Abdelrahman Said
777 lines
42 KiB
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777 lines
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<link rel="chapter" href="igraph-Separators.html" title="Chapter 24. Vertex separators">
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<link rel="chapter" href="igraph-Community.html" title="Chapter 25. Detecting community structure">
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<link rel="chapter" href="igraph-Graphlets.html" title="Chapter 26. Graphlets">
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<link rel="chapter" href="igraph-Embedding.html" title="Chapter 28. Embedding of graphs">
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<link rel="chapter" href="igraph-Layout.html" title="Chapter 29. Generating layouts for graph drawing">
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<link rel="chapter" href="igraph-Processes.html" title="Chapter 30. Processes on graphs">
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<link rel="chapter" href="igraph-Foreign.html" title="Chapter 31. Reading and writing graphs from and to files">
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<link rel="chapter" href="igraph-Nongraph.html" title="Chapter 33. Non-graph related functions">
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<div class="chapter">
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<div class="titlepage"><div><div><h1 class="title">
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<a name="igraph-Separators"></a>Chapter 24. Vertex separators</h1></div></div></div>
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<div class="toc"><dl class="toc">
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<dt><span class="section"><a href="igraph-Separators.html#igraph_is_separator">1. <code class="function">igraph_is_separator</code> — Would removing this set of vertices disconnect the graph?</a></span></dt>
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<dt><span class="section"><a href="igraph-Separators.html#igraph_is_minimal_separator">2. <code class="function">igraph_is_minimal_separator</code> — Decides whether a set of vertices is a minimal separator.</a></span></dt>
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<dt><span class="section"><a href="igraph-Separators.html#igraph_all_minimal_st_separators">3. <code class="function">igraph_all_minimal_st_separators</code> — List all vertex sets that are minimal (s,t) separators for some s and t.</a></span></dt>
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<dt><span class="section"><a href="igraph-Separators.html#igraph_minimum_size_separators">4. <code class="function">igraph_minimum_size_separators</code> — Find all minimum size separating vertex sets.</a></span></dt>
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<dt><span class="section"><a href="igraph-Separators.html#igraph_even_tarjan_reduction">5. <code class="function">igraph_even_tarjan_reduction</code> — Even-Tarjan reduction of a graph.</a></span></dt>
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</dl></div>
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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="igraph_is_separator"></a>1. <code class="function">igraph_is_separator</code> — Would removing this set of vertices disconnect the graph?</h2></div></div></div>
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<a class="indexterm" name="id-1.25.2.2"></a><p>
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</p>
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<div class="informalexample"><pre class="programlisting">
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igraph_error_t igraph_is_separator(const igraph_t *graph,
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const igraph_vs_t candidate,
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igraph_bool_t *res);
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</pre></div>
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<p>
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</p>
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<p>
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A vertex set <code class="constant">S</code> is a separator if there are vertices <code class="constant">u</code> and <code class="constant">v</code>
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in the graph such that all paths between <code class="constant">u</code> and <code class="constant">v</code> pass through
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some vertices in <code class="constant">S</code>.
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</p>
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<p><b>Arguments: </b>
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</p>
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<div class="variablelist"><table border="0" class="variablelist">
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<colgroup>
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<col align="left" valign="top">
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<col>
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</colgroup>
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<tbody>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>graph</code></em>:</span></p></td>
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<td><p>
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The input graph. It may be directed, but edge
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directions are ignored.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>candidate</code></em>:</span></p></td>
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<td><p>
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The candidate separator.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>res</code></em>:</span></p></td>
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<td><p>
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Pointer to a boolean variable, the result is stored here.
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</p></td>
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</tr>
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</tbody>
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</table></div>
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<p>
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</p>
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<p><b>Returns: </b></p>
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<div class="variablelist"><table border="0" class="variablelist">
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<colgroup>
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<col align="left" valign="top">
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<col>
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</colgroup>
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<tbody><tr>
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<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
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<td><p>
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Error code.
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</p></td>
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</tr></tbody>
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</table></div>
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<p>
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Time complexity: O(|V|+|E|), linear in the number vertices and edges.
