An edge-ordered graph is a simple graph equipped with a total order on its edges. Distinct numerical labels can represent the order, but their magnitudes are irrelevant. An isomorphism must preserve both adjacency and the relative order of edges.
For example, in a three-edge path graph, the middle edge can be smallest, middle, or largest in the edge order. These give three nonisomorphic edge-ordered paths, since reversal interchanges the two end edges.
For a forbidden edge-ordered graph 
, one version of graph saturation
requires that adding any missing edge in any position
in the order creates a copy of 
. Other versions require this only for insertion as the smallest
edge, or for at least one insertion position. These
are different conditions, with corresponding minimum edge
counts satisfying 
(Bošković and Keszegh 2026).
