We introduce a natural temporal analogue of Eulerian circuits and prove that, in contrast with the static case, it is NP-hard to determine whether a given temporal graph is temporally Eulerian even if strong restrictions are placed on the structure of the underlying graph and each edge is active at only three times. However, we do obtain an FPT-algorithm with respect to a new parameter called interval-membership-width which restricts the times assigned to different edges; we believe that this parameter will be of independent interest for other temporal graph problems. Our techniques also allow us to resolve two open question of Akrida, Mertzios and Spirakis [CIAC 2019] concerning a related problem of exploring temporal stars. Furthermore, we introduce a vertex-variant of interval-membership-width (which can be arbitrarily larger than its edge-counterpart) and use it to obtain an FPT-time algorithm for a natural vertex-exploration problem that remains hard even when interval-membership-width is bounded.

翻译：我们引入了欧莱安电路的自然时间模拟,并证明与静态情况相反,很难确定某个特定时间图是否属时间性的欧莱安,即使对底图结构设置了严格的限制,而且每个边缘仅活跃3次。然而,我们确实获得了一个称为间隙会合-边缘的新参数的FPT-algorithm,该参数限制不同边缘的时间分配;我们认为,这一参数对其他时间图问题具有独立的兴趣。我们的技术还使我们能够解决阿克里达、默奇奥和斯皮拉基斯(CIAC 2019)两个与探索时间恒星有关的问题。此外,我们引入了一个间会合线的顶点变量(可任意大于其边角),并使用它来获得FPT-时间算法处理一个天然的脊椎解问题,即使间会合点被捆绑起来,这个问题也仍然很困难。