<abstract><p>Utilizing two ideas of neutrosophic subsets (NS) and triangular norms, we introduce a new type of graph as valued-inverse Dombi neutrosophic graphs. The valued-inverse Dombi neutrosophic graphs are a generalization of inverse neutrosophic graphs and are dual to Dombi neutrosophic graphs. We present the concepts of (complete) strong valued-inverse Dombi neutrosophic graphs and analyze the complement of (complete) strong valued-inverse Dombi neutrosophic graphs and self-valued complemented valued-inverse Dombi neutrosophic graphs. Since the valued-inverse Dombi neutrosophic graphs depend on real values, solving the non-equation and the concept of homomorphism play a prominent role in determining the complete, strong, complementarity and self-complementarity of valued-inverse Dombi neutrosophic graphs. We introduce the truth membership order, indeterminacy membership order, falsity membership order, truth membership size, indeterminacy membership size and indeterminacy membership size of any given valued-inverse Dombi neutrosophic graph, which play a major role in the application of valued inverse Dombi neutrosophic graphs in complex networks. An application of a valued-inverse Dombi neutrosophic graph is also described in this study.</p></abstract>