Perfect Domination of Rectangular Grids

Let m and n be positive integers. The algorithmic search for perfect dominating sets of the rectangular grid Gm;n satisfying an initial condition S 0 de ned as an admissible subset of a side Gm;1 of Gm;n such that S \ Gm;1 = S 0 is considered. A binary decision algorithm that generates all such perfect dominating sets is presented, and some related questions and conjectures are posed, leading to consideration of their periodic extendibility to the plane lattice as well as to extremal properties of the grid-depth n for a xed grid-width m. 1 Perfect Dominating Sets A vertex subset S of a graph G is called a perfect dominating set, (PDS), of G if any vertex v of G not in S is adjacent to exactly one element of S. The problem of nding an isolated PDS, called an e cient dominating set, ([6, 9]), or a 1-perfect code, of a graph, is NP-complete, ([10]). The problem of nding a minimal PDS of a planar graph is also NP-complete, ([8], where a di erent terminology is used: what here is called a PDS is referred to as a semiperfect dominating set, while the name `PDS' is reserved for 1-perfect codes). The study of PDS's of n-cubes was addressed in [12, 7, 3, 5, 1, 2, 11, 4]. 1.1 Initial Conditions for PDS's It is natural to consider the problem of existence of a mathematical object given some initial condition(s). For example, in looking for PDS's of a graph G with a distinguished subgraph G (say a cycle G bounding the in nite face of a 2-connected plane graph G), such a condition could be a vertex subset S of G so we have the problem of existence of a PDS S of G such that S\G = S. Proposition 1.1 Given graphs G G and a vertex subset S of G, the following conditions are pairwise equivalent and necessary for the existence of a PDS S of G such that S \G = S: 1. no two components of the induced subgraph G[S] are at distance 2; 2. G[S] has its components at distances 3; This work was performed in part while the author was on sabbatical leave from the University of Puerto Rico for the academic year 1999-2000 and visiting the Mathematics Institute of the Denmark Technical University, from March 1 to May 31, 2000.