Computing the strong metric dimension for co-maximal ideal graphs of commutative rings

Abstract

Let $R$ be a commutative ring with identity. The co-maximal ideal graph of $R$, denoted by $\Gamma (R)$, is a simple graph whose vertices are proper ideals of $R$ which are not contained in the Jacobson radical of $R$ and two distinct vertices $I,J$ are adjacent if and only if $I+J=R$. In this paper, we use Gallai${}^{{}^{,}}$s Theorem and the concept of strong resolving graph to compute the strong metric dimension for co-maximal ideal graphs of commutative rings. Explicit formulae for the strong metric dimension, depending on whether the ring is reduced or not, are established.