Spanning 2-Connected Subgraphs in Alphabet Graphs, Special Classes of Grid Graphs

A grid graph G is a finite induced subgraph of the infinite 2-dimensional grid defined by Z × Z and all edges between pairs of vertices from Z × Z at Euclidean distance precisely 1. A natural drawing of G is obtained by drawing its vertices in R2 according to their coordinates. Apart from the outer face, all (inner) faces with area exceeding one (not bounded by a 4-cycle) in a natural drawing of G are called the holes of G. We define 26 classes of grid graphs called alphabet graphs, with no or a few holes. We determine which of the alphabet 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 alphabet graphs.