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</p>
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<div class="hideshow" onClick="toggle(this, event)">
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<div class="example">
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<a name="id-1.25.2.8.1"></a><p class="title"><b>Example 24.1. File <code class="code">examples/simple/igraph_is_separator.c</code></b></p>
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<div class="example-contents">
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<pre class="programlisting"><span class="strong"><strong>#include</strong></span> <igraph.h>
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<span class="strong"><strong>#include</strong></span> <stdio.h>
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<span class="strong"><strong>#define</strong></span> <span class="strong"><strong>FAIL</strong></span>(msg, error) <span class="strong"><strong>do</strong></span> { <span class="strong"><strong>printf</strong></span>(msg "\n") ; <span class="strong"><strong>return</strong></span> error; } <span class="strong"><strong>while</strong></span> (0)
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int <span class="strong"><strong>main</strong></span>(void) {
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igraph_t graph;
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igraph_vector_int_t sep;
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igraph_bool_t result;
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<span class="emphasis"><em>/* Initialize the library. */</em></span>
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<span class="strong"><strong><a class="link" href="igraph-Basic.html#igraph_setup" title="4.1. igraph_setup — Initializes the igraph library.">igraph_setup</a></strong></span>();
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<span class="emphasis"><em>/* Simple star graph, remove the center */</em></span>
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<span class="strong"><strong><a class="link" href="igraph-Generators.html#igraph_star" title="4.1. igraph_star — Creates a star graph, every vertex connects only to the center.">igraph_star</a></strong></span>(&graph, 10, IGRAPH_STAR_UNDIRECTED, 0);
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<span class="strong"><strong><a class="link" href="igraph-Separators.html#igraph_is_separator" title="1. igraph_is_separator — Would removing this set of vertices disconnect the graph?">igraph_is_separator</a></strong></span>(&graph, <span class="strong"><strong><a class="link" href="igraph-Iterators.html#igraph_vss_1" title="4.3. igraph_vss_1 — Vertex set with a single vertex (immediate version).">igraph_vss_1</a></strong></span>(0), &result);
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<span class="strong"><strong>if</strong></span> (!result) {
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<span class="strong"><strong>FAIL</strong></span>("Center of star graph failed.", 1);
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}
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<span class="emphasis"><em>/* Same graph, but another vertex */</em></span>
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<span class="strong"><strong><a class="link" href="igraph-Separators.html#igraph_is_separator" title="1. igraph_is_separator — Would removing this set of vertices disconnect the graph?">igraph_is_separator</a></strong></span>(&graph, <span class="strong"><strong><a class="link" href="igraph-Iterators.html#igraph_vss_1" title="4.3. igraph_vss_1 — Vertex set with a single vertex (immediate version).">igraph_vss_1</a></strong></span>(6), &result);
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<span class="strong"><strong>if</strong></span> (result) {
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<span class="strong"><strong>FAIL</strong></span>("Non-center of star graph failed.", 2);
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}
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<span class="emphasis"><em>/* Same graph, all vertices but the center */</em></span>
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<span class="strong"><strong><a class="link" href="igraph-Separators.html#igraph_is_separator" title="1. igraph_is_separator — Would removing this set of vertices disconnect the graph?">igraph_is_separator</a></strong></span>(&graph, <span class="strong"><strong><a class="link" href="igraph-Iterators.html#igraph_vss_range" title="4.5. igraph_vss_range — An interval of vertices (immediate version).">igraph_vss_range</a></strong></span>(1, 10), &result);
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<span class="strong"><strong>if</strong></span> (result) {
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<span class="strong"><strong>FAIL</strong></span>("All non-central vertices of star graph failed.", 5);
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}
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<span class="emphasis"><em>/* Same graph, all vertices */</em></span>
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<span class="strong"><strong><a class="link" href="igraph-Separators.html#igraph_is_separator" title="1. igraph_is_separator — Would removing this set of vertices disconnect the graph?">igraph_is_separator</a></strong></span>(&graph, <span class="strong"><strong><a class="link" href="igraph-Iterators.html#igraph_vss_range" title="4.5. igraph_vss_range — An interval of vertices (immediate version).">igraph_vss_range</a></strong></span>(0, 10), &result);
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<span class="strong"><strong>if</strong></span> (result) {
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<span class="strong"><strong>FAIL</strong></span>("All vertices of star graph failed.", 6);
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}
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<span class="strong"><strong><a class="link" href="igraph-Basic.html#igraph_destroy" title="5.1.4. igraph_destroy — Frees the memory allocated for a graph object.">igraph_destroy</a></strong></span>(&graph);
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<span class="emphasis"><em>/* Karate club */</em></span>
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<span class="strong"><strong><a class="link" href="igraph-Generators.html#igraph_famous" title="8.1. igraph_famous — Create a famous graph by simply providing its name.">igraph_famous</a></strong></span>(&graph, "zachary");
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<span class="strong"><strong>igraph_vector_int_init</strong></span>(&sep, 0);
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<span class="strong"><strong>igraph_vector_int_push_back</strong></span>(&sep, 32);
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<span class="strong"><strong>igraph_vector_int_push_back</strong></span>(&sep, 33);
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<span class="strong"><strong><a class="link" href="igraph-Separators.html#igraph_is_separator" title="1. igraph_is_separator — Would removing this set of vertices disconnect the graph?">igraph_is_separator</a></strong></span>(&graph, <span class="strong"><strong><a class="link" href="igraph-Iterators.html#igraph_vss_vector" title="4.4. igraph_vss_vector — Vertex set based on a vector (immediate version).">igraph_vss_vector</a></strong></span>(&sep), &result);
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<span class="strong"><strong>if</strong></span> (!result) {
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<span class="strong"><strong>FAIL</strong></span>("Karate network (32,33) failed", 3);
