Let \(G\) be a finite group. The product order relatively prime graph (PORP graph) \(\Gamma_{porp}(G)\) is the graph with vertex set \(V(\Gamma_{porp}(G))=G\) and two distinct vertices \(a\) and \(b\) are adjacent if \((o(a),o(ab))=1\) or \((o(b),o(ab))=1\). In this paper, we investigate the product order relatively prime graph associated with the dihedral group \(D_n\). We study the graph for two important classes of dihedral groups, namely \(D_{p^\alpha}\), where \(p\) is a prime and \(\alpha\ge1\), and \(D_{pq}\), where \(p\) and \(q\) are distinct odd primes. We determine the graph structure and establish several structural properties, including the degree sequence, total number of edges, independence number, clique number and structural decomposition. Furthermore, we investigate graph-theoretic properties such as planarity, bipartiteness, Eulericity, Hamiltonicity, and characterize the conditions under which the graph is complete.
