Parameterized Complexity of Connected Even/Odd Subgraph Problems

Fedor V. Fomin (1) , Petr A. Golovach (2)

1 UiB - Department of Informatics [Bergen]
2 School of Engineering and Computing Sciences

Cai and Yang initiated the systematic parameterized complexity study of the following set of problems around Eulerian graphs. For a given graph G and integer k, the task is to decide if G contains a (connected) subgraph with k vertices (edges) with all vertices of even (odd) degrees. They succeed to establish the parameterized complexity of all cases except two, when we ask about - a connected k-edge subgraph with all vertices of odd degrees, the problem known as k-Edge Connected Odd Subgraph; and - a connected k- vertex induced subgraph with all vertices of even degrees, the problem known as k-Vertex Eulerian Subgraph. We resolve both open problems and thus complete the characterization of even/odd subgraph problems from parameterized complexity perspective. We show that k-Edge Connected Odd Subgraph is FPT and that k-Vertex Eulerian Subgraph is W[1]-hard. Our FPT algorithm is based on a novel combinatorial result on the treewidth of minimal connected odd graphs with even amount of edges.

Mots clés: Parameterized complexity; Euler graph; even graph; odd graph; treewidth

Domaines: Complexité [cs.CC]; Algorithme et structure de données [cs.DS]

Fedor V. Fomin, Petr A. Golovach. Parameterized Complexity of Connected Even/Odd Subgraph Problems. STACS'12 (29th Symposium on Theoretical Aspects of Computer Science), Feb 2012, Paris, France. pp.432-440. ⟨hal-00678168⟩