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}
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<span class="strong"><strong>igraph_vector_int_resize</strong></span>(&sep, 5);
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<span class="strong"><strong><a class="link" href="igraph-Data-structures.html#VECTOR" title="2.4.1. VECTOR — Accessing an element of a vector.">VECTOR</a></strong></span>(sep)[0] = 8;
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<span class="strong"><strong><a class="link" href="igraph-Data-structures.html#VECTOR" title="2.4.1. VECTOR — Accessing an element of a vector.">VECTOR</a></strong></span>(sep)[1] = 9;
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<span class="strong"><strong><a class="link" href="igraph-Data-structures.html#VECTOR" title="2.4.1. VECTOR — Accessing an element of a vector.">VECTOR</a></strong></span>(sep)[2] = 19;
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<span class="strong"><strong><a class="link" href="igraph-Data-structures.html#VECTOR" title="2.4.1. VECTOR — Accessing an element of a vector.">VECTOR</a></strong></span>(sep)[3] = 30;
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<span class="strong"><strong><a class="link" href="igraph-Data-structures.html#VECTOR" title="2.4.1. VECTOR — Accessing an element of a vector.">VECTOR</a></strong></span>(sep)[4] = 31;
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<span class="strong"><strong><a class="link" href="igraph-Separators.html#igraph_is_separator" title="1. igraph_is_separator — Would removing this set of vertices disconnect the graph?">igraph_is_separator</a></strong></span>(&graph, <span class="strong"><strong><a class="link" href="igraph-Iterators.html#igraph_vss_vector" title="4.4. igraph_vss_vector — Vertex set based on a vector (immediate version).">igraph_vss_vector</a></strong></span>(&sep), &result);
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<span class="strong"><strong>if</strong></span> (result) {
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<span class="strong"><strong>FAIL</strong></span>("Karate network (8,9,19,30,31) failed", 4);
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}
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<span class="strong"><strong><a class="link" href="igraph-Basic.html#igraph_destroy" title="5.1.4. igraph_destroy — Frees the memory allocated for a graph object.">igraph_destroy</a></strong></span>(&graph);
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<span class="strong"><strong>igraph_vector_int_destroy</strong></span>(&sep);
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<span class="strong"><strong>return</strong></span> 0;
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}
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</pre>
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<p></p>
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</div>
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</div>
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<br class="example-break">
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</div>
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<p>
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</p>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="igraph_is_minimal_separator"></a>2. <code class="function">igraph_is_minimal_separator</code> — Decides whether a set of vertices is a minimal separator.</h2></div></div></div>
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<a class="indexterm" name="id-1.25.3.2"></a><p>
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</p>
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<div class="informalexample"><pre class="programlisting">
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igraph_error_t igraph_is_minimal_separator(const igraph_t *graph,
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const igraph_vs_t candidate,
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igraph_bool_t *res);
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</pre></div>
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<p>
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</p>
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<p>
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A vertex separator <code class="constant">S</code> is minimal is no proper subset of <code class="constant">S</code>
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is also a separator.
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</p>
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<p><b>Arguments: </b>
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</p>
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<div class="variablelist"><table border="0" class="variablelist">
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<colgroup>
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<col align="left" valign="top">
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<col>
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</colgroup>
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<tbody>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>graph</code></em>:</span></p></td>
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<td><p>
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The input graph. It may be directed, but edge
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directions are ignored.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>candidate</code></em>:</span></p></td>
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<td><p>
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The candidate minimal separators.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>res</code></em>:</span></p></td>
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<td><p>
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Pointer to a boolean variable, the result is stored
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here.
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</p></td>
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</tr>
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</tbody>
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</table></div>
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<p>
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</p>
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<p><b>Returns: </b></p>
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<div class="variablelist"><table border="0" class="variablelist">
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<colgroup>
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<col align="left" valign="top">
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<col>
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</colgroup>
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<tbody><tr>
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<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
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<td><p>
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Error code.
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</p></td>
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</tr></tbody>
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</table></div>
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<p>
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Time complexity: O(|V|+|E|), linear in the number vertices and edges.
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</p>
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<div class="hideshow" onClick="toggle(this, event)">
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<div class="example">
