<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@a51232dd240645ddb792f489339f19b4" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="False" data-init="VerticalStudentView" data-request-token="078fc852dbe011efb03d029b57e3cc89">
<h2 class="hd hd-2 unit-title">Introduction</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@html+block@e7f0c7b023eb42bb97ea9c45c707c3dd">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-block-type="html" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@html+block@e7f0c7b023eb42bb97ea9c45c707c3dd" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="False" data-init="XBlockToXModuleShim" data-request-token="078fc852dbe011efb03d029b57e3cc89">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Read <a href="/assets/courseware/v1/0b1a1890958b6227874af6c69cfcd652/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Session19.pdf" target="[object Object]">Chapter 11.1–11.4 (PDF)</a> of <em>Mathematics for Computer Science</em> for 2.8 Degrees & Isomorphism.</p>
<p>View the <a href="/assets/courseware/v1/3fe4ea512400d98130a87a16f52de982/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_cp19.pdf" target="[object Object]">Section 2.8 In-Class Questions (PDF)</a></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@705c5363db2b47a18ae56f134f1077ad" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="False" data-init="VerticalStudentView" data-request-token="078fc852dbe011efb03d029b57e3cc89">
<h2 class="hd hd-2 unit-title">Lecture Video | Degrees</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@video+block@590194c71cff4d5fa51d2f59cccbcd30">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-block-type="video" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@video+block@590194c71cff4d5fa51d2f59cccbcd30" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="False" data-init="XBlockToXModuleShim" data-request-token="078fc852dbe011efb03d029b57e3cc89">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Degrees</h3>
<div
id="video_590194c71cff4d5fa51d2f59cccbcd30"
class="video closed"
data-metadata='{"autoAdvance": false, "transcriptLanguages": {"en": "English"}, "savedVideoPosition": 0.0, "recordedYoutubeIsAvailable": true, "speed": null, "transcriptTranslationUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@590194c71cff4d5fa51d2f59cccbcd30/handler/transcript/translation/__lang__", "start": 0.0, "showCaptions": "true", "ytTestTimeout": 1500, "saveStateEnabled": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "generalSpeed": 1.0, "transcriptAvailableTranslationsUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@590194c71cff4d5fa51d2f59cccbcd30/handler/transcript/available_translations", "completionPercentage": 0.95, "end": 0.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": ["https://ia800207.us.archive.org/32/items/MIT6.042JS15/MIT6_042JS15_simpledegrees_video_ipod.mp4"], "ytMetadataEndpoint": "", "prioritizeHls": false, "autohideHtml5": false, "autoplay": false, "poster": null, "publishCompletionUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@590194c71cff4d5fa51d2f59cccbcd30/handler/publish_completion", "saveStateUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@590194c71cff4d5fa51d2f59cccbcd30/handler/xmodule_handler/save_user_state", "duration": 0.0, "captionDataDir": null, "completionEnabled": false, "streams": "1.00:TIpnudrzvgg"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="590194c71cff4d5fa51d2f59cccbcd30"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_590194c71cff4d5fa51d2f59cccbcd30">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_590194c71cff4d5fa51d2f59cccbcd30">
<div class="wrapper-download-video">
<h4 class="hd hd-5">Video</h4>
<a class="btn-link video-sources video-download-button" href="https://ia800207.us.archive.org/32/items/MIT6.042JS15/MIT6_042JS15_simpledegrees_video_ipod.mp4">
Download video file
</a>
</div>
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@590194c71cff4d5fa51d2f59cccbcd30/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@590194c71cff4d5fa51d2f59cccbcd30/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:OCW+6.042J+2T2019+type@html+block@e60d94bacbdb4932918fccc3952452eb">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-block-type="html" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@html+block@e60d94bacbdb4932918fccc3952452eb" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="False" data-init="XBlockToXModuleShim" data-request-token="078fc852dbe011efb03d029b57e3cc89">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Download a copy of the slides for <a href="/assets/courseware/v1/8c6d1a11a50fce8c7b9ad398f5565510/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Degrees.pdf" target="[object Object]">Degrees (PDF)</a></p>
