Editing to Eulerian graphs

•We prove that editing to Eulerian graphs is polynomial-time solvable.•We prove the same result for directed graphs.•We fully classify the complexity of two generalizations of this problem.•We classify variants where vertex deletions are additionally permitted.

The Eulerian Editing problem asks, given a graph G and an integer k, whether G can be modified into an Eulerian graph using at most k edge additions and edge deletions. We show that this problem is polynomial-time solvable for both undirected and directed graphs. We generalize these results for problems with degree parity constraints and degree balance constraints, respectively. We also consider the variants where vertex deletions are permitted. Combined with known results, this leads to full complexity classifications for both undirected and directed graphs and for every subset of the three graph operations.