, Volume 68, Issue 1, pp 41-61

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Parameterized Complexity of Eulerian Deletion Problems

  • Marek CyganAffiliated withInstitute of Informatics, University of Warsaw
  • , Dániel MarxAffiliated withInstitut für Informatik, Humboldt-Universität zu Berlin
  • , Marcin PilipczukAffiliated withInstitute of Informatics, University of Warsaw
  • , Michał PilipczukAffiliated withDepartment of Informatics, University of Bergen Email author 
  • , Ildikó SchlotterAffiliated withDepartment of Computer Science and Information Theory, Budapest University of Technology and Economics


We study a family of problems where the goal is to make a graph Eulerian, i.e., connected and with all the vertices having even degrees, by a minimum number of deletions. We completely classify the parameterized complexity of various versions: undirected or directed graphs, vertex or edge deletions, with or without the requirement of connectivity, etc. The collection of results shows an interesting contrast: while the node-deletion variants remain intractable, i.e., W[1]-hard for all the studied cases, edge-deletion problems are either fixed-parameter tractable or polynomial-time solvable. Of particular interest is a randomized FPT algorithm for making an undirected graph Eulerian by deleting the minimum number of edges, based on a novel application of the color coding technique. For versions that remain NP-complete but fixed-parameter tractable we consider also possibilities of polynomial kernelization; unfortunately, we prove that this is not possible unless NP⊆coNP/poly.


Fixed-parameter tractability Kernelization Eulerian graph Deletion distance