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<a name="id-1.25.3.8.1"></a><p class="title"><b>Example 24.2. File <code class="code">examples/simple/igraph_is_minimal_separator.c</code></b></p>
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<div class="example-contents">
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<pre class="programlisting"><span class="strong"><strong>#include</strong></span> <igraph.h>
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<span class="strong"><strong>#include</strong></span> <stdio.h>
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<span class="strong"><strong>#define</strong></span> <span class="strong"><strong>FAIL</strong></span>(msg, error) <span class="strong"><strong>do</strong></span> { <span class="strong"><strong>printf</strong></span>(msg "\n") ; <span class="strong"><strong>return</strong></span> error; } <span class="strong"><strong>while</strong></span> (0)
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int <span class="strong"><strong>main</strong></span>(void) {
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igraph_t graph;
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igraph_vector_int_t sep;
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igraph_bool_t result;
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<span class="emphasis"><em>/* Initialize the library. */</em></span>
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<span class="strong"><strong><a class="link" href="igraph-Basic.html#igraph_setup" title="4.1. igraph_setup — Initializes the igraph library.">igraph_setup</a></strong></span>();
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<span class="emphasis"><em>/* Simple star graph, remove the center */</em></span>
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<span class="strong"><strong><a class="link" href="igraph-Generators.html#igraph_star" title="4.1. igraph_star — Creates a star graph, every vertex connects only to the center.">igraph_star</a></strong></span>(&graph, 10, IGRAPH_STAR_UNDIRECTED, 0);
|
||
<span class="strong"><strong><a class="link" href="igraph-Separators.html#igraph_is_minimal_separator" title="2. igraph_is_minimal_separator — Decides whether a set of vertices is a minimal separator.">igraph_is_minimal_separator</a></strong></span>(&graph, <span class="strong"><strong><a class="link" href="igraph-Iterators.html#igraph_vss_1" title="4.3. igraph_vss_1 — Vertex set with a single vertex (immediate version).">igraph_vss_1</a></strong></span>(0), &result);
|
||
<span class="strong"><strong>if</strong></span> (!result) {
|
||
<span class="strong"><strong>FAIL</strong></span>("Center of star graph failed.", 1);
|
||
}
|
||
|
||
<span class="emphasis"><em>/* Same graph, but another vertex */</em></span>
|
||
<span class="strong"><strong><a class="link" href="igraph-Separators.html#igraph_is_minimal_separator" title="2. igraph_is_minimal_separator — Decides whether a set of vertices is a minimal separator.">igraph_is_minimal_separator</a></strong></span>(&graph, <span class="strong"><strong><a class="link" href="igraph-Iterators.html#igraph_vss_1" title="4.3. igraph_vss_1 — Vertex set with a single vertex (immediate version).">igraph_vss_1</a></strong></span>(6), &result);
|
||
<span class="strong"><strong>if</strong></span> (result) {
|
||
<span class="strong"><strong>FAIL</strong></span>("Non-center of star graph failed.", 2);
|
||
}
|
||
<span class="strong"><strong><a class="link" href="igraph-Basic.html#igraph_destroy" title="5.1.4. igraph_destroy — Frees the memory allocated for a graph object.">igraph_destroy</a></strong></span>(&graph);
|
||
|
||
<span class="emphasis"><em>/* Karate club */</em></span>
|
||
<span class="strong"><strong><a class="link" href="igraph-Generators.html#igraph_famous" title="8.1. igraph_famous — Create a famous graph by simply providing its name.">igraph_famous</a></strong></span>(&graph, "zachary");
|
||
<span class="strong"><strong>igraph_vector_int_init</strong></span>(&sep, 0);
|
||
<span class="strong"><strong>igraph_vector_int_push_back</strong></span>(&sep, 32);
|
||
<span class="strong"><strong>igraph_vector_int_push_back</strong></span>(&sep, 33);
|
||
<span class="strong"><strong><a class="link" href="igraph-Separators.html#igraph_is_minimal_separator" title="2. igraph_is_minimal_separator — Decides whether a set of vertices is a minimal separator.">igraph_is_minimal_separator</a></strong></span>(&graph, <span class="strong"><strong><a class="link" href="igraph-Iterators.html#igraph_vss_vector" title="4.4. igraph_vss_vector — Vertex set based on a vector (immediate version).">igraph_vss_vector</a></strong></span>(&sep), &result);
|
||
<span class="strong"><strong>if</strong></span> (!result) {
|
||
<span class="strong"><strong>FAIL</strong></span>("Karate network (32,33) failed", 3);
|
||
}
|
||
|
||
<span class="strong"><strong>igraph_vector_int_resize</strong></span>(&sep, 5);
|
||
<span class="strong"><strong><a class="link" href="igraph-Data-structures.html#VECTOR" title="2.4.1. VECTOR — Accessing an element of a vector.">VECTOR</a></strong></span>(sep)[0] = 8;
|
||
<span class="strong"><strong><a class="link" href="igraph-Data-structures.html#VECTOR" title="2.4.1. VECTOR — Accessing an element of a vector.">VECTOR</a></strong></span>(sep)[1] = 9;
|
||
<span class="strong"><strong><a class="link" href="igraph-Data-structures.html#VECTOR" title="2.4.1. VECTOR — Accessing an element of a vector.">VECTOR</a></strong></span>(sep)[2] = 19;
|
||
<span class="strong"><strong><a class="link" href="igraph-Data-structures.html#VECTOR" title="2.4.1. VECTOR — Accessing an element of a vector.">VECTOR</a></strong></span>(sep)[3] = 30;
|
||
<span class="strong"><strong><a class="link" href="igraph-Data-structures.html#VECTOR" title="2.4.1. VECTOR — Accessing an element of a vector.">VECTOR</a></strong></span>(sep)[4] = 31;
|
||
<span class="strong"><strong><a class="link" href="igraph-Separators.html#igraph_is_minimal_separator" title="2. igraph_is_minimal_separator — Decides whether a set of vertices is a minimal separator.">igraph_is_minimal_separator</a></strong></span>(&graph, <span class="strong"><strong><a class="link" href="igraph-Iterators.html#igraph_vss_vector" title="4.4. igraph_vss_vector — Vertex set based on a vector (immediate version).">igraph_vss_vector</a></strong></span>(&sep), &result);
|
||
<span class="strong"><strong>if</strong></span> (result) {
|
||
<span class="strong"><strong>FAIL</strong></span>("Karate network (8,9,19,30,31) failed", 4);
|
||
}
|
||
|
||
<span class="strong"><strong><a class="link" href="igraph-Basic.html#igraph_destroy" title="5.1.4. igraph_destroy — Frees the memory allocated for a graph object.">igraph_destroy</a></strong></span>(&graph);
|
||
<span class="strong"><strong>igraph_vector_int_destroy</strong></span>(&sep);
|
||
|
||
<span class="strong"><strong>return</strong></span> 0;
|
||
}
|
||
</pre>
|
||
<p></p>
|
||
</div>
|
||
</div>
|
||
<br class="example-break">
|
||
</div>
|
||
<p>
|
||
</p>
|
||
</div>
|
||
<div class="section">
|
||
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
||
<a name="igraph_all_minimal_st_separators"></a>3. <code class="function">igraph_all_minimal_st_separators</code> — List all vertex sets that are minimal (s,t) separators for some s and t.</h2></div></div></div>
|
||
<a class="indexterm" name="id-1.25.4.2"></a><p>
|
||
</p>
|
||
<div class="informalexample"><pre class="programlisting">
|
||
igraph_error_t igraph_all_minimal_st_separators(
|
||
const igraph_t *graph, igraph_vector_int_list_t *separators
|
||
);
|
||
</pre></div>
|
||
<p>
|
||
</p>
|
||
<p>
|
||
|
||
|
||
|
||
This function lists all vertex sets that are minimal (s,t)
|
||
separators for some (s,t) vertex pair.
