  Robertson and Seymour's celebrated Graph Minor Theorem states that graphs are
well-quasi-ordered by the minor relation. Unlike the minor relation, the
topological minor relation does not well-quasi-order graphs in general. Among
all known infinite antichains with respect to the topological containment,
subdivisions of a graph obtained from an arbitrarily long path by duplicating
each edge can be found. In the 1980's Robertson conjectured that this is the
only obstruction. Formally, he conjectured that for every positive integer $k$,
graphs that do not contain the graph obtained from a path of length $k$ by
duplicating each edge as a topological minor are well-quasi-ordered by the
topological minor relation. The case $k=1$ implies Kruskal's Tree Theorem, and
the case $k=2$ implies a conjecture of V\'{a}zsonyi on subcubic graphs.
  This series of papers dedicates a proof of Robertson's conjecture. We prove
Robertson's conjecture for graphs of bounded tree-width in this paper. It is an
essential step toward the complete proof of Robertson's conjecture, and the
machinery developed in this paper will be applied in future papers of the
series. This bounded tree-width case proved in this paper implies all known
results about well-quasi-ordering graphs by the topological minor relation that
can be proved without using the Graph Minor Theorem, and our proof in this
paper is self-contained.
