Embedding one-dimensional continua into T×I

In this paper, we characterize one-dimensional continua in terms of graph-chains and then use these chains to construct embeddings into the product of a simple triod and an arc. We develop a sufficiency condition for this construction to work, and show that if this condition does not hold, answering the question of embeddability into the product of a triod and an arc requires an answer to the more difficult problem of determining embeddability into the plane.