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 ${\sf mos}(G)$ and $\chi_{{\sf odd}}(G)$, respectively. We prove that deciding whether $\chi_{{\sf odd}}(G) \leq q$ is polynomial-time solvable if $q \leq 2$, and NP-complete otherwise. We provide algorithms in time $2^{O({\sf rw})} \cdot n^{O(1)}$ and $2^{O(q \cdot {\sf rw})} \cdot n^{O(1)}$ to compute ${\sf mos}(G)$ and to decide whether $\chi_{{\sf odd}}(G) \leq q$ on $n$-vertex graphs of rank-width at most ${\sf rw}$, 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.

翻译：我们研究了查找问题的复杂性, 给出了一个 G$ 的图形, 最大诱导的 G$ 和所有度的奇数( 称为奇数子图), 以及最小数量的 用于分割 $V( G) 的奇数子。 我们分别将这些参数称为 $@ sf mos}( G) 和 $\ chi ⁇ sf od{ ( G) 和 $\ chi ⁇ sf wid} ( G) q)\\ leq q q q q $( 奇数) 。 我们证明, 决定 美元=2 leq 2 美元和 NP- 完成的多元值 。 我们在时间上提供 $O (sf)\ (s)\ 美元 (sf) 美元 (sf) lax leq q $( q) q) 和 NPP- wex 美元 (w}}\\\\ 美元 (cd) lax lax lex- lex level- legy level legy lections legrof, legy lesh lex lex lesh