A new problem related to Eulerian graphs

TL;DR

This paper introduces the concept of (semi) S-Eulerian subgraphs within graphs, proves the NP-Completeness of determining such subgraphs, and identifies efficient solutions for connected subgraphs.

Contribution

It defines (semi) S-Eulerian subgraphs, proves the NP-Completeness of their detection, and shows linear-time solutions for connected subgraphs.

Findings

Determining (semi) S-Eulerian subgraphs is NP-Complete.

The problem is solvable in linear time for connected subgraphs.

The paper introduces a new graph-theoretic problem related to Eulerian trails.

Abstract

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 $H$.