We study the complexity of the problems of finding, given a graph G, a largest induced subgraph of G with all degrees odd (called an odd subgraph), and the smallest number of odd subgraphs that partition V(G). We call these parameters mos(G) and χodd(G)
Nearby in the stack
, respectively. We prove that deciding whether
χodd(G)≤q
is polynomial-time solvable if
q≤2
, and NP-complete otherwise. We provide algorithms in time
2O(rw)⋅nO(1)
and
2O(q⋅rw)⋅nO(1)
to compute
mos(G)
and to decide whether
χodd(G)≤q
on
n
-vertex graphs of rank-width at most
rw
, respectively, and we prove that the dependency on rank-width is asymptotically optimal under the ETH. Finally, we give some tight bounds for these parameters on restricted graph classes or in relation to other parameters.