Geodesic
Decycling a graph by the removal of a matching: new algorithmic and structural aspects in some classes of graphs
Discrete mathematics & theoretical computer science, Tome 20 (2018) no. 2

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A graph $G$ is {\em matching-decyclable} if it has a matching $M$ such that $G-M$ is acyclic. Deciding whether $G$ is matching-decyclable is an NP-complete problem even if $G$ is 2-connected, planar, and subcubic. In this work we present results on matching-decyclability in the following classes: Hamiltonian subcubic graphs, chordal graphs, and distance-hereditary graphs. In Hamiltonian subcubic graphs we show that deciding matching-decyclability is NP-complete even if there are exactly two vertices of degree two. For chordal and distance-hereditary graphs, we present characterizations of matching-decyclability that lead to $O(n)$-time recognition algorithms. 
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     author = {Protti, F\'abio and Souza, U\'everton S.},
     title = {Decycling a graph by the removal of a matching: new algorithmic and structural aspects in some classes of graphs},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
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     number = {2},
     year = {2018},
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     language = {en},
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Protti, Fábio; Souza, Uéverton S. Decycling a graph by the removal of a matching: new algorithmic and structural aspects in some classes of graphs. Discrete mathematics & theoretical computer science, Tome 20 (2018) no. 2. doi: 10.23638/DMTCS-20-2-15

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