Decomposition dimension of Cartesian product of some graphs

Abstract

For an ordered [Formula: see text]-decomposition [Formula: see text] of a connected graph [Formula: see text], the [Formula: see text]-representation of an edge [Formula: see text] is the [Formula: see text]-tuple [Formula: see text], where [Formula: see text] represents the distance from [Formula: see text] to [Formula: see text]. A decomposition [Formula: see text] is resolving if every two distinct edges of [Formula: see text] have distinct representations. The minimum [Formula: see text] for which [Formula: see text] has a resolving [Formula: see text]-decomposition is its decomposition dimension [Formula: see text]. In this paper, decomposition dimension of Cartesian product of paths, cycles and stars is studied.