|
||
|
||
</p>
|
||
<p>
|
||
Note that some vertex sets returned by this function may not be minimal
|
||
with respect to disconnecting the graph (or increasing the number of
|
||
connected components). Take for example the 5-vertex graph with edges
|
||
<code class="literal">0-1-2-3-4-1</code>. This function returns the vertex sets
|
||
<code class="literal">{1}</code>, <code class="literal">{2,4}</code> and <code class="literal">{1,3}</code>.
|
||
Notice that <code class="literal">{1,3}</code> is not minimal with respect to disconnecting
|
||
the graph, as <code class="literal">{1}</code> would be sufficient for that. However, it is
|
||
minimal with respect to separating vertices <code class="constant">2</code> and <code class="constant">4</code>.
|
||
|
||
</p>
|
||
<p>
|
||
See more about the implemented algorithm in
|
||
Anne Berry, Jean-Paul Bordat and Olivier Cogis: Generating All the
|
||
Minimal Separators of a Graph, In: Peter Widmayer, Gabriele Neyer
|
||
and Stephan Eidenbenz (editors): Graph-theoretic concepts in
|
||
computer science, 1665, 167--172, 1999. Springer.
|
||
<a class="ulink" href="https://doi.org/10.1007/3-540-46784-X_17" target="_top">https://doi.org/10.1007/3-540-46784-X_17</a>
|
||
|
||
</p>
|
||
<p><b>Arguments: </b>
|
||
</p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>graph</code></em>:</span></p></td>
|
||
<td><p>
|
||
The input graph. It may be directed, but edge
|
||
directions are ignored.
|
||
</p></td>
|
||
</tr>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>separators</code></em>:</span></p></td>
|
||
<td><p>
|
||
Pointer to a list of integer vectors, the separators
|
||
will be stored here.
|
||
</p></td>
|
||
</tr>
|
||
</tbody>
|
||
</table></div>
|
||
<p>
|
||
</p>
|
||
<p><b>Returns: </b></p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody><tr>
|
||
<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
|
||
<td><p>
|
||
Error code.
|
||
</p></td>
|
||
</tr></tbody>
|
||
</table></div>
|
||
<p>
|
||
|
||
</p>
|
||
<p><b>See also: </b></p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody><tr>
|
||
<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
|
||
<td><p>
|
||
<a class="link" href="igraph-Separators.html#igraph_minimum_size_separators" title="4. igraph_minimum_size_separators — Find all minimum size separating vertex sets."><code class="function">igraph_minimum_size_separators()</code></a>
|
||
</p></td>
|
||
</tr></tbody>
|
||
</table></div>
|
||
<p>
|
||
|
||
Time complexity: O(n|V|^3), |V| is the number of vertices, n is the
|
||
number of separators.
|
||
|
||
</p>
|
||
<div class="hideshow" onClick="toggle(this, event)">
|
||
<div class="example">
|
||
<a name="id-1.25.4.12.1"></a><p class="title"><b>Example 24.3. File <code class="code">examples/simple/igraph_minimal_separators.c</code></b></p>
|
||
<div class="example-contents">
|
||
<pre class="programlisting"><span class="strong"><strong>#include</strong></span> <igraph.h>
|
||
<span class="strong"><strong>#include</strong></span> <stdio.h>
|
||
|
||
int <span class="strong"><strong>main</strong></span>(void) {
|
||
igraph_t graph;
|
||
igraph_vector_int_list_t separators;
|
||
igraph_int_t i, n;
|
||
|
||
<span class="emphasis"><em>/* Initialize the library. */</em></span>
|
||
<span class="strong"><strong><a class="link" href="igraph-Basic.html#igraph_setup" title="4.1. igraph_setup — Initializes the igraph library.">igraph_setup</a></strong></span>();
|
||
|
||
<span class="strong"><strong><a class="link" href="igraph-Generators.html#igraph_famous" title="8.1. igraph_famous — Create a famous graph by simply providing its name.">igraph_famous</a></strong></span>(&graph, "zachary");
|
||
<span class="strong"><strong>igraph_vector_int_list_init</strong></span>(&separators, 0);
|
||
<span class="strong"><strong><a class="link" href="igraph-Separators.html#igraph_all_minimal_st_separators" title="3. igraph_all_minimal_st_separators — List all vertex sets that are minimal (s,t) separators for some s and t.">igraph_all_minimal_st_separators</a></strong></span>(&graph, &separators);
|
||
|
||
