NON-SEPARATING PATHS IN GRAPHS
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When developing a theory for 3-connected graphs, Tutte showed that for any 3-connected graph G and any three vertices a, b, c of G, G-c has an a-b path P such that G-P is connected. We call paths non-separating if their removal results in a graph satisfying a certain connectivity constraint. There is a series of work on non-separating paths in graphs and their applications. For any graph G and distinct vertices a,b,c,d in V(G), we give a structural characterization for G not containing a path A from a to b and avoiding c and d such that removing A from G results in a 2-connected graph. Using this structure theorem, we construct a 7-connected such graph. We will also discuss potential applications to other problems, including the 3-linkage conjecture made by Thomassen in 1980. This is based on joint work with Shijie Xie and Xingxing Yu.