Computing the -Clique Metric Dimension of Graphs via (Edge) Corona Products and Integer Linear Programming Model
IF 1.3
4区 数学
Q1 MATHEMATICS
Zeinab Shahmiri, Mostafa Tavakoli
{"title":"Computing the -Clique Metric Dimension of Graphs via (Edge) Corona Products and Integer Linear Programming Model","authors":"Zeinab Shahmiri, Mostafa Tavakoli","doi":"10.1155/2024/3241718","DOIUrl":null,"url":null,"abstract":"Let <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)\"></path></g></svg> be a graph with <svg height=\"6.1673pt\" style=\"vertical-align:-0.2063904pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 6.6501 6.1673\" width=\"6.6501pt\" 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></svg> vertices and <span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 26.71 12.5794\" width=\"26.71pt\" 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(.0091,0,0,-0.0091,8.619,3.132)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,19.079,0)\"></path></g></svg><span></span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"30.2921838 -9.28833 20.53 12.5794\" width=\"20.53pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,30.342,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,34.853,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,47.908,0)\"></path></g></svg><span></span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"54.454183799999996 -9.28833 10.208 12.5794\" width=\"10.208pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,54.504,0)\"><use xlink:href=\"#g113-89\"></use></g></svg></span> is an <span><svg height=\"9.49473pt\" style=\"vertical-align:-0.2063999pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 3.60972 9.49473\" width=\"3.60972pt\" 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=\"#g113-109\"></use></g></svg>-</span>clique of <span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 13.5529 11.5564\" width=\"13.5529pt\" 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><g transform=\"matrix(.013,0,0,-0.013,8.892,0)\"></path></g></svg>.</span> A vertex <span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 16.359 11.5564\" width=\"16.359pt\" 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,-0.013,9.495,0)\"></path></g></svg><span></span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"19.9411838 -9.28833 27.354 11.5564\" width=\"27.354pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,19.991,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,29.215,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,33.713,0)\"><use xlink:href=\"#g113-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,42.605,0)\"><use xlink:href=\"#g113-42\"></use></g></svg></span> is said to resolve a pair of cliques <span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 16.898 11.5564\" width=\"16.898pt\" 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=\"#g113-124\"></use></g><g transform=\"matrix(.013,0,0,-0.013,4.511,0)\"><use xlink:href=\"#g113-89\"></use></g><g transform=\"matrix(.013,0,0,-0.013,13.934,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"19.027183800000003 -9.28833 13.201 11.5564\" width=\"13.201pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,19.077,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,27.558,0)\"><use xlink:href=\"#g113-126\"></use></g></svg></span> in <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> if <span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 27.303 12.5794\" width=\"27.303pt\" 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(.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=\"#g113-45\"></use></g></svg><span></span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"29.4321838 -9.28833 25.748 12.5794\" width=\"25.748pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,29.482,0)\"><use xlink:href=\"#g113-89\"></use></g><g transform=\"matrix(.013,0,0,-0.013,39.428,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,55.23,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,47.558,0)\"><use xlink:href=\"#g117-34\"></use></g></svg><span></span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"58.799183799999994 -9.28833 27.303 12.5794\" width=\"27.303pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,58.849,0)\"><use xlink:href=\"#g113-101\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,65.999,3.132)\"><use xlink:href=\"#g50-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,72.827,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,77.325,0)\"><use xlink:href=\"#g185-40\"></use></g><g transform=\"matrix(.013,0,0,-0.013,83.188,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"88.28118380000001 -9.28833 13.33 12.5794\" width=\"13.33pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,88.331,0)\"><use xlink:href=\"#g113-90\"></use></g><g transform=\"matrix(.013,0,0,-0.013,96.812,0)\"><use xlink:href=\"#g113-42\"></use></g></svg></span> where <svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 14.1045 12.5794\" width=\"14.1045pt\" 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></svg> is the distance function 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> For a pair of cliques <span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 16.898 11.5564\" width=\"16.898pt\" 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=\"#g113-124\"></use></g><g transform=\"matrix(.013,0,0,-0.013,4.511,0)\"><use xlink:href=\"#g113-89\"></use></g><g transform=\"matrix(.013,0,0,-0.013,13.934,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"19.027183800000003 -9.28833 13.201 11.5564\" width=\"13.201pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,19.077,0)\"><use xlink:href=\"#g113-90\"></use></g><g transform=\"matrix(.013,0,0,-0.013,27.558,0)\"><use xlink:href=\"#g113-126\"></use></g></svg>,</span></span> the resolving neighbourhood of <svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 10.0819 8.68572\" width=\"10.0819pt\" 