1Department of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente P.O. Box 217, 7500 AE Enschede, The Netherlands
2Department of Discrete and Statistical Sciences, Auburn University, Auburn, Alabama 36849, USA
A grid graph is a finite induced subgraph of the infinite 2-dimensional grid defined by and all edges between pairs of vertices from at Euclidean distance precisely . An -rectangular grid graph is induced by all vertices with coordinates from to and from to , respectively. A natural drawing of a (rectangular) grid graph is obtained by drawing its vertices in 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 -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 -connected subgraph with as few edges as possible for all these graphs.
Keywords: (rectangular) grid graph, Hamilton cycle, spanning 2- connected subgraph