PAT/source/html/a1122.html

Hamiltonian Cycle
<div class="ques-view"><p>The "Hamilton cycle problem" is to find a simple cycle that contains every vertex in a graph. Such a cycle is called a "Hamiltonian cycle".</p>
<p>In this problem, you are supposed to tell if a given cycle is a Hamiltonian cycle.</p>
<h3 id="input-specification-">Input Specification:</h3>
<p>Each input file contains one test case. For each case, the first line contains 2 positive integers <span class="katex"><span class="katex-mathml"><math><mrow><mi>N</mi></mrow>N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span> (<span class="katex"><span class="katex-mathml"><math><mrow><mn>2</mn><mo>&lt;</mo><mi>N</mi><mo>≤</mo><mn>2</mn><mn>0</mn><mn>0</mn></mrow>2&lt; N \le 200</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.8193em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">2</span><span class="mrel">&lt;</span><span class="mord mathit" style="margin-right:0.10903em;">N</span><span class="mrel">≤</span><span class="mord mathrm">2</span><span class="mord mathrm">0</span><span class="mord mathrm">0</span></span></span></span>), the number of vertices, and <span class="katex"><span class="katex-mathml"><math><mrow><mi>M</mi></mrow>M</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">M</span></span></span></span>, the number of edges in an undirected graph. Then <span class="katex"><span class="katex-mathml"><math><mrow><mi>M</mi></mrow>M</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">M</span></span></span></span> lines follow, each describes an edge in the format <code>Vertex1 Vertex2</code>, where the vertices are numbered from 1 to <span class="katex"><span class="katex-mathml"><math><mrow><mi>N</mi></mrow>N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span>. The next line gives a positive integer <span class="katex"><span class="katex-mathml"><math><mrow><mi>K</mi></mrow>K</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.07153em;">K</span></span></span></span> which is the number of queries, followed by <span class="katex"><span class="katex-mathml"><math><mrow><mi>K</mi></mrow>K</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.07153em;">K</span></span></span></span> lines of queries, each in the format:</p>
<p><span class="katex"><span class="katex-mathml"><math><mrow><mi>n</mi></mrow>n</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">n</span></span></span></span> <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>V</mi><mn>1</mn></msub></mrow>V_1</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.22222em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>V</mi><mn>2</mn></msub></mrow>V_2</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.22222em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> ... <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>V</mi><mi>n</mi></msub></mrow>V_n</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.22222em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span></p>
<p>where <span class="katex"><span class="katex-mathml"><math><mrow><mi>n</mi></mrow>n</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">n</span></span></span></span> is the number of vertices in the list, and <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow>V_i</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.22222em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span>'s are the vertices on a path.</p>
<h3 id="output-specification-">Output Specification:</h3>
<p>For each query, print in a line <code>YES</code> if the path does form a Hamiltonian cycle, or <code>NO</code> if not.</p>
<h3 id="sample-input-">Sample Input:</h3>
<pre><code class="lang-in">6 10
6 2
3 4
1 5
2 5
3 1
4 1
1 6
6 3
1 2
4 5
6
7 5 1 4 3 6 2 5
6 5 1 4 3 6 2
9 6 2 1 6 3 4 5 2 6
4 1 2 5 1
7 6 1 3 4 5 2 6
7 6 1 2 5 4 3 1
</code></pre>
<h3 id="sample-output-">Sample Output:</h3>
<pre><code class="lang-out">YES
NO
NO
NO
YES
NO
</code></pre>
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