Document Zbl 1342.68150

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I. Hereditary properties.</strong> <i>(English)</i> <a class="label nowrap" href="/1342.68150">Zbl 1342.68150</a> </h2> <div class="source"> <a href="/serials/735" title="Journal Profile">Theor. Comput. Sci.</a> <a href="/?q=in%3A355960" title="Articles in this Issue">636, 85-94 (2016)</a>. </div> <div class="abstract">Summary: We study the complexity of deciding whether a given digraph $D$ has a vertex-partition into two disjoint subdigraphs with given structural properties. Let $\mathcal{H}$ and $\mathcal{E}$ denote the following two sets of natural properties of digraphs: $\mathcal{H} = \{\text{acyclic}$, complete, arcless, oriented (no 2-cycle), semicomplete, symmetric, $\text{tournament}\}$ and $\mathcal{E} = \{\text{strongly connected}$, connected, minimum out-degree at least 1, minimum in-degree at least 1, minimum semi-degree at least 1, minimum degree at least 1, having an out-branching, $\text{having an in-branching}\}$. In this paper, we determine the complexity of deciding, for any fixed pair of positive integers $k_1$, $k_2$, whether a given digraph has a vertex partition into two digraphs $D_1$, $D_2$ such that $|V(D_i)| \geq k_i$ and $D_i$ has property $\mathbb{P}_i$ for $i = 1, 2$ when $\mathbb{P}_1 \in \mathcal{H}$ and $\mathbb{P}_2 \in \mathcal{H} \cup \mathcal{E}$. We also classify the complexity of the same problems when restricted to strongly connected digraphs.<br class="zbmathjax-paragraph"> The complexity of the 2-partition problems where both $\mathbb{P}_1$ and $\mathbb{P}_2$ are in $\mathcal{E}$ is determined in Part II [the authors et al., Theor. Comput. Sci. 640, 1–19 (2016; <a href="/1345.68168">Zbl 1345.68168</a>)].</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A6588481">Cited in <strong>4</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A6588481">Cited in <strong>10</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A68Q25" title="MSC2020">68Q25</a> </td> <td class="space"> Analysis of algorithms and problem complexity </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C20" title="MSC2020">05C20</a> </td> <td class="space"> Directed graphs (digraphs), tournaments </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C70" title="MSC2020">05C70</a> </td> <td class="space"> Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68Q17" title="MSC2020">68Q17</a> </td> <td class="space"> Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3ANP-completeness">NP-completeness</a>; <a href="/?q=ut%3Apolynomial">polynomial</a>; <a href="/?q=ut%3Apartition">partition</a>; <a href="/?q=ut%3Asplitting+digraphs">splitting digraphs</a>; <a href="/?q=ut%3Aacyclic">acyclic</a>; <a href="/?q=ut%3Asemicomplete+digraph">semicomplete digraph</a>; <a href="/?q=ut%3Atournament">tournament</a>; <a href="/?q=ut%3Aout-branching">out-branching</a>; <a href="/?q=ut%3Afeedback+vertex+set">feedback vertex set</a>; <a href="/?q=ut%3A2-partition">2-partition</a>; <a href="/?q=ut%3Aminimum+degree">minimum degree</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1345.68168">Zbl 1345.68168</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1342.68150" data-ciurl="/ci/06588481" data-biburl="/bibtex/06588481.bib" data-amsurl="/amsrefs/06588481.bib" data-xmlurl="/xml/06588481.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06588481.pdf" title="Zbl 1342.68150 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.tcs.2016.05.029" aria-label="DOI for “Finding good 2-partitions of digraphs. I. Hereditary properties”" title="10.1016/j.tcs.2016.05.029">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Alon, N., Splitting digraphs, Combin. Probab. Comput., 15, 933-937 (2006) · <a href="/1116.05033" class="nowrap">Zbl 1116.05033</a></td> </tr><tr> <td>[2]</td> <td class="space">Bang-Jensen, J.; Cohen, N.; Havet, F., Finding good 2-partitions of digraphs II. Enumerable properties, Theoret. Comput. Sci. (2016), to appear · <a href="/1345.68168" class="nowrap">Zbl 1345.68168</a></td> </tr><tr> <td>[3]</td> <td class="space">Bang-Jensen, J.; Gutin, G., Digraphs: Theory, Algorithms and Applications (2009), Springer-Verlag: Springer-Verlag London · <a href="/1170.05002" class="nowrap">Zbl 1170.05002</a></td> </tr><tr> <td>[4]</td> <td class="space">Bensmail, J., On the complexity of partitioning a graph into a few connected subgraphs, J. Comb. 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G., Bipartition of graph under degree constraints, Sci. China Math., 58, 869-874 (2015) · <a href="/1317.05152" class="nowrap">Zbl 1317.05152</a></td> </tr><tr> <td>[16]</td> <td class="space">Misra, N.; Philip, G.; Raman, V.; Saurabh, S.; Sikdar, S., FPT algorithms for connected feedback vertex set, J. Comb. Optim., 24, 131-146 (2012) · <a href="/1258.05060" class="nowrap">Zbl 1258.05060</a></td> </tr><tr> <td>[17]</td> <td class="space">Neumann-Lara, V., The dichromatic number of a digraph, J. Combin. Theory Ser. B, 33, 265-270 (1982) · <a href="/0506.05031" class="nowrap">Zbl 0506.05031</a></td> </tr><tr> <td>[18]</td> <td class="space">Reid, K. B., Two complementary circuits in two-connected tournaments, (Cycles in Graphs. Cycles in Graphs, Burnaby, B.C., 1982. Cycles in Graphs. Cycles in Graphs, Burnaby, B.C., 1982, North-Holland Math. 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