The r-fold-n-point-splitting operation is an important operation in Graph Theory defined by Slater [15]. Later, Ghafari [6] extended 3-fold-n-point-splitting operation in binary matroids and obtained the result for Eulerian matroids whose 3-fold is Eulerian. In this paper, we give another approach to extend 3-fold-3-point-splitting in binary matroids in terms of splitting and haracterize binary gammoid whose 3-fold-3-point spitting is binary gammoid.

In this paper, we examine the thermodynamic geometry of four-dimensional quasi-topological black holes by computing the Ruppeiner scalar curvature R which serves as an empirical tool to describe the nature of interactions among black hole microstructures. In four dimensions, we write novel black hole solutions within the framework of generalized quasi-topological gravity, extended through a fundamental p-form field. Temperature, entropy, and thermodynamic volume are explicitly expressed using the extended first law. The nature of the interactions between the microstructure is then revealed by computing R, where positive curvature indicates repulsion dominant interactions and negative curvature indicates the dominance of attraction. Our approach uses divergences and sign changing nature of R to identify critical points and phase transitions. Further, our analysis reveals a notably streamlined thermodynamic behavior, a single zero-crossing of curvature R, marking a second-order phase transition and offering direct insight into the underlying microstructure interactions.

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