n = <span class="strong"><strong>igraph_vector_int_list_size</strong></span>(&separators);
|
||
<span class="strong"><strong>for</strong></span> (i = 0; i < n; i++) {
|
||
igraph_bool_t res;
|
||
igraph_vector_int_t *sep = <span class="strong"><strong>igraph_vector_int_list_get_ptr</strong></span>(&separators, i);
|
||
|
||
<span class="strong"><strong><a class="link" href="igraph-Separators.html#igraph_is_separator" title="1. igraph_is_separator — Would removing this set of vertices disconnect the graph?">igraph_is_separator</a></strong></span>(&graph, <span class="strong"><strong><a class="link" href="igraph-Iterators.html#igraph_vss_vector" title="4.4. igraph_vss_vector — Vertex set based on a vector (immediate version).">igraph_vss_vector</a></strong></span>(sep), &res);
|
||
<span class="strong"><strong>if</strong></span> (!res) {
|
||
<span class="strong"><strong>printf</strong></span>("Vertex set %" IGRAPH_PRId " is not a separator!\n", i);
|
||
<span class="strong"><strong>igraph_vector_int_print</strong></span>(sep);
|
||
<span class="strong"><strong>return</strong></span> 1;
|
||
}
|
||
}
|
||
|
||
<span class="strong"><strong><a class="link" href="igraph-Basic.html#igraph_destroy" title="5.1.4. igraph_destroy — Frees the memory allocated for a graph object.">igraph_destroy</a></strong></span>(&graph);
|
||
<span class="strong"><strong>igraph_vector_int_list_destroy</strong></span>(&separators);
|
||
|
||
<span class="strong"><strong>return</strong></span> 0;
|
||
}
|
||
</pre>
|
||
<p></p>
|
||
</div>
|
||
</div>
|
||
<br class="example-break">
|
||
</div>
|
||
<p>
|
||
</p>
|
||
</div>
|
||
<div class="section">
|
||
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
||
<a name="igraph_minimum_size_separators"></a>4. <code class="function">igraph_minimum_size_separators</code> — Find all minimum size separating vertex sets.</h2></div></div></div>
|
||
<a class="indexterm" name="id-1.25.5.2"></a><p>
|
||
</p>
|
||
<div class="informalexample"><pre class="programlisting">
|
||
igraph_error_t igraph_minimum_size_separators(
|
||
const igraph_t *graph, igraph_vector_int_list_t *separators
|
||
);
|
||
</pre></div>
|
||
<p>
|
||
</p>
|
||
<p>
|
||
|
||
|
||
|
||
This function lists all separator vertex sets of minimum size.
|
||
A vertex set is a separator if its removal disconnects the graph.
|
||
|
||
</p>
|
||
<p>
|
||
If the graph is already disconnected, no separators are returned.
|
||
Note that this convention differs from that used by some other
|
||
funtions such as <a class="link" href="igraph-Separators.html#igraph_all_minimal_st_separators" title="3. igraph_all_minimal_st_separators — List all vertex sets that are minimal (s,t) separators for some s and t."><code class="function">igraph_all_minimal_st_separators()</code></a>.
|
||
|
||
</p>
|
||
<p>
|
||
Complete graphs have no vertex separators.
|
||
|
||
</p>
|
||
<p>
|
||
The implementation is based on the following paper:
|
||
Arkady Kanevsky: Finding all minimum-size separating vertex sets in
|
||
a graph, Networks 23, 533--541, 1993.
|
||
<a class="ulink" href="https://doi.org/10.1002/net.3230230604" target="_top">https://doi.org/10.1002/net.3230230604</a>
|
||
|
||
</p>
|
||
<p><b>Arguments: </b>
|
||
</p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>graph</code></em>:</span></p></td>
|
||
<td><p>
|
||
The input graph, which must be undirected.
|
||
</p></td>
|
||
</tr>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>separators</code></em>:</span></p></td>
|
||
<td><p>
|
||
An initialized list of integer vectors, the separators
|
||
are stored here. It is a list of pointers to <span class="type">igraph_vector_int_t</span>
|
||
objects. Each vector will contain the IDs of the vertices in
|
||
the separator. The separators are returned in an arbitrary order.
|
||
</p></td>
|
||
</tr>
|
||
</tbody>
|
||
</table></div>
|
||
<p>
|
||
</p>
|
||
<p><b>Returns: </b></p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody><tr>
|
||
<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
|
||
<td><p>
|
||
Error code.
|
||
</p></td>
|
||
</tr></tbody>
|
||
</table></div>
|
||
<p>
|
||
|
||
Time complexity: TODO.
