On immersions of uncountable graphs

In his paper on well-quasi-ordering infinite trees (Proc. Cambridge Philos. Soc. 61 (1965) 697), Nash-Williams proposed the conjecture that the class of all graphs (finite or infinite) is well-quasi-ordered by the immersion relation (which is denoted here by ⩽1). In addition, in a subsequent paper, Nash-Williams discussed a weaker version of his original conjecture to the effect that the class of graphs is well-quasi-ordered with respect to a relation ⩽2 which, roughly speaking, is obtained by redefining H⩽1G so that distinct vertices of H can be mapped into the same vertex of G. It is the purpose of the present note to disprove Nash-Williams’ two immersion conjectures.