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-{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2024-12-13T05:03:43","documenter_version":"1.8.0"}}
+{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2024-12-13T17:50:31","documenter_version":"1.8.0"}}

dev/generated/DominatingSet/index.html

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 Space complexity: 2^8.0
 Read-write complexity: 2^11.333714426093971</code></pre><h2 id="Solving-properties"><a class="docs-heading-anchor" href="#Solving-properties">Solving properties</a><a id="Solving-properties-1"></a><a class="docs-heading-anchor-permalink" href="#Solving-properties" title="Permalink"></a></h2><h3 id="Counting-properties"><a class="docs-heading-anchor" href="#Counting-properties">Counting properties</a><a id="Counting-properties-1"></a><a class="docs-heading-anchor-permalink" href="#Counting-properties" title="Permalink"></a></h3><h5 id="Domination-polynomial"><a class="docs-heading-anchor" href="#Domination-polynomial">Domination polynomial</a><a id="Domination-polynomial-1"></a><a class="docs-heading-anchor-permalink" href="#Domination-polynomial" title="Permalink"></a></h5><p>The graph polynomial for the dominating set problem is known as the domination polynomial (see <a href="https://arxiv.org/abs/0905.2251">arXiv:0905.2251</a>). It is defined as</p><p class="math-container">\[D(G, x) = \sum_{k=0}^{\gamma(G)} d_k x^k,\]</p><p>where <span>$d_k$</span> is the number of dominating sets of size <span>$k$</span> in graph <span>$G=(V, E)$</span>.</p><pre><code class="language-julia hljs">domination_polynomial = solve(problem, GraphPolynomial())[]</code></pre>10&#8729;x<sup>3</sup> &#43; 75&#8729;x<sup>4</sup> &#43; 192&#8729;x<sup>5</sup> &#43; 200&#8729;x<sup>6</sup> &#43; 120&#8729;x<sup>7</sup> &#43; 45&#8729;x<sup>8</sup> &#43; 10&#8729;x<sup>9</sup> &#43; x<sup>10</sup><p>The domination number <span>$\gamma(G)$</span> can be computed with the <a href="../../ref/#GenericTensorNetworks.SizeMin"><code>SizeMin</code></a> property:</p><pre><code class="language-julia hljs">domination_number = solve(problem, SizeMin())[]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">3.0ₜ</code></pre><p>Similarly, we have its counting <a href="../../ref/#GenericTensorNetworks.CountingMin"><code>CountingMin</code></a>:</p><pre><code class="language-julia hljs">counting_min_dominating_set = solve(problem, CountingMin())[]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(3.0, 10.0)ₜ</code></pre><h3 id="Configuration-properties"><a class="docs-heading-anchor" href="#Configuration-properties">Configuration properties</a><a id="Configuration-properties-1"></a><a class="docs-heading-anchor-permalink" href="#Configuration-properties" title="Permalink"></a></h3><h5 id="finding-minimum-dominating-set"><a class="docs-heading-anchor" href="#finding-minimum-dominating-set">finding minimum dominating set</a><a id="finding-minimum-dominating-set-1"></a><a class="docs-heading-anchor-permalink" href="#finding-minimum-dominating-set" title="Permalink"></a></h5><p>One can enumerate all minimum dominating sets with the <a href="../../ref/#GenericTensorNetworks.ConfigsMin"><code>ConfigsMin</code></a> property in the program.</p><pre><code class="language-julia hljs">min_configs = read_config(solve(problem, ConfigsMin())[])
 
-all(c-&gt;is_dominating_set(graph, c), min_configs)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">true</code></pre><pre><code class="language-julia hljs">show_configs(graph, locations, reshape(collect(min_configs), 2, 5); padding_left=20)</code></pre><img src="414bca8c.svg" alt="Example block output"/><p>Similarly, if one is only interested in computing one of the minimum dominating sets, one can use the graph property <a href="../../ref/#GenericTensorNetworks.SingleConfigMin"><code>SingleConfigMin</code></a>.</p><hr/><p><em>This page was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../Coloring/">« Coloring problem</a><a class="docs-footer-nextpage" href="../Satisfiability/">Satisfiability problem »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Friday 13 December 2024 05:03">Friday 13 December 2024</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+all(c-&gt;is_dominating_set(graph, c), min_configs)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">true</code></pre><pre><code class="language-julia hljs">show_configs(graph, locations, reshape(collect(min_configs), 2, 5); padding_left=20)</code></pre><img src="411c2ae6.svg" alt="Example block output"/><p>Similarly, if one is only interested in computing one of the minimum dominating sets, one can use the graph property <a href="../../ref/#GenericTensorNetworks.SingleConfigMin"><code>SingleConfigMin</code></a>.</p><hr/><p><em>This page was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../Coloring/">« Coloring problem</a><a class="docs-footer-nextpage" href="../Satisfiability/">Satisfiability problem »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Friday 13 December 2024 17:50">Friday 13 December 2024</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>