In 1976 Eberhart, Fúgate, and Gordh proved that the weakly confluent image of a graph is a graph. A much weaker condition on the map is introduced called partial confluence, and it is shown that the image of a graph is a graph if and only if the map is partially confluent.
In addition, it is shown that certain properties of one-dimensional continua are preserved by partially confluent maps, generalizing theorems of Cook and Lelek, Tymchatyn and Lelek, and Grace and Vought. Also, some continua in addition to graphs are shown to be the images of partially confluent maps only.
Copyright © 1987 American Mathematical Society. This article first appeared in Proceedings of the American Mathematical Society 101:3 (1987), 563-570.
Nall, Van C. "Maps Which Preserve Graphs." Proceedings of the American Mathematical Society 101, no. 3 (1987): 563-570. doi:10.1090/S0002-9939-1987-0908670-X.