A closure for Hamilton-connectedness in 1,3,Γ₃\-free graphs

We introduce a closure technique for Hamilton-connectedness of 1,3,Γ₃\-free graphs, where Γ₃ is the graph obtained by joining two vertex-disjoint triangles with a path of length $3$. The closure turns a claw-free graph into a line graph of a multigraph while preserving its (non)-Hamilton-connectedness. The most technical parts of the proof are computer-assisted. The main application of the closure is given in a subsequent paper showing that every $3$-connected 1,3,Γ₃\-free graph is Hamilton-connected, thus resolving one of the two last open cases in the characterization of pairs of connected forbidden subgraphs implying Hamilton-connectedness.