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=\"#g113-89\"></use></g></svg> and <span><svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 8.6074 8.68572\" width=\"8.6074pt\" 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=\"#g113-90\"></use></g></svg>,</span> denoted by <span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 31.812 12.5794\" width=\"31.812pt\" 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(.0091,0,0,-0.0091,8.086,3.132)\"><use xlink:href=\"#g50-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,14.914,0)\"><use xlink:href=\"#g113-124\"></use></g><g transform=\"matrix(.013,0,0,-0.013,19.425,0)\"><use xlink:href=\"#g113-89\"></use></g><g transform=\"matrix(.013,0,0,-0.013,28.848,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"33.942183799999995 -9.28833 13.23 12.5794\" width=\"13.23pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,33.992,0)\"><use xlink:href=\"#g113-90\"></use></g><g transform=\"matrix(.013,0,0,-0.013,42.473,0)\"><use xlink:href=\"#g113-126\"></use></g></svg>,</span></span> is the collection of all vertices which resolve the pair <span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 16.898 11.5564\" width=\"16.898pt\" 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=\"#g113-124\"></use></g><g transform=\"matrix(.013,0,0,-0.013,4.511,0)\"><use xlink:href=\"#g113-89\"></use></g><g transform=\"matrix(.013,0,0,-0.013,13.934,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"19.027183800000003 -9.28833 13.201 11.5564\" width=\"13.201pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,19.077,0)\"><use xlink:href=\"#g113-90\"></use></g><g transform=\"matrix(.013,0,0,-0.013,27.558,0)\"><use xlink:href=\"#g113-126\"></use></g></svg>.</span></span> A subset <svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 6.25863 8.8423\" width=\"6.25863pt\" 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></svg> of <svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 27.3063 11.5564\" width=\"27.3063pt\" 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=\"#g113-87\"></use></g><g transform=\"matrix(.013,0,0,-0.013,9.224,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,13.722,0)\"><use xlink:href=\"#g113-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,22.614,0)\"><use xlink:href=\"#g113-42\"></use></g></svg> is called an <span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 10.96 11.5564\" width=\"10.96pt\" 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=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,4.498,0)\"><use xlink:href=\"#g113-109\"></use></g><g transform=\"matrix(.013,0,0,-0.013,7.996,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"13.089183799999999 -9.28833 11.218 11.5564\" width=\"11.218pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,13.139,0)\"><use xlink:href=\"#g113-108\"></use></g><g transform=\"matrix(.013,0,0,-0.013,19.679,0)\"><use xlink:href=\"#g113-42\"></use></g></svg>-</span></span>clique metric generator for <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> if <span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 35.231 12.5794\" width=\"35.231pt\" 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)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,3.419,0)\"><use xlink:href=\"#g113-83\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,11.505,3.132)\"><use xlink:href=\"#g50-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,18.333,0)\"><use xlink:href=\"#g113-124\"></use></g><g transform=\"matrix(.013,0,0,-0.013,22.844,0)\"><use xlink:href=\"#g113-89\"></use></g><g transform=\"matrix(.013,0,0,-0.013,32.267,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"37.360183799999994 -9.28833 23.529 12.5794\" width=\"23.529pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,37.41,0)\"><use xlink:href=\"#g113-90\"></use></g><g transform=\"matrix(.013,0,0,-0.013,45.892,0)\"><use xlink:href=\"#g113-126\"></use></g><g transform=\"matrix(.013,0,0,-0.013,53.308,0)\"></path></g></svg><span></span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"63.7951838 -9.28833 20.818 12.5794\" width=\"20.818pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,63.845,0)\"><use xlink:href=\"#g113-84\"></use></g><g transform=\"matrix(.013,0,0,-0.013,69.981,0)\"><use xlink:href=\"#g113-9\"></use></g><g transform=\"matrix(.013,0,0,-0.013,77.032,0)\"></path></g></svg><span></span><svg height=\"12.5794pt\" style=\"vertical-align:-3.29107pt\" version=\"1.1\" viewbox=\"88.2441838 -9.28833 6.908 12.5794\" width=\"6.908pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,88.294,0)\"><use xlink:href=\"#g113-108\"></use></g></svg></span> for each pair of distinct <span><svg height=\"9.49473pt\" style=\"vertical-align:-0.2063999pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 3.60972 9.49473\" width=\"3.60972pt\" 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=\"#g113-109\"></use></g></svg>-</span>cliques <svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 10.0819 8.68572\" width=\"10.0819pt\" 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=\"#g113-89\"></use></g></svg> and <svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 8.6074 8.68572\" width=\"8.6074pt\" 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=\"#g113-90\"></use></g></svg> 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 <span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 10.96 11.5564\" width=\"10.96pt\" 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=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,4.498,0)\"><use xlink:href=\"#g113-109\"></use></g><g transform=\"matrix(.013,0,0,-0.013,7.996,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"13.089183799999999 -9.28833 11.218 11.5564\" width=\"11.218pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,13.139,0)\"><use xlink:href=\"#g113-108\"></use></g><g transform=\"matrix(.013,0,0,-0.013,19.679,0)\"><use xlink:href=\"#g113-42\"></use></g></svg>-</span></span>clique metric dimension 