A new problem related to Eulerian graphs

Let $G$ be a graph, and $H$ be a finite subgraph of $G$. We say that $H$ is a (semi) $S$-Eulerian subgraph if there exists a closed (open) trail $T$ in $G$ such that each edge of $H$ appears in $T$. We show that the problem of determining whether a subgraph $H$ of a finite graph $G$ is (semi) $S$-Eulerian is NP-Complete. Moreover, we show that both versions of the problem become linear in time if we restrict ourselves to connected subgraphs $H$.