|
||
|
||
</p>
|
||
<div class="hideshow" onClick="toggle(this, event)">
|
||
<div class="example">
|
||
<a name="id-1.25.5.11.1"></a><p class="title"><b>Example 24.4. File <code class="code">examples/simple/igraph_minimum_size_separators.c</code></b></p>
|
||
<div class="example-contents">
|
||
<pre class="programlisting"><span class="strong"><strong>#include</strong></span> <igraph.h>
|
||
|
||
int <span class="strong"><strong>main</strong></span>(void) {
|
||
<span class="strong"><strong><a class="link" href="igraph-Basic.html#igraph_setup" title="4.1. igraph_setup — Initializes the igraph library.">igraph_setup</a></strong></span>();
|
||
igraph_t g;
|
||
igraph_vector_int_list_t sep;
|
||
|
||
<span class="strong"><strong><a class="link" href="igraph-Generators.html#igraph_small" title="2.2. igraph_small — Shorthand to create a small graph, giving the edges as arguments.">igraph_small</a></strong></span>(&g, 7, IGRAPH_UNDIRECTED,
|
||
1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0,
|
||
-1);
|
||
<span class="strong"><strong>igraph_vector_int_list_init</strong></span>(&sep, 0);
|
||
<span class="strong"><strong><a class="link" href="igraph-Separators.html#igraph_minimum_size_separators" title="4. igraph_minimum_size_separators — Find all minimum size separating vertex sets.">igraph_minimum_size_separators</a></strong></span>(&g, &sep);
|
||
|
||
<span class="strong"><strong>for</strong></span> (igraph_int_t i = 0; i < <span class="strong"><strong>igraph_vector_int_list_size</strong></span>(&sep); i++) {
|
||
igraph_vector_int_t* v = <span class="strong"><strong>igraph_vector_int_list_get_ptr</strong></span>(&sep, i);
|
||
<span class="strong"><strong>igraph_vector_int_print</strong></span>(v);
|
||
}
|
||
|
||
<span class="strong"><strong>igraph_vector_int_list_destroy</strong></span>(&sep);
|
||
<span class="strong"><strong><a class="link" href="igraph-Basic.html#igraph_destroy" title="5.1.4. igraph_destroy — Frees the memory allocated for a graph object.">igraph_destroy</a></strong></span>(&g);
|
||
|
||
<span class="strong"><strong>return</strong></span> 0;
|
||
}
|
||
</pre>
|
||
<p></p>
|
||
</div>
|
||
</div>
|
||
<br class="example-break">
|
||
</div>
|
||
<p>
|
||
</p>
|
||
</div>
|
||
<div class="section">
|
||
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
||
<a name="igraph_even_tarjan_reduction"></a>5. <code class="function">igraph_even_tarjan_reduction</code> — Even-Tarjan reduction of a graph.</h2></div></div></div>
|
||
<a class="indexterm" name="id-1.25.6.2"></a><p>
|
||
</p>
|
||
<div class="informalexample"><pre class="programlisting">
|
||
igraph_error_t igraph_even_tarjan_reduction(const igraph_t *graph, igraph_t *graphbar,
|
||
igraph_vector_t *capacity);
|
||
</pre></div>
|
||
<p>
|
||
</p>
|
||
<p>
|
||
|
||
|
||
|
||
A digraph is created with twice as many vertices and edges. For each
|
||
original vertex <code class="constant">i</code>, two vertices <code class="literal">i' = i</code> and
|
||
<code class="literal">i'' = i' + n</code> are created,
|
||
with a directed edge from <code class="literal">i'</code> to <code class="literal">i''</code>.
|
||
For each original directed edge from <code class="constant">i</code> to <code class="constant">j</code>, two new edges are created,
|
||
from <code class="literal">i'</code> to <code class="literal">j''</code> and from <code class="literal">i''</code>
|
||
to <code class="literal">j'</code>.
|
||
|
||
</p>
|
||
<p>This reduction is used in the paper (observation 2):
|
||
Arkady Kanevsky: Finding all minimum-size separating vertex sets in
|
||
a graph, Networks 23, 533--541, 1993.
|
||
|
||
</p>
|
||
<p>The original paper where this reduction was conceived is
|
||
Shimon Even and R. Endre Tarjan: Network Flow and Testing Graph
|
||
Connectivity, SIAM J. Comput., 4(4), 507–518.
|
||
<a class="ulink" href="https://doi.org/10.1137/0204043" target="_top">https://doi.org/10.1137/0204043</a>
|
||
|
||
</p>
|
||
<p><b>Arguments: </b>
|
||
</p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>graph</code></em>:</span></p></td>
|
||
<td><p>
|
||
A graph. Although directness is not checked, this function
|
||
is commonly used only on directed graphs.
|
||
</p></td>
|
||
</tr>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>graphbar</code></em>:</span></p></td>
|
||
<td><p>
|
||
Pointer to a new directed graph that will contain the
|
||
reduction, with twice as many vertices and edges.
|
||
</p></td>
|
||
</tr>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>capacity</code></em>:</span></p></td>
|
||
<td><p>
|
||
Pointer to an initialized vector or a null pointer. If
|
||
not a null pointer, then it will be filled the capacity from
|
||
the reduction: the first |E| elements are 1, the remaining |E|
|
||
are equal to |V| (which is used to indicate infinity).
|
||
</p></td>
|
||
</tr>
|
||
</tbody>
|
||
</table></div>
|
||
<p>
|
||
</p>
|
||
<p><b>Returns: </b></p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody><tr>
|
||
<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
|
||
<td><p>
|
||
Error code.
|
||
</p></td>
|
||
</tr></tbody>
|
||
</table></div>
|
||
<p>
|
||
|
||
Time complexity: O(|E|+|V|).
