An explicit construction of optimal dominating and [1, 2]–dominating sets in grid

Abstract

A dominating set in a graph G is a subset of vertices D such that every vertex in is a neighbor of some vertex of D. The domination number of G is the minimum size of a dominating set of G and it is denoted by γ(G). A dominating set with cardinality γ(G) is called optimal dominating set. Also, a subset D of a graph G is a [1, 2]-set if, each vertex is adjacent to either one or two vertices in D and the minimum cardinality of [1, 2]-dominating set of G, is denoted by . Chang’s conjecture says that for every and this conjecture has been proven by Goncalves et al. This paper presents an explicit constructing method to find an optimal dominating set for grid graph Gm,n where in O (size of answer). In addition, we will show that where holds in response to an open question posed by Chellali et al.