Trees having many minimal dominating sets
We disprove a conjecture by Skupień that every tree of order n has at most 2n/2 minimal dominating sets. We construct a family of trees of both parities of the order for which the number of minimal dominating sets exceeds 1.4167n. We also provide an algorithm for listing all minimal dominating sets...
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| Veröffentlicht in: | Information processing letters 2013-04, Vol.113 (8), p.276-279 |
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| creator | Krzywkowski, Marcin |
| description | We disprove a conjecture by Skupień that every tree of order n has at most 2n/2 minimal dominating sets. We construct a family of trees of both parities of the order for which the number of minimal dominating sets exceeds 1.4167n. We also provide an algorithm for listing all minimal dominating sets of a tree in time O(1.4656n). This implies that every tree has at most 1.4656n minimal dominating sets.
► We disprove a conjecture that every tree of order n has at most 2n/2 minimal dominating sets. ► We establish 1.4167n to be a lower bound on the running time of an algorithm for listing all m-d sets of a given tree. ► We provide an algorithm for listing all m-d sets of a tree of order n in time O(1.4656n). ► The above implies that every tree has at most 1.4656n minimal dominating sets. |
| doi_str_mv | 10.1016/j.ipl.2013.01.020 |
| format | Article |
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► We disprove a conjecture that every tree of order n has at most 2n/2 minimal dominating sets. ► We establish 1.4167n to be a lower bound on the running time of an algorithm for listing all m-d sets of a given tree. ► We provide an algorithm for listing all m-d sets of a tree of order n in time O(1.4656n). ► The above implies that every tree has at most 1.4656n minimal dominating sets.</description><identifier>ISSN: 0020-0190</identifier><identifier>EISSN: 1872-6119</identifier><identifier>DOI: 10.1016/j.ipl.2013.01.020</identifier><identifier>CODEN: IFPLAT</identifier><language>eng</language><publisher>Amsterdam: Elsevier B.V</publisher><subject>Algorithms ; Combinatorial bound ; Combinatorial problems ; Exponential algorithm ; Graph theory ; Listing algorithm ; Minimal dominating set ; Set theory ; Studies ; Tree</subject><ispartof>Information processing letters, 2013-04, Vol.113 (8), p.276-279</ispartof><rights>2013 Elsevier B.V.</rights><rights>Copyright Elsevier Sequoia S.A. Apr 30, 2013</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c325t-24f8b827497a026278fc39846586ecc05b4f43fff3698c3f84be45667180cebf3</citedby><cites>FETCH-LOGICAL-c325t-24f8b827497a026278fc39846586ecc05b4f43fff3698c3f84be45667180cebf3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.ipl.2013.01.020$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,780,784,3548,27923,27924,45994</link.rule.ids></links><search><creatorcontrib>Krzywkowski, Marcin</creatorcontrib><title>Trees having many minimal dominating sets</title><title>Information processing letters</title><description>We disprove a conjecture by Skupień that every tree of order n has at most 2n/2 minimal dominating sets. We construct a family of trees of both parities of the order for which the number of minimal dominating sets exceeds 1.4167n. We also provide an algorithm for listing all minimal dominating sets of a tree in time O(1.4656n). This implies that every tree has at most 1.4656n minimal dominating sets.
► We disprove a conjecture that every tree of order n has at most 2n/2 minimal dominating sets. ► We establish 1.4167n to be a lower bound on the running time of an algorithm for listing all m-d sets of a given tree. ► We provide an algorithm for listing all m-d sets of a tree of order n in time O(1.4656n). ► The above implies that every tree has at most 1.4656n minimal dominating sets.</description><subject>Algorithms</subject><subject>Combinatorial bound</subject><subject>Combinatorial problems</subject><subject>Exponential algorithm</subject><subject>Graph theory</subject><subject>Listing algorithm</subject><subject>Minimal dominating set</subject><subject>Set theory</subject><subject>Studies</subject><subject>Tree</subject><issn>0020-0190</issn><issn>1872-6119</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2013</creationdate><recordtype>article</recordtype><recordid>eNp9UMtOwzAQtBBIlMIHcIvEiUPCru04jjihipdUiUs5W4m7BkdtUuy0Uv8eR-XMaVejmd2ZYewWoUBA9dAVfrcpOKAoAAvgcMZmqCueK8T6nM0gQTlgDZfsKsYOAJQU1YzdrwJRzL6bg--_sm3TH7Ot7_222WTrIW3NOOGRxnjNLlyziXTzN-fs8-V5tXjLlx-v74unZW4FL8ecS6dbzStZVw1wxSvtrKi1VKVWZC2UrXRSOOeEqrUVTsuWZKlUhRostU7M2d3p7i4MP3uKo-mGfejTS4MCFda8VDyx8MSyYYgxkDO7kFyHo0EwUyOmM6kRMzViAE2KnzSPJw0l-wdPwUTrqbe09oHsaNaD_0f9C8krZtY</recordid><startdate>20130430</startdate><enddate>20130430</enddate><creator>Krzywkowski, Marcin</creator><general>Elsevier B.V</general><general>Elsevier Sequoia S.A</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20130430</creationdate><title>Trees having many minimal dominating sets</title><author>Krzywkowski, Marcin</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c325t-24f8b827497a026278fc39846586ecc05b4f43fff3698c3f84be45667180cebf3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2013</creationdate><topic>Algorithms</topic><topic>Combinatorial bound</topic><topic>Combinatorial problems</topic><topic>Exponential algorithm</topic><topic>Graph theory</topic><topic>Listing algorithm</topic><topic>Minimal dominating set</topic><topic>Set theory</topic><topic>Studies</topic><topic>Tree</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Krzywkowski, Marcin</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Information processing letters</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Krzywkowski, Marcin</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Trees having many minimal dominating sets</atitle><jtitle>Information processing letters</jtitle><date>2013-04-30</date><risdate>2013</risdate><volume>113</volume><issue>8</issue><spage>276</spage><epage>279</epage><pages>276-279</pages><issn>0020-0190</issn><eissn>1872-6119</eissn><coden>IFPLAT</coden><abstract>We disprove a conjecture by Skupień that every tree of order n has at most 2n/2 minimal dominating sets. We construct a family of trees of both parities of the order for which the number of minimal dominating sets exceeds 1.4167n. We also provide an algorithm for listing all minimal dominating sets of a tree in time O(1.4656n). This implies that every tree has at most 1.4656n minimal dominating sets.
► We disprove a conjecture that every tree of order n has at most 2n/2 minimal dominating sets. ► We establish 1.4167n to be a lower bound on the running time of an algorithm for listing all m-d sets of a given tree. ► We provide an algorithm for listing all m-d sets of a tree of order n in time O(1.4656n). ► The above implies that every tree has at most 1.4656n minimal dominating sets.</abstract><cop>Amsterdam</cop><pub>Elsevier B.V</pub><doi>10.1016/j.ipl.2013.01.020</doi><tpages>4</tpages></addata></record> |
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| ispartof | Information processing letters, 2013-04, Vol.113 (8), p.276-279 |
| issn | 0020-0190 1872-6119 |
| language | eng |
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| source | ScienceDirect Journals (5 years ago - present) |
| subjects | Algorithms Combinatorial bound Combinatorial problems Exponential algorithm Graph theory Listing algorithm Minimal dominating set Set theory Studies Tree |
| title | Trees having many minimal dominating sets |
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