Document Zbl 1479.05080

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Geom.</a> <a href="/?q=in%3A473350" title="Articles in this Issue">100, Article ID 101808, 14 p. (2022)</a>. </div> <div class="abstract">Summary: A unit disk graph $G$ on a given set $P$ of points in the plane is a geometric graph where an edge exists between two points $p, q \in P$ if and only if $| p q | \leq 1$. A spanning subgraph $G^\prime$ of $G$ is a $k$-hop spanner if and only if for every edge $p q \in G$, there is a path between $p$, $q$ in $G^\prime$ with at most $k$ edges. We obtain the following results for unit disk graphs in the plane.<br class="zbmathjax-paragraph"><br class="zbmathjax-paragraph">(i) Every $n$-vertex unit disk graph has a 5-hop spanner with at most $5.5n$ edges. We analyze the family of spanners constructed by <span class="zbmathjax-textit">A. Biniaz</span> [ibid. 89, Article ID 101622, 12 p. (2020; <a href="/1450.05083">Zbl 1450.05083</a>)] and improve the upper bound on the number of edges from $9n$ to $5.5n$. <br class="zbmathjax-paragraph">(ii) Using a new construction, we show that every $n$-vertex unit disk graph has a 3-hop spanner with at most $11n$ edges. <br class="zbmathjax-paragraph">(iii) Every $n$-vertex unit disk graph has a 2-hop spanner with $O(n \log n)$ edges. This is the first nontrivial construction of 2-hop spanners. <br class="zbmathjax-paragraph">(iv) For every sufficiently large positive integer $n$, there exists a set $P$ of $n$ points on a circle, such that every plane hop spanner on $P$ has hop stretch factor at least 4. Previously, no lower bound greater than 2 was known. <br class="zbmathjax-paragraph">(v) For every finite point set on a circle, there exists a plane (i.e., crossing-free) 4-hop spanner. As such, this provides a tight bound for points on a circle. <br class="zbmathjax-paragraph">(vi) The maximum degree of $k$-hop spanners cannot be bounded from above by a function of $k$ for any positive integer $k$.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7445249">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C12" title="MSC2020">05C12</a> </td> <td class="space"> Distance in graphs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C07" title="MSC2020">05C07</a> </td> <td class="space"> Vertex degrees </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C99" title="MSC2020">05C99</a> </td> <td class="space"> Graph theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68R10" title="MSC2020">68R10</a> </td> <td class="space"> Graph theory (including graph drawing) in computer science </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Agraph+approximation">graph approximation</a>; <a href="/?q=ut%3A%5C%28+%5Cepsilon+%5C%29-net">$ \epsilon $-net</a>; <a href="/?q=ut%3Ahop-spanner">hop-spanner</a>; <a href="/?q=ut%3Aunit+disk+graph">unit disk graph</a>; <a href="/?q=ut%3Alower+bound">lower bound</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1450.05083">Zbl 1450.05083</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 1479.05080" data-ciurl="/ci/07445249" data-biburl="/bibtex/07445249.bib" data-amsurl="/amsrefs/07445249.bib" data-xmlurl="/xml/07445249.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07445249.pdf" title="Zbl 1479.05080 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.comgeo.2021.101808" aria-label="DOI for “Sparse hop spanners for unit disk graphs”" title="10.1016/j.comgeo.2021.101808">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2002.07840"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Althöfer, I.; Das, G.; Dobkin, D. 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