{"title":"Algorithmic Complexity and Bounds for Domination Subdivision Numbers of Graphs","authors":"Fu-Tao Hu, Chang-Xu Zhang, Shu-Cheng Yang","doi":"10.1155/2024/3795448","DOIUrl":null,"url":null,"abstract":"Let <span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 20.155 11.5564\" width=\"20.155pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,12.524,0)\"></path></g></svg><span></span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"23.7371838 -9.28833 14.99 11.5564\" width=\"14.99pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,23.787,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,28.285,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,35.813,0)\"></path></g></svg><span></span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"40.9061838 -9.28833 12.769 11.5564\" width=\"12.769pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,40.956,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,48.964,0)\"></path></g></svg></span> be a simple graph. A subset <span><svg height=\"9.75571pt\" style=\"vertical-align:-1.11981pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 21.221 9.75571\" width=\"21.221pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,13.59,0)\"></path></g></svg><span></span><svg height=\"9.75571pt\" style=\"vertical-align:-1.11981pt\" version=\"1.1\" viewbox=\"24.803183800000003 -8.6359 9.417 9.75571\" width=\"9.417pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,24.853,0)\"><use xlink:href=\"#g113-87\"></use></g></svg></span> is a dominating set if every vertex not in <svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 10.095 8.68572\" width=\"10.095pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-69\"></use></g></svg> is adjacent to a vertex in <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 10.095 8.68572\" width=\"10.095pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-69\"></use></g></svg>.</span> The domination number of <span><svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.02496 8.8423\" width=\"9.02496pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-72\"></use></g></svg>,</span> denoted by <span><svg height=\"12.7178pt\" style=\"vertical-align:-3.42947pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 24.5921 12.7178\" width=\"24.5921pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,6.516,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,11.014,0)\"><use xlink:href=\"#g113-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,19.906,0)\"><use xlink:href=\"#g113-42\"></use></g></svg>,</span> is the smallest cardinality of a dominating set of <span><svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.02496 8.8423\" width=\"9.02496pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-72\"></use></g></svg>.</span> The domination subdivision number <svg height=\"14.8369pt\" style=\"vertical-align:-5.528999pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.3079 35.0582 14.8369\" width=\"35.0582pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,4.875,0)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,11.783,3.132)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,17.004,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,21.502,0)\"><use xlink:href=\"#g113-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,30.394,0)\"><use xlink:href=\"#g113-42\"></use></g></svg> of <svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.02496 8.8423\" width=\"9.02496pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-72\"></use></g></svg> is the minimum number of edges that must be subdivided (each edge can be subdivided at most once) in order to increase the domination number. In 2000, Haynes et al. showed that <span><svg height=\"14.8369pt\" style=\"vertical-align:-5.528999pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.3079 46.155 14.8369\" width=\"46.155pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g190-116\"></use></g><g transform=\"matrix(.013,0,0,-0.013,4.875,0)\"><use xlink:href=\"#g190-101\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,11.783,3.132)\"><use xlink:href=\"#g50-225\"></use></g><g transform=\"matrix(.013,0,0,-0.013,17.004,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,21.502,0)\"><use xlink:href=\"#g113-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,30.394,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,38.524,0)\"></path></g></svg><span></span><svg height=\"14.8369pt\" style=\"vertical-align:-5.528999pt\" version=\"1.1\" viewbox=\"49.7361838 -9.3079 39.374 14.8369\" width=\"39.374pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,49.786,0)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,56.936,3.132)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,63.764,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,68.262,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,74.126,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,81.529,0)\"></path></g></svg><span></span><svg height=\"14.8369pt\" style=\"vertical-align:-5.528999pt\" version=\"1.1\" viewbox=\"92.01618380000001 -9.3079 48.959 14.8369\" width=\"48.959pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,92.066,0)\"><use xlink:href=\"#g113-101\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,99.216,3.132)\"><use xlink:href=\"#g50-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,106.044,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,110.542,0)\"><use xlink:href=\"#g185-40\"></use></g><g transform=\"matrix(.013,0,0,-0.013,116.405,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,123.808,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,134.345,0)\"></path></g></svg></span> for any edge <span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 23.353 11.5564\" width=\"23.353pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,6.994,0)\"><use xlink:href=\"#g185-40\"></use></g><g transform=\"matrix(.013,0,0,-0.013,16.489,0)\"></path></g></svg><span></span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"26.9351838 -9.28833 26.166 11.5564\" width=\"26.166pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,26.985,0)\"><use xlink:href=\"#g113-70\"></use></g><g transform=\"matrix(.013,0,0,-0.013,34.993,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,39.491,0)\"><use xlink:href=\"#g113-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,48.384,0)\"><use xlink:href=\"#g113-42\"></use></g></svg></span> with <span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 41.231 12.5794\" width=\"41.231pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-101\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,7.15,3.132)\"><use xlink:href=\"#g50-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,13.978,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,18.476,0)\"><use xlink:href=\"#g113-118\"></use></g><g transform=\"matrix(.013,0,0,-0.013,25.47,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,33.6,0)\"></path></g></svg><span></span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"44.8131838 -9.28833 6.478 