A General Lower Bound for the Domination Number of Cylindrical Graphs

Abstract

In this paper we present a lower bound for the domination number of the Cartesian product of a path and a cycle, that is tight if the length of the cycle is a multiple of five. This bound improves the natural lower bound obtained by using the domination number of the Cartesian product of two paths, that is the best one known so far.A dominating set in a graph G is a vertex subset S such that every vertex in V (G) \ S has a neighbor in S. The domination number of G is the minimum size of a dominating set of G and it is denoted by γ(G).The domination number is a classical graph parameter that has a challenging development in Cartesian product graphs. As proof of this, the computation of the domination number of the grid graphs, that is, the Cartesian product of two paths, was an open problem for almost thirty years. The final paper concerning this problem [12] closes a list of partial results regarding exact values for particular cases [1,5,7,15] and general upper and lower bounds [4,7,13], among other papers.Contrary to grids, the complete computation of the domination number of cylindrical graphs, that is, the Cartesian product of a path and a cycle, is still open. Currently, there are some partial results, see for instance [8,16,19], showing different techniques to compute exact values of particular cases, where one of the parameters, the path length or the cycle length, is small enough.We devote this paper to computing a general lower bound for the domination number of the cylindrical graphs, that is tight if the length of the cycle is a multiple of five. Our strategy to obtain this bound is similar to the approach in [13], that uses the concept of wasted domination. On the other hand, we found inspiration in [16] to design a (min, +) matrix multiplication algorithm that computes the minimum wasted domination needed for our purpose.