Operational properties of sequentially distance preserving graphs

In the study of combinatorial commutative algebra, especially in the graph-theoretic characterization of simplicial complexes, sequentially distance preserving graphs are widely used and play the important role in solving the corresponding problems. In the report, aimed at sequentially distance preserving graphs, the graph operations including connection and union were used as the main methods to characterize whether the sequentially distance preserving property is kept under the corresponding graph operations. Specifically, the mathematical induction and the detailed analysis on the distance preserving property under graph connection were integrated to prove that the connection graph of any two graphs remains sequentially distance preserving. The analyses for distance preserving subgraphs and the distance preserving property under graph union were used to propose an equivalent condition for maintaining the sequentially distance preserving property after union, a union graph is sequentially distance preserving if and only if both original graphs are sequentially distance preserving, and for at least one of them, the terminal vertex of its distance preserving sequence is the common vertex of the two original graphs. During the research process, the technical roadmap based on graph structure analysis and inductive reasoning was established, which provides the theoretical basis for the complete characterization of sequentially distance preserving graphs and the technical method for the graph-theoretic characterization of simplicial complexes.