We study a family of problems where the goal is to make a graph Eulerian, i.e., connected and with all the vertices having even degrees, by a minimum number of deletions. We completely classify the parameterized complexity of various versions: undirected or directed graphs, vertex or edge deletions, with or without the requirement of connectivity, etc. The collection of results shows an interesting contrast: while the node-deletion variants remain intractable, i.e., W[1]-hard for all the studied cases, edge-deletion problems are either fixed-parameter tractable or polynomial-time solvable. Of particular interest is a randomized FPT algorithm for making an undirected graph Eulerian by deleting the minimum number of edges, based on a novel application of the color coding technique. For versions that remain NP-complete but fixed-parameter tractable we consider also possibilities of polynomial kernelization; unfortunately, we prove that this is not possible unless NP⊆coNP/poly.

中文翻译：

---

欧拉删除问题的参数化复杂度

我们研究了一系列问题，其目标是通过最少的删除次数制作一个欧拉图，即连接所有顶点的度数为偶数。我们对各种版本的参数化复杂性进行了完全分类：无向图或有向图、顶点或边删除、是否需要连通性等。结果集合显示了一个有趣的对比：而节点删除变体仍然难以处理，即， W[1]-hard 对于所有研究的情况，边删除问题要么是固定参数易处理的，要么是多项式时间可解决的。特别令人感兴趣的是一种随机 FPT 算法，该算法基于颜色编码技术的新应用，通过删除最少边数来制作无向图欧拉。对于保持 NP 完全但固定参数易处理的版本，我们还考虑多项式核化的可能性；不幸的是，我们证明除非 NP⊆coNP/poly，否则这是不可能的。