<p><a href="/assets/courseware/v1/210f46a1540f911ea1b61cb4bc8954f1/asset-v1:OCW+6.042J+2T2019+type@asset+block/Degrees_2.8_Lectrans.pdf" target="[object Object]">Lecture video transcript (PDF)</a></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@d14676ae057b40c792124bbede7b586e" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="False" data-init="VerticalStudentView" data-request-token="078fc852dbe011efb03d029b57e3cc89">
<h2 class="hd hd-2 unit-title">Exercise | Counting Degrees & Edges</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@77da1d35907a4db79f1c1c2c290be129">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@77da1d35907a4db79f1c1c2c290be129" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="True" data-init="XBlockToXModuleShim" data-request-token="078fc852dbe011efb03d029b57e3cc89">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_77da1d35907a4db79f1c1c2c290be129" class="problems-wrapper" role="group"
aria-labelledby="77da1d35907a4db79f1c1c2c290be129-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@77da1d35907a4db79f1c1c2c290be129" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@77da1d35907a4db79f1c1c2c290be129/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="2"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="77da1d35907a4db79f1c1c2c290be129-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@77da1d35907a4db79f1c1c2c290be129-problem-progress" tabindex="-1">
Counting Degrees &amp; Edges
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@77da1d35907a4db79f1c1c2c290be129-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>1. How many edges does a graph have if the degrees of its vertices are 4, 3, 3, 2, and 2?</p> <div id="inputtype_77da1d35907a4db79f1c1c2c290be129_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_77da1d35907a4db79f1c1c2c290be129_2_1" id="input_77da1d35907a4db79f1c1c2c290be129_2_1" aria-describedby="status_77da1d35907a4db79f1c1c2c290be129_2_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_77da1d35907a4db79f1c1c2c290be129_2_1"/>
<span class="status unanswered" id="status_77da1d35907a4db79f1c1c2c290be129_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_77da1d35907a4db79f1c1c2c290be129_2_1" class="answer"/>
</div>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_77da1d35907a4db79f1c1c2c290be129_3_1">
<fieldset aria-describedby="status_77da1d35907a4db79f1c1c2c290be129_3_1">
<legend id="77da1d35907a4db79f1c1c2c290be129_3_1-legend" class="response-fieldset-legend field-group-hd">2. Which of the following must be true for any simple graph <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>G</mi>
</math></legend>
<div class="field">
<input type="checkbox" name="input_77da1d35907a4db79f1c1c2c290be129_3_1[]" id="input_77da1d35907a4db79f1c1c2c290be129_3_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="77da1d35907a4db79f1c1c2c290be129_3_1-choice_0-label" for="input_77da1d35907a4db79f1c1c2c290be129_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_77da1d35907a4db79f1c1c2c290be129_3_1"> The sum of the degree of every vertex in <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>G</mi>
</math> is even.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_77da1d35907a4db79f1c1c2c290be129_3_1[]" id="input_77da1d35907a4db79f1c1c2c290be129_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="77da1d35907a4db79f1c1c2c290be129_3_1-choice_1-label" for="input_77da1d35907a4db79f1c1c2c290be129_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_77da1d35907a4db79f1c1c2c290be129_3_1"> There are an even number of vertices with odd degree.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_77da1d35907a4db79f1c1c2c290be129_3_1[]" id="input_77da1d35907a4db79f1c1c2c290be129_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="77da1d35907a4db79f1c1c2c290be129_3_1-choice_2-label" for="input_77da1d35907a4db79f1c1c2c290be129_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_77da1d35907a4db79f1c1c2c290be129_3_1"> There are an even number of vertices with even degree.
