Exploring Metric and Strong Metric Dimensions in Inclusion Ideal Graphs of Commutative Rings

TL;DR

This paper investigates the metric and strong metric dimensions of inclusion ideal graphs derived from commutative rings, providing structural insights and explicit calculations for these graph invariants.

Contribution

It introduces the analysis of metric and strong metric dimensions in inclusion ideal graphs of commutative rings, including structural characterizations and explicit computations.

Findings

Determined the metric dimension of inclusion ideal graphs.

Characterized the structure of the resolving graph of these graphs.

Computed the strong metric dimension for specific cases.

Abstract

The inclusion ideal graph of a commutative unitary ring $R$ is the (undirected) graph $In(R)$ whose vertices all non-trivial ideals of $R$ and two distinct vertices are adjacent if and only if one of them is a proper subset of the other one. In this paper, the metric dimension of $In(R)$ is discussed. Moreover, the structure of the resolving graph of $In(R)$ is characterized and as an application, we compute the strong metric dimension of $In(R)$.