Strong metric dimension in annihilating-ideal graph of commutative rings (Q6602395)

A vertex \(w\) of a connected graph \(G\) strongly resolves two vertices \(u, v\) of \(G\) if there exists a shortest path from \(u\) to \(w\) containing \(v\) or a shortest path from \(v\) to \(w\) containing \(u\). A set \(S\) of vertices is a strong resolving set for \(G\) if every pair of vertices of \(G\) is strongly resolved by some vertex of \(S\). The smallest cardinality of a strong resolving set for \(G\) is called the strong metric dimension of \(G\).\N\NAn ideal \(I\) of a commutative ring \(R\) is called an annihilating-ideal if there exists \(r \in R \setminus \{0\}\) such that \(Ir = (0)\). The annihilating-ideal graph of a ring \(R\) is the ideal version of the zero-divisor graph. Its vertices are the non-zero annihilating-ideals and there is an edge between two distinct vertices \(I\) and \(J\) if and only if \(IJ = (0)\). The authors in this paper study the strong metric dimension of the annihilating-ideal graph. They find an explicit formula for the strong metric dimension for non-local reduced rings and for Quasi-Frobenius rings in which ann\((M) \subseteq M\) for every maximal ideal \(M\).