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The Dung Abstract Argumentation Framework (AAF) is an effective formalism for modelling disputes between two or more agents. Generally, the Dung AF is extended to include some unique interactions between agents. This has further been explained with the Bipolar Argumentation Framework (BAF). In the academic space, the use of AAF is highly signified. We can use the AF as a means to resolve disagreements that allows for the determination of a winning argument. In general, there can be imperfect ontologies that affect how reasoning is defined. Typical logic-based AFs apply to the incoherent/uncertain ontologies. However, Dung demonstrated a stable extension of AF to support an “acceptable standard of behavior”. This talk will align with present endeavors on extending the Dung AAF to consider the notion of conflict-freeness in relation to persistence over a hypergraph. With a generic type of argumentation, there are some methods that can exploit certain complex decision procedures. Argument and attack relations within the Dung AAF, thus are further defined to obtain a graphical formula of Kripke groundedness. The incorporating of multiple levels of knowledge aligns with a computational linguistics aspect for the defining of a classification criteria for AAF. In the construction, I will provide some treatment of ‘good’ model-theoretic properties that bridge AAF with Zarankiewicz’s problem to introduce how arguments are consistent with bipartite hypergraphs. The Zarankiewicz problem appears with the communication complexity on AF graphs.