图的支配细分数的算法复杂性和边界
IF 1.3
4区 数学
Q1 MATHEMATICS
Fu-Tao Hu, Chang-Xu Zhang, Shu-Cheng Yang
{"title":"图的支配细分数的算法复杂性和边界","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":"{\"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}","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
摘要
假设是一个简单图。如果不在 的每个顶点都与在 的顶点相邻，那么这个子集就是一个支配集。支配数 ，表示为 ，是支配集的最小心数。 支配细分数 ，是为增加支配数而必须细分的最小边数（每条边最多可细分一次）。2000 年，Haynes 等人的研究表明，对于任何有且的边，都是阶数不小于 3 的连通图。在本文中，我们将上述约束改进为 ，并进一步证明了判断是否为 NP-hard的决策问题。此外，我们还展示了一些图的支配细分数的边界或精确值。本文章由计算机程序翻译，如有差异，请以英文原文为准。
Algorithmic Complexity and Bounds for Domination Subdivision Numbers of Graphs
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.
求助全文
通过发布文献求助，成功后即可免费获取论文全文。
去求助
来源期刊
Journal of Mathematics
Mathematics-General Mathematics
CiteScore
2.50
自引率
14.30%
发文量
0
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.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX
EndNote
RefMan
NoteFirst
NoteExpress
求助内容：
应助结果提醒方式：请完成安全验证×
微信好友
朋友圈
QQ好友
复制链接
取消
已复制链接
快去分享给好友吧!
我知道了
点击右上角分享
相关文献
京公网安备 11010802042870号
