangular grid graph is induced by all vertices with coordinates from \( 1 \) to \( m \) and from \( 1 \) 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.