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=\"12.599pt\" style=\"vertical-align:-3.2911pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.3079 66.7761 12.599\" width=\"66.7761pt\" 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=\"#g113-109\"></use></g><g transform=\"matrix(.013,0,0,-0.013,6.404,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,16.94,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,22.439,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,29.303,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,32.787,0)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,43.534,3.132)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,48.681,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,53.179,0)\"><use xlink:href=\"#g113-72\"></use></g><g transform=\"matrix(.013,0,0,-0.013,62.071,0)\"><use xlink:href=\"#g113-42\"></use></g></svg>,</span> is defined as <span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 45.403 11.5564\" width=\"45.403pt\" 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,-0.013,10.647,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,14.131,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,21.322,0)\"><use xlink:href=\"#g113-124\"></use></g><g transform=\"matrix(.013,0,0,-0.013,25.833,0)\"><use xlink:href=\"#g113-9\"></use></g><g transform=\"matrix(.013,0,0,-0.013,29.252,0)\"><use xlink:href=\"#g113-84\"></use></g><g transform=\"matrix(.013,0,0,-0.013,35.388,0)\"><use xlink:href=\"#g113-9\"></use></g><g transform=\"matrix(.013,0,0,-0.013,42.439,0)\"><use xlink:href=\"#g113-59\"></use></g></svg><span></span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"48.985183799999994 -9.28833 6.335 11.5564\" width=\"6.335pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,49.035,0)\"><use xlink:href=\"#g113-84\"></use></g></svg></span> is an <span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 10.96 11.5564\" width=\"10.96pt\" 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=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,4.498,0)\"><use xlink:href=\"#g113-109\"></use></g><g transform=\"matrix(.013,0,0,-0.013,7.996,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"13.089183799999999 -9.28833 11.218 11.5564\" width=\"11.218pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,13.139,0)\"><use xlink:href=\"#g113-108\"></use></g><g transform=\"matrix(.013,0,0,-0.013,19.679,0)\"><use xlink:href=\"#g113-42\"></use></g></svg>-</span></span>clique metric generator of <span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 13.5529 11.5564\" width=\"13.5529pt\" 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><g transform=\"matrix(.013,0,0,-0.013,8.892,0)\"><use xlink:href=\"#g113-126\"></use></g></svg>.</span> In this paper, the <span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 10.96 11.5564\" width=\"10.96pt\" 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=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,4.498,0)\"><use xlink:href=\"#g113-109\"></use></g><g transform=\"matrix(.013,0,0,-0.013,7.996,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"13.089183799999999 -9.28833 11.218 11.5564\" width=\"11.218pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,13.139,0)\"><use xlink:href=\"#g113-108\"></use></g><g transform=\"matrix(.013,0,0,-0.013,19.679,0)\"><use xlink:href=\"#g113-42\"></use></g></svg>-</span></span>clique metric dimension of corona and edge corona of two graphs are computed. In addition, an integer linear programming model is presented for the <span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 10.96 11.5564\" width=\"10.96pt\" 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=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,4.498,0)\"><use xlink:href=\"#g113-109\"></use></g><g transform=\"matrix(.013,0,0,-0.013,7.996,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"13.089183799999999 -9.28833 11.218 11.5564\" width=\"11.218pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,13.139,0)\"><use xlink:href=\"#g113-108\"></use></g><g transform=\"matrix(.013,0,0,-0.013,19.679,0)\"><use xlink:href=\"#g113-42\"></use></g></svg>-</span></span>clique metric basis for a given graph <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> and its <span><svg height=\"9.49473pt\" style=\"vertical-align:-0.2063999pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 3.60972 9.49473\" width=\"3.60972pt\" 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=\"#g113-109\"></use></g></svg>-</span>cliques.","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"4 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-04-30","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/10.1155/2024/3241718","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 graph with vertices and is an -clique of . A vertex is said to resolve a pair of cliques in if where is the distance function of . For a pair of cliques , the resolving neighbourhood of and , denoted by , is the collection of all vertices which resolve the pair . A subset of is called an -clique metric generator for if for each pair of distinct -cliques and of . The -clique metric dimension of , denoted by , is defined as is an -clique metric generator of . In this paper, the -clique metric dimension of corona and edge corona of two graphs are computed. In addition, an integer linear programming model is presented for the -clique metric basis for a given graph and its -cliques.通过（边）日冕乘积和整数线性规划模型计算图的-斜度量维度
让 是一个有顶点的图，并且 是 的-clique。 如果 是 的距离函数，则称顶点解析了一对cliques。 对于一对cliques， 和 的解析邻域，用 表示，是解析了这对cliques的所有顶点的集合。对于每一对不同的 和 的-cliques，如果 的子集被称为 的-clique度量生成器，则 的-clique度量维度 ，用 ，表示，定义为 是 的-clique度量生成器。 本文计算了两个图的-clique度量维度的corona和边corona。此外，本文还提出了一个整数线性规划模型，用于计算给定图形及其-cliques 的-clique度量基础。
本文章由计算机程序翻译，如有差异，请以英文原文为准。
求助全文
约1分钟内获得全文
求助全文
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.
相关文献
×
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
京公网安备 11010802042870号