</label>
</div>
<span id="answer_77da1d35907a4db79f1c1c2c290be129_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_77da1d35907a4db79f1c1c2c290be129_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_77da1d35907a4db79f1c1c2c290be129_solution_1"/>
</div><div class="solution-span">
<span id="solution_77da1d35907a4db79f1c1c2c290be129_solution_2"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Counting Degrees &amp; Edges" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_77da1d35907a4db79f1c1c2c290be129" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_77da1d35907a4db79f1c1c2c290be129">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="77da1d35907a4db79f1c1c2c290be129-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show Answer</span></button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="77da1d35907a4db79f1c1c2c290be129-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="77da1d35907a4db79f1c1c2c290be129-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="77da1d35907a4db79f1c1c2c290be129-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@872422245dcc41b095fc49cbc88a731a" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="False" data-init="VerticalStudentView" data-request-token="078fc852dbe011efb03d029b57e3cc89">
<h2 class="hd hd-2 unit-title">Lecture Video | Isomorphism</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@video+block@f4cab5863bc64b47b65b26bb02d08837">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-block-type="video" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@video+block@f4cab5863bc64b47b65b26bb02d08837" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="False" data-init="XBlockToXModuleShim" data-request-token="078fc852dbe011efb03d029b57e3cc89">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Isomorphism</h3>
<div
id="video_f4cab5863bc64b47b65b26bb02d08837"
class="video closed"
data-metadata='{"autoAdvance": false, "transcriptLanguages": {"en": "English"}, "savedVideoPosition": 0.0, "recordedYoutubeIsAvailable": true, "speed": null, "transcriptTranslationUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@f4cab5863bc64b47b65b26bb02d08837/handler/transcript/translation/__lang__", "start": 0.0, "showCaptions": "true", "ytTestTimeout": 1500, "saveStateEnabled": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "generalSpeed": 1.0, "transcriptAvailableTranslationsUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@f4cab5863bc64b47b65b26bb02d08837/handler/transcript/available_translations", "completionPercentage": 0.95, "end": 0.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": ["https://ia800207.us.archive.org/32/items/MIT6.042JS15/MIT6_042JS15_simple_isomorphism_video_ipod.mp4"], "ytMetadataEndpoint": "", "prioritizeHls": false, "autohideHtml5": false, "autoplay": false, "poster": null, "publishCompletionUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@f4cab5863bc64b47b65b26bb02d08837/handler/publish_completion", "saveStateUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@f4cab5863bc64b47b65b26bb02d08837/handler/xmodule_handler/save_user_state", "duration": 0.0, "captionDataDir": null, "completionEnabled": false, "streams": "1.00:hVerxuP4cFg"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="f4cab5863bc64b47b65b26bb02d08837"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_f4cab5863bc64b47b65b26bb02d08837">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_f4cab5863bc64b47b65b26bb02d08837">
<div class="wrapper-download-video">
<h4 class="hd hd-5">Video</h4>
<a class="btn-link video-sources video-download-button" href="https://ia800207.us.archive.org/32/items/MIT6.042JS15/MIT6_042JS15_simple_isomorphism_video_ipod.mp4">
Download video file
</a>
</div>
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@f4cab5863bc64b47b65b26bb02d08837/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@f4cab5863bc64b47b65b26bb02d08837/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:OCW+6.042J+2T2019+type@html+block@7d0ae485edaa43e69c74ad09bd9d5189">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-block-type="html" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@html+block@7d0ae485edaa43e69c74ad09bd9d5189" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="False" data-init="XBlockToXModuleShim" data-request-token="078fc852dbe011efb03d029b57e3cc89">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Download a copy of the slides for <a href="/assets/courseware/v1/29acee3409f0f1dd76e9ed6cd546b90f/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Isomorphism.pdf" target="[object Object]">Isomorphism (PDF)</a></p>
<p><a href="/assets/courseware/v1/bb664907751e462f79fb0265758d1863/asset-v1:OCW+6.042J+2T2019+type@asset+block/Isomorphism_2.8_Lectrans.pdf" target="[object Object]">Lecture video transcript (PDF)</a></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@ecccde397f2c411295d9456e181296b3" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="False" data-init="VerticalStudentView" data-request-token="078fc852dbe011efb03d029b57e3cc89">
<h2 class="hd hd-2 unit-title">Exercise | Isomorphism</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@c1b4987a3ca8473e98c5b4d0fe825aae">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@c1b4987a3ca8473e98c5b4d0fe825aae" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="True" data-init="XBlockToXModuleShim" data-request-token="078fc852dbe011efb03d029b57e3cc89">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<p>Could not format HTML for problem. Contact course staff in the discussion forum for assistance.</p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@f17c7587ea854c5cadfaeb2fdcda41ab" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="False" data-init="VerticalStudentView" data-request-token="078fc852dbe011efb03d029b57e3cc89">
