### American Journal of Mathematical and Computer Modelling

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### Modification in Two-Connected Graph with Gallai’s Property in 2-Dimensional and 3-Dimensional Graph Containing 19 Vertices

The graph theory plays an important role in the network analysis, social networking as well as in many engineering fields such as electrical circuits, artificial intelligence, architecture, making the design or pattern of roads, buildings, shopping mall and etc. Due to this wide range application human enjoying her life with peacefully, Graph theory creates a way for human being to connect among themselves by social network. All above applications based on graph or molecule which may be the planer, non-planer and Peterson graph or etc. Peterson graph is the most important and reasonable example of Hypo-Hamiltonian graph. In the earlier, it was found as a hypo-traceable graph (graph which has not Hamiltonian graph. Naeem et al has worked on “A Two-Connected Graph with Gallai’s Property” In his research paper he has applied the property and has found the longest path and cycle in the graph. In this research paper we will develop the 3-dimensional graph of computational molecule contains 19 vertices and will split it into three different planes (xy, xz and yz-plane), and will find the longest path, longest cycle the molecule. The designed graphs can be useful in various fields of science and technology including computational geometry, networking, theoretical computer science and circuit designing.

Gallai Property, Hamiltonian Path, Hamiltonian Cycle, Hypo Hamiltonian Graph, Graph Theory, Traceable Graph

Rabnawaz Mallah, Inayatullah Soomro, Sarang Latif, Dost Muhammad, Altaf Hussain. (2023). Modification in Two-Connected Graph with Gallai’s Property in 2-Dimensional and 3-Dimensional Graph Containing 19 Vertices. American Journal of Mathematical and Computer Modelling, 8(1), 1-5. https://doi.org/10.11648/j.ajmcm.20230801.11