|
||
|
||
</p>
|
||
<div class="hideshow" onClick="toggle(this, event)">
|
||
<div class="example">
|
||
<a name="id-1.25.6.10.1"></a><p class="title"><b>Example 24.5. File <code class="code">examples/simple/even_tarjan.c</code></b></p>
|
||
<div class="example-contents">
|
||
<pre class="programlisting"><span class="strong"><strong>#include</strong></span> <igraph.h>
|
||
<span class="strong"><strong>#include</strong></span> <limits.h>
|
||
|
||
int <span class="strong"><strong>main</strong></span>(void) {
|
||
|
||
igraph_t g, gbar;
|
||
igraph_int_t k1, k2 = INT_MAX;
|
||
igraph_real_t tmpk;
|
||
igraph_int_t i, j, n;
|
||
<a class="link" href="igraph-Flows.html#igraph_maxflow_stats_t" title="1.4. igraph_maxflow_stats_t — Data structure holding statistics from the push-relabel maximum flow solver.">igraph_maxflow_stats_t</a> stats;
|
||
|
||
<span class="emphasis"><em>/* Initialize the library. */</em></span>
|
||
<span class="strong"><strong><a class="link" href="igraph-Basic.html#igraph_setup" title="4.1. igraph_setup — Initializes the igraph library.">igraph_setup</a></strong></span>();
|
||
|
||
<span class="strong"><strong><a class="link" href="igraph-Generators.html#igraph_famous" title="8.1. igraph_famous — Create a famous graph by simply providing its name.">igraph_famous</a></strong></span>(&g, "meredith");
|
||
<span class="strong"><strong><a class="link" href="igraph-Separators.html#igraph_even_tarjan_reduction" title="5. igraph_even_tarjan_reduction — Even-Tarjan reduction of a graph.">igraph_even_tarjan_reduction</a></strong></span>(&g, &gbar, <span class="emphasis"><em>/*capacity=*/</em></span> NULL);
|
||
|
||
<span class="strong"><strong><a class="link" href="igraph-Flows.html#igraph_vertex_connectivity" title="3.4. igraph_vertex_connectivity — The vertex connectivity of a graph.">igraph_vertex_connectivity</a></strong></span>(&g, &k1, <span class="emphasis"><em>/* checks= */</em></span> false);
|
||
|
||
n = <span class="strong"><strong><a class="link" href="igraph-Basic.html#igraph_vcount" title="5.2.1. igraph_vcount — The number of vertices in a graph.">igraph_vcount</a></strong></span>(&g);
|
||
<span class="strong"><strong>for</strong></span> (i = 0; i < n; i++) {
|
||
<span class="strong"><strong>for</strong></span> (j = i + 1; j < n; j++) {
|
||
igraph_bool_t conn;
|
||
<span class="strong"><strong><a class="link" href="igraph-Structural.html#igraph_are_adjacent" title="1.1. igraph_are_adjacent — Decides whether two vertices are adjacent.">igraph_are_adjacent</a></strong></span>(&g, i, j, &conn);
|
||
<span class="strong"><strong>if</strong></span> (conn) {
|
||
<span class="strong"><strong>continue</strong></span>;
|
||
}
|
||
<span class="strong"><strong><a class="link" href="igraph-Flows.html#igraph_maxflow_value" title="1.2. igraph_maxflow_value — Maximum flow in a network with the push/relabel algorithm.">igraph_maxflow_value</a></strong></span>(&gbar, &tmpk,
|
||
<span class="emphasis"><em>/* source= */</em></span> i + n,
|
||
<span class="emphasis"><em>/* target= */</em></span> j,
|
||
<span class="emphasis"><em>/* capacity= */</em></span> 0,
|
||
&stats);
|
||
<span class="strong"><strong>if</strong></span> (tmpk < k2) {
|
||
k2 = tmpk;
|
||
}
|
||
}
|
||
}
|
||
|
||
<span class="strong"><strong><a class="link" href="igraph-Basic.html#igraph_destroy" title="5.1.4. igraph_destroy — Frees the memory allocated for a graph object.">igraph_destroy</a></strong></span>(&gbar);
|
||
<span class="strong"><strong><a class="link" href="igraph-Basic.html#igraph_destroy" title="5.1.4. igraph_destroy — Frees the memory allocated for a graph object.">igraph_destroy</a></strong></span>(&g);
|
||
|
||
<span class="strong"><strong>if</strong></span> (k1 != k2) {
|
||
<span class="strong"><strong>printf</strong></span>("k1 = %" IGRAPH_PRId " while k2 = %" IGRAPH_PRId "\n", k1, k2);
|
||
<span class="strong"><strong>return</strong></span> 1;
|
||
}
|
||
|
||
<span class="strong"><strong>return</strong></span> 0;
|
||
}
|
||
</pre>
|
||
<p></p>
|
||
</div>
|
||
</div>
|
||
<br class="example-break">
|
||
</div>
|
||
<p>
|
||
</p>
|
||
</div>
|
||
</div>
|
||
<table class="navigation-footer" width="100%" summary="Navigation footer" cellpadding="2" cellspacing="0"><tr valign="middle">
|
||
<td align="left"><a accesskey="p" href="igraph-Flows.html"><b>← Chapter 23. Maximum flows, minimum cuts and related measures</b></a></td>
|
||
<td align="right"><a accesskey="n" href="igraph-Community.html"><b>Chapter 25. Detecting community structure →</b></a></td>
|
||
</tr></table>
|
||
</body>
|
||
</html>
|