<h2 class="hd hd-2 unit-title">Exercise | Extreme Graphs</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@eee92383c0cb4e87a8cf422f5e99baae">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@eee92383c0cb4e87a8cf422f5e99baae" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="True" data-init="XBlockToXModuleShim" data-request-token="078fc852dbe011efb03d029b57e3cc89">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_eee92383c0cb4e87a8cf422f5e99baae" class="problems-wrapper" role="group"
aria-labelledby="eee92383c0cb4e87a8cf422f5e99baae-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@eee92383c0cb4e87a8cf422f5e99baae" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@eee92383c0cb4e87a8cf422f5e99baae/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="2"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="eee92383c0cb4e87a8cf422f5e99baae-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@eee92383c0cb4e87a8cf422f5e99baae-problem-progress" tabindex="-1">
Extreme Graphs
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@eee92383c0cb4e87a8cf422f5e99baae-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>For a simple connected graph with a total degree of 44, what is </p>
<div id="inputtype_eee92383c0cb4e87a8cf422f5e99baae_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_eee92383c0cb4e87a8cf422f5e99baae_2_1" id="label_eee92383c0cb4e87a8cf422f5e99baae_2_1">1. The minimum number of vertices?</label>
<input type="text" name="input_eee92383c0cb4e87a8cf422f5e99baae_2_1" id="input_eee92383c0cb4e87a8cf422f5e99baae_2_1" aria-describedby="status_eee92383c0cb4e87a8cf422f5e99baae_2_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_eee92383c0cb4e87a8cf422f5e99baae_2_1"/>
<span class="status unanswered" id="status_eee92383c0cb4e87a8cf422f5e99baae_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_eee92383c0cb4e87a8cf422f5e99baae_2_1" class="answer"/>
</div>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_eee92383c0cb4e87a8cf422f5e99baae_3_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_eee92383c0cb4e87a8cf422f5e99baae_3_1" id="label_eee92383c0cb4e87a8cf422f5e99baae_3_1">2. The maximum number of vertices?</label>
<input type="text" name="input_eee92383c0cb4e87a8cf422f5e99baae_3_1" id="input_eee92383c0cb4e87a8cf422f5e99baae_3_1" aria-describedby="status_eee92383c0cb4e87a8cf422f5e99baae_3_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_eee92383c0cb4e87a8cf422f5e99baae_3_1"/>
<span class="status unanswered" id="status_eee92383c0cb4e87a8cf422f5e99baae_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_eee92383c0cb4e87a8cf422f5e99baae_3_1" class="answer"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_eee92383c0cb4e87a8cf422f5e99baae_solution_1"/>
</div><div class="solution-span">
<span id="solution_eee92383c0cb4e87a8cf422f5e99baae_solution_2"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Extreme Graphs" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_eee92383c0cb4e87a8cf422f5e99baae" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_eee92383c0cb4e87a8cf422f5e99baae">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="eee92383c0cb4e87a8cf422f5e99baae-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show Answer</span></button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="eee92383c0cb4e87a8cf422f5e99baae-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="eee92383c0cb4e87a8cf422f5e99baae-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="eee92383c0cb4e87a8cf422f5e99baae-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@730e86c82a68484d96752184960a406b" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="False" data-init="VerticalStudentView" data-request-token="078fc852dbe011efb03d029b57e3cc89">
<h2 class="hd hd-2 unit-title">Exercise | Isomorphic Graphs</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@c7a0db306b4e431494fbeef0aa325881">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@c7a0db306b4e431494fbeef0aa325881" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="True" data-init="XBlockToXModuleShim" data-request-token="078fc852dbe011efb03d029b57e3cc89">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_c7a0db306b4e431494fbeef0aa325881" class="problems-wrapper" role="group"
aria-labelledby="c7a0db306b4e431494fbeef0aa325881-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@c7a0db306b4e431494fbeef0aa325881" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@c7a0db306b4e431494fbeef0aa325881/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="c7a0db306b4e431494fbeef0aa325881-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@c7a0db306b4e431494fbeef0aa325881-problem-progress" tabindex="-1">
Isomorphic Graphs
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@c7a0db306b4e431494fbeef0aa325881-problem-progress"></div>
<div class="problem">
<div>
<p><img height="222" width="428" src="/assets/courseware/v1/eccd87dc42e050b7d40ae0030f136824/asset-v1:OCW+6.042J+2T2019+type@asset+block/6.042_Unit_II_2.86.jpg" alt="Isomorphic graphs"/></p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_c7a0db306b4e431494fbeef0aa325881_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_c7a0db306b4e431494fbeef0aa325881_2_1" id="label_c7a0db306b4e431494fbeef0aa325881_2_1">The two graphs above are isomorphic, which means that there exists an edge-preserving bijection from the set of vertices of the graph on the left to the set of vertices of the graph on the right.
