Editing to Eulerian Graphs

We investigate the problem of modifying a graph into a connected graph in which the degree of each vertex satisfies a prescribed parity constraint. Let ea, ed and vd denote the operations edge addition, edge deletion and vertex deletion respectively. For any S ⊆ {ea, ed, vd}, we define Connected Degree Parity Editing(S) (CDPE(S)) to be the problem that takes as input a graph G, an integer k and a function δ : V (G)→ {0, 1}, and asks whether G can be modified into a connected graph H with dH(v) ≡ δ(v) (mod 2) for each v ∈ V (H), using at most k operations from S. We prove that if S = {ea} or S = {ea, ed}, then CDPE(S) can be solved in polynomial time; if {vd} ⊆ S ⊆ {ea, ed, vd}, then CDPE(S) is NP-complete and W[1]-hard when parameterized by k, even if δ ≡ 0. Together with known results by Cai and Yang and by Cygan, Marx, Pilipczuk, Pilipczuk and Schlotter, our results completely classify the classical and parameterized complexity of the CDPE(S) problem for all S ⊆ {ea, ed, vd}. We obtain the same classification for a natural variant of the CDPE(S) problem on directed graphs, where the target is a weakly connected digraph in which the difference between the inand out-degree of every vertex equals a prescribed value. As an important implication of our results, we obtain polynomial-time algorithms for Eulerian Editing problem and its directed variant. To the best of our knowledge, the only other natural non-trivial graph class H for which the H-Editing problem is known to be polynomialtime solvable is the class of split graphs. 1998 ACM Subject Classification G.2.2 Graph Theory, F.2.2 Nonnumerical Algorithms and Problems