3 years ago

# Eliminating Odd Cycles by Removing a Matching.

Carlos V.G.C. Lima, Jayme L. Szwarcfiter, Dieter Rautenbach, Uéverton S. Souza

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 outerplanar graphs. Finally, a $2^{vc(G)}\cdot n$ algorithm, and a kernel having at most $2\cdot nd(G)$ vertices are presented, where $vc(G)$ and $nd(G)$ are the vertex cover number and the neighborhood diversity of the input graph, respectively.

Publisher URL: http://arxiv.org/abs/1710.07741

DOI: arXiv:1710.07741v1

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