The concept of minimal resolving partition and resolving set plays a pivotal role in diverse areas such as robot navigation, networking, optimization, mastermind games and coin weighing. It is hard to compute exact values of partition dimension for a graphic metric space, (G, dG) and networks. In this article, we give the sharp upper bounds and lower bounds for the partition dimension of generalized Möbius ladders, Mm, n, for all n≥3 and m≥2.

In this article, we develop the theory of SAGBI bases in G-algebras and create a criterion through which we can check if a set of polynomials in a G-algebra is a SAGBI basis or not. Moreover, we will construct an algorithm to compute SAGBI bases from a subset of polynomials contained in a subalgebra of a G-algebra.

Let $R$ be a commutative ring such that $R=R_1\times \cdots \times R_n$. In this paper, we give a method to compute (strong) Sagbi bases for subalgebras of a polynomial ring over $R$ provided that these bases are computable in a polynomial ring over $R_i$ for $1\leq i \leq n$. As an application, we prove the existence of strong Sagbi bases for subalgebras in a polynomial ring over a principal ideal ring.