The networks of the present day communication systems, be it a public road transportation system or a MANET or an Adhoc Network, frequently face a lot of uncertainties in particular regarding traffic jam, flood or water logging or PWD maintenance work (in case of public road network), attack or damage from inter nal or external agents, sudden failure of one or few no des. Consequently, at a real instant of time, the e xisting links/arcs of a given network (graph) are not alway s in their original/excellent condition physically or logically, rather in a weaker condition, or even so metimes disabled or blocked temporarily and waiting for maintenance/repair; and hence ultimately causing de lay in communication or transportation. We do not t ake any special consideration if few of the links be in a better condition at the real time of communicati on, we consider only such cases where few links are in inf erior condition (partially or fully damaged). The c lassical Dijkstra’s algorithm to find the shortest path in g raphs is applicable only if we assume that all the links of the concerned graph are available at their original (id eal) condition at that real time of communication, but at real time scenario it is not the case. Consequently, the mathematically calculated shortest path extracted by using Dijkstra’s algorithm may become costlier (even in-f easible in some cases) in terms of time and/or in t erms of other overhead costs; whereas some other path may be the most efficient or most optimal. Many real lif e situations of communication network or transportati on network cannot be modeled into graphs, but can be well modeled into multigraphs because of the scope of de aling with multiple links (or arcs) connecting a pa ir of nodes. The classical Dijkstra’s algorithm to find t he shortest path in graphs is not applicable to mul tigraphs. In this study the authors make a refinement of the cla ssical Dijkstra’s algorithm to make it applicable t o directed multigraphs having few links partially or fully dam aged. We call such type of multigraphs by GRTmultigraphs and the modified algorithm is called by Dijkstra’s Algorithm for GRT-Multigraphs (DA-GRTM, in short). The DA-GRTM outputs the shortest paths and the corresponding min cost in a GRT-multigraph at real time and thus the solution is a real time solu tion, not an absolute solution. It is claimed that DA-GRTM will play a major role in the present day communica tion systems which are in many cases giant networks, in particular in those networks which cannot be modeled into graphs but into multigraphs.