Abstract
We study the parameterized complexity of the following Euler subgraph problems: (a) Large Euler Subgraph: For a given graph $G$ and integer parameter $k$, does $G$ contain an induced Eulerian subgraph with at least $k$ vertices? (b) Long Circuit: For a given graph $G$ and integer parameter $k$, does $G$ contain an Eulerian subgraph with at least $k$ edges? Our main algorithmic result is that Large Euler Subgraph is fixed parameter tractable (FPT) on undirected graphs. The complexity of the problem changes drastically on directed graphs, and we obtain the following complexity dichotomy: Large Euler Subgraph is NP-hard for every fixed $k>3$ and is solvable in polynomial time for $k\leq 3$. For Long Circuit, we prove that the problem is FPT on directed and undirected graphs.
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