John S. Caughman, Charles L. Dunn, Joshua D. Laison, Nancy Ann Neudauer, Colin L. Starr
{"title":"矩形可见性图的面积、周长、高度和宽度","authors":"John S. Caughman, Charles L. Dunn, Joshua D. Laison, Nancy Ann Neudauer, Colin L. Starr","doi":"10.1007/s10878-023-01084-9","DOIUrl":null,"url":null,"abstract":"<p>A rectangle visibility graph (RVG) is represented by assigning to each vertex a rectangle in the plane with horizontal and vertical sides in such a way that edges in the graph correspond to unobstructed horizontal and vertical lines of sight between their corresponding rectangles. To discretize, we consider only rectangles whose corners have integer coordinates. For any given RVG, we seek a representation with smallest bounding box as measured by its area, perimeter, height, or width (height is assumed not to exceed width). We derive a number of results regarding these parameters. Using these results, we show that these four measures are distinct, in the sense that there exist graphs <span>\\(G_1\\)</span> and <span>\\(G_2\\)</span> with <span>\\({{\\,\\textrm{area}\\,}}(G_1) < {{\\,\\textrm{area}\\,}}(G_2)\\)</span> but <span>\\({{\\,\\textrm{perim}\\,}}(G_2) < {{\\,\\textrm{perim}\\,}}(G_1)\\)</span>, and analogously for all other pairs of these parameters. We further show that there exists a graph <span>\\(G_3\\)</span> with representations <span>\\(S_1\\)</span> and <span>\\(S_2\\)</span> such that <span>\\({{\\,\\textrm{area}\\,}}(G_3)={{\\,\\textrm{area}\\,}}(S_1)<{{\\,\\textrm{area}\\,}}(S_2)\\)</span> but <span>\\({{\\,\\textrm{perim}\\,}}(G_3)={{\\,\\textrm{perim}\\,}}(S_2)<{{\\,\\textrm{perim}\\,}}(S_1)\\)</span>. In other words, <span>\\(G_3\\)</span> requires distinct representations to minimize area and perimeter. Similarly, such graphs exist to demonstrate the independence of all other pairs of these parameters. Among graphs with <span>\\(n \\le 6\\)</span> vertices, the empty graph <span>\\(E_n\\)</span> requires largest area. But for graphs with <span>\\(n=7\\)</span> and <span>\\(n=8\\)</span> vertices, we show that the complete graphs <span>\\(K_7\\)</span> and <span>\\(K_8\\)</span> require larger area than <span>\\(E_7\\)</span> and <span>\\(E_8\\)</span>, respectively. Using this, we show that for all <span>\\(n \\ge 8\\)</span>, the empty graph <span>\\(E_n\\)</span> does not have largest area, perimeter, height, or width among all RVGs on <i>n</i> vertices.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Area, perimeter, height, and width of rectangle visibility graphs\",\"authors\":\"John S. Caughman, Charles L. Dunn, Joshua D. Laison, Nancy