The Classification of Finitely Spreading Graphs

Thomassen introduced the concept of a finitely spreading graph: an infinite graph whose edges can be oriented, each in one, both, or neither direction, so that every vertex has finite out‐degree and every ray has a forward oriented tail. He conjectured that a graph is finitely spreading if and only if it is bounded in the sense of Halin—equivalently (see [ 5 ]), if it contains none of three specified infinitely spreading graphs. We prove Thomassen's conjecture in amended form, adding a fourth minimal obstruction to the three conjectured ones.