Eliminating Odd Cycles by Removing a Matching ∗ Carlos V.G.C. Lima 1,† , Dieter Rautenbach 2,† , U´ everton S. Souza 3,† , and Jayme L. Szwarcfiter 4† 1 Departament of Coputer Science, Federal University of Minas Gerais, Belo Horizonte, Brazil 2 Institute of Optimization and Operations Research, Ulm University, Ulm, Germany 3 Institute of Computing, Fluminense Federal University, Niter´ oi, Brazil 4 PESC, COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil Abstract We study the problem of determining whether a given graph G =( V, E ) admits a matching M whose removal destroys all odd cycles of G (or equivalently whether G − M is bipartite). This problem is also equivalent to determine whether G admits a (1, 1)-coloring, which is a 2-coloring of V (G) in which each color class induces a graph of maximum degree at most 1. We show that such a decision problem is NP-complete even for planar graphs of maximum degree 4, but can be solved in linear-time in graphs of maximum degree 3. We also present polynomial time algorithms for (claw, paw)-free graphs, graphs containing only triangles as odd cycles, graphs with bounded dominating sets, P 5 -free graphs, and chordal graphs. In addition, we show that the problem is fixed-parameter tractable when parameterized by clique-width, which implies polynomial time solvability for many interesting graph classes of such as distance-hereditary graphs and outer- planar graphs. Finally a 2 vc(G) · n algorithm, and a kernel having at most 2 · nd(G) vertices are presented, where vc(G) and nd(G) are the vertex cover number and the neighborhood diversity of the input graph, respectively. Keywords: Odd decycling matching · (1, 1)-coloring · Planar graphs · Parameterized complexity 1 Introduction Given a graph G =( V, E ) and a graph property Π, the Π edge-deletion problem consists in determining the mini- mum number of edges required to be removed in order to obtain a graph satisfying Π [12]. Given an integer k ≥ 0, the Π edge-deletion decision problem asks for a set F ⊆ E (G) with |F |≤ k, such that the obtained graph by the re- moval of F satisfies Π. Both versions have received widely attention on the study of their complexity, where we can cite [2, 12, 23, 26, 29, 33, 37, 38] and references therein for applications. When the obtained graph is required to be bipar- tite, the corresponding edge- (vertex-) deletion problem is called edge (vertex) bipartization [1, 14, 22] or edge (vertex) frustration [39]. Choi, Nakajima, and Rim [14] showed that the edge bipartization decision problem is NP-complete even for cubic graphs. Furma´ nczyk, Kubale, and Radziszowski [22] considered vertex bipartization of cubic graphs by the removal of an independent set. In this paper we study the analogous edge deletion decision problem, that is, the problem of determining whether a finite, simple, and undirected graph G admits a removal of a set of edges that is a matching in G in order to obtain a bipartite graph. Formally, for a set M of edges of a graph G =( V, E ), let G − M be the graph with vertex set V (G) and edge set E (G) \ M. For a matching M ⊆ E (G), we say that M is an odd decycling matching of G if G − M is bipartite. Let BM denote the set of all graphs admitting an odd decycling matching. We deal with the complexity of the following decision problem. ODD DECYCLING MATCHING Input: A finite, simple, and undirected graph G. Question: Does G ∈ BM ? A more restricted version of this problem is considered by Schaefer [35]. He deals with the problem of determining whether a given graph G admits a 2-coloring of the vertices so that each vertex has exactly one neighbor with same color as itself. We can see that the removal of the set of edges whose endvertices have same color, which is a perfect matching of G, generates a bipartite graph. Schaefer proved that such a problem is NP-complete even for planar cubic graphs. With respect to the minimization version, the edge-deletion decision problem in order to obtain a bipartite graph is analogous to SIMPLE MAX CUT, which was proved to be NP-complete by Garey, Johnson and Stockmeyer [23]. Yannakakis [37] proved its NP-completeness even for cubic graphs. * Partially supported by CAPES, FAPERJ, and CNPq/DAAD2015SWE/290021/2015-4. † Email addresses: carloslima@dcc.ufmg.br (Lima), dieter.rautenbach@uni-ulm.de (Rautenbach), ueverton@ic.uff.br (Souza), and jayme@cos.ufrj.br (Szwarcfiter) 1 arXiv:1710.07741v1 [cs.DM] 21 Oct 2017