12.5794\" width=\"6.478pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,44.863,0)\"></path></g></svg></span> and <span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 40.1 12.5794\" width=\"40.1pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-101\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,7.15,3.132)\"><use xlink:href=\"#g50-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,13.978,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,18.476,0)\"><use xlink:href=\"#g185-40\"></use></g><g transform=\"matrix(.013,0,0,-0.013,24.339,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,32.469,0)\"><use xlink:href=\"#g117-94\"></use></g></svg><span></span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"43.6821838 -9.28833 6.474 12.5794\" width=\"6.474pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,43.732,0)\"><use xlink:href=\"#g113-51\"></use></g></svg></span> where <svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.02496 8.8423\" width=\"9.02496pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-72\"></use></g></svg> is a connected graph with order no less than 3. In this paper, we improve the above bound to <span><svg height=\"14.8369pt\" style=\"vertical-align:-5.528999pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.3079 46.155 14.8369\" width=\"46.155pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g190-116\"></use></g><g transform=\"matrix(.013,0,0,-0.013,4.875,0)\"><use xlink:href=\"#g190-101\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,11.783,3.132)\"><use xlink:href=\"#g50-225\"></use></g><g transform=\"matrix(.013,0,0,-0.013,17.004,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,21.502,0)\"><use xlink:href=\"#g113-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,30.394,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,38.524,0)\"><use xlink:href=\"#g117-93\"></use></g></svg><span></span><svg height=\"14.8369pt\" style=\"vertical-align:-5.528999pt\" version=\"1.1\" viewbox=\"49.7361838 -9.3079 40.505 14.8369\" width=\"40.505pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,49.786,0)\"><use xlink:href=\"#g113-101\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,56.936,3.132)\"><use xlink:href=\"#g50-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,63.764,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,68.262,0)\"><use xlink:href=\"#g113-118\"></use></g><g transform=\"matrix(.013,0,0,-0.013,75.257,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,82.66,0)\"><use xlink:href=\"#g117-36\"></use></g></svg><span></span><svg height=\"14.8369pt\" style=\"vertical-align:-5.528999pt\" version=\"1.1\" viewbox=\"93.14718380000001 -9.3079 89.115 14.8369\" width=\"89.115pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,93.197,0)\"><use xlink:href=\"#g113-101\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,100.347,3.132)\"><use xlink:href=\"#g50-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,107.175,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,111.673,0)\"><use xlink:href=\"#g185-40\"></use></g><g transform=\"matrix(.013,0,0,-0.013,117.536,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,124.939,0)\"><use xlink:href=\"#g117-33\"></use></g><g transform=\"matrix(.013,0,0,-0.013,135.476,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,138.895,0)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,148.957,3.132)\"><use xlink:href=\"#g50-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,155.785,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,160.283,0)\"><use xlink:href=\"#g113-118\"></use></g><g transform=\"matrix(.013,0,0,-0.013,167.277,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,174.681,0)\"></path></g></svg><span></span><span><svg height=\"14.8369pt\" style=\"vertical-align:-5.528999pt\" version=\"1.1\" viewbox=\"185.1671838 -9.3079 55.569 14.8369\" width=\"55.569pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,185.217,0)\"><use xlink:href=\"#g113-79\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,195.279,3.132)\"><use xlink:href=\"#g50-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,202.107,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,206.605,0)\"><use xlink:href=\"#g185-40\"></use></g><g transform=\"matrix(.013,0,0,-0.013,212.469,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,216.967,0)\"><use xlink:href=\"#g113-9\"></use></g><g transform=\"matrix(.013,0,0,-0.013,223.291,0)\"><use xlink:href=\"#g117-33\"></use></g><g transform=\"matrix(.013,0,0,-0.013,233.827,0)\"><use xlink:href=\"#g113-50\"></use></g></svg>,</span></span> and furthermore, we show the decision problem for determining whether <span><svg height=\"14.8369pt\" style=\"vertical-align:-5.528999pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.3079 46.155 14.8369\" width=\"46.155pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g190-116\"></use></g><g transform=\"matrix(.013,0,0,-0.013,4.875,0)\"><use xlink:href=\"#g190-101\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,11.783,3.132)\"><use xlink:href=\"#g50-225\"></use></g><g transform=\"matrix(.013,0,0,-0.013,17.004,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,21.502,0)\"><use xlink:href=\"#g113-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,30.394,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,38.524,0)\"><use xlink:href=\"#g117-34\"></use></g></svg><span></span><svg height=\"14.8369pt\" style=\"vertical-align:-5.528999pt\" version=\"1.1\" viewbox=\"49.7361838 -9.3079 6.459 14.8369\" width=\"6.459pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,49.786,0)\"><use xlink:href=\"#g113-50\"></use></g></svg></span> is NP-hard. Moreover, we show some bounds or exact values for domination subdivision numbers of some graphs.","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"39 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1155/2024/3795448","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let be a simple graph. A subset is a dominating set if every vertex not in is adjacent to a vertex in . The domination number of , denoted by , is the smallest cardinality of a dominating set of . The domination subdivision number of is the minimum number of edges that must be subdivided (each edge can be subdivided at most once) in order to increase the domination number. In 2000, Haynes et al. showed that for any edge with and where is a connected graph with order no less than 3. In this paper, we improve the above bound to , and furthermore, we show the decision problem for determining whether is NP-hard. Moreover, we show some bounds or exact values for domination subdivision numbers of some graphs.
期刊介绍：
Journal of Mathematics is a broad scope journal that publishes original research articles as well as review articles on all aspects of both pure and applied mathematics. As well as original research, Journal of Mathematics also publishes focused review articles that assess the state of the art, and identify upcoming challenges and promising solutions for the community.
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