<br/>
<br/>
How many such bijections are there?</label>
<input type="text" name="input_c7a0db306b4e431494fbeef0aa325881_2_1" id="input_c7a0db306b4e431494fbeef0aa325881_2_1" aria-describedby="status_c7a0db306b4e431494fbeef0aa325881_2_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_c7a0db306b4e431494fbeef0aa325881_2_1"/>
<span class="status unanswered" id="status_c7a0db306b4e431494fbeef0aa325881_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_c7a0db306b4e431494fbeef0aa325881_2_1" class="answer"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_c7a0db306b4e431494fbeef0aa325881_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Isomorphic Graphs" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_c7a0db306b4e431494fbeef0aa325881" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_c7a0db306b4e431494fbeef0aa325881">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="c7a0db306b4e431494fbeef0aa325881-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show Answer</span></button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="c7a0db306b4e431494fbeef0aa325881-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="c7a0db306b4e431494fbeef0aa325881-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="c7a0db306b4e431494fbeef0aa325881-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@bd3d10d2391040ecace9b1eec3aaeb5a" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="False" data-init="VerticalStudentView" data-request-token="078fc852dbe011efb03d029b57e3cc89">
<h2 class="hd hd-2 unit-title">Exercise | Non-Isomorphic Graphs</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@d010d2bec8f648da8c6bc5a0ca1c6a65">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@d010d2bec8f648da8c6bc5a0ca1c6a65" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="True" data-init="XBlockToXModuleShim" data-request-token="078fc852dbe011efb03d029b57e3cc89">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_d010d2bec8f648da8c6bc5a0ca1c6a65" class="problems-wrapper" role="group"
aria-labelledby="d010d2bec8f648da8c6bc5a0ca1c6a65-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@d010d2bec8f648da8c6bc5a0ca1c6a65" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@d010d2bec8f648da8c6bc5a0ca1c6a65/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="d010d2bec8f648da8c6bc5a0ca1c6a65-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@d010d2bec8f648da8c6bc5a0ca1c6a65-problem-progress" tabindex="-1">
Non-Isomorphic Graphs
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@d010d2bec8f648da8c6bc5a0ca1c6a65-problem-progress"></div>
<div class="problem">
<div>
<p><img height="220" width="433" src="/assets/courseware/v1/c76289843d8aa1a0f478349226ea3216/asset-v1:OCW+6.042J+2T2019+type@asset+block/6.042_Unit_II_2.87.jpg" alt="Non-Isomorphic Graphs"/></p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1">
<fieldset aria-describedby="status_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1">
<legend id="d010d2bec8f648da8c6bc5a0ca1c6a65_2_1-legend" class="response-fieldset-legend field-group-hd">Let's call the graph on the left <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>G</mi>
</math> and the graph on the right <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>H</mi>
</math>. The two graphs are not isomorphic. Indicate which of the following assertions can prove this fact.</legend>
<div class="field">
<input type="checkbox" name="input_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1[]" id="input_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="d010d2bec8f648da8c6bc5a0ca1c6a65_2_1-choice_0-label" for="input_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>H</mi>
</math> has a vertex of degree 4, but <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>G</mi>
</math> does not.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1[]" id="input_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="d010d2bec8f648da8c6bc5a0ca1c6a65_2_1-choice_1-label" for="input_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</math> has degree 3, but <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>v</mi>
<mn>1</mn>
</msub>
</math> has degree 2.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1[]" id="input_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="d010d2bec8f648da8c6bc5a0ca1c6a65_2_1-choice_2-label" for="input_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>G</mi>
</math> has four vertices of degree 3, but <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>H</mi>
</math> has only two vertices of degree 3.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1[]" id="input_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="d010d2bec8f648da8c6bc5a0ca1c6a65_2_1-choice_3-label" for="input_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>G</mi>
</math> has a vertex named <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</math>, but <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>H</mi>
</math> does not.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1[]" id="input_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="d010d2bec8f648da8c6bc5a0ca1c6a65_2_1-choice_4-label" for="input_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>G</mi>
</math> has three vertices that form a right triangle, but no three vertices of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>H</mi>
</math> form a right triangle.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1[]" id="input_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="d010d2bec8f648da8c6bc5a0ca1c6a65_2_1-choice_5-label" for="input_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>G</mi>
</math> has an edge that is incident to two vertices of degree 3, but <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>H</mi>
</math> does not.
</label>
</div>
<span id="answer_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_d010d2bec8f648da8c6bc5a0ca1c6a65_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_d010d2bec8f648da8c6bc5a0ca1c6a65_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Non-Isomorphic Graphs" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_d010d2bec8f648da8c6bc5a0ca1c6a65" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_d010d2bec8f648da8c6bc5a0ca1c6a65">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="d010d2bec8f648da8c6bc5a0ca1c6a65-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show Answer</span></button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="d010d2bec8f648da8c6bc5a0ca1c6a65-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="d010d2bec8f648da8c6bc5a0ca1c6a65-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="d010d2bec8f648da8c6bc5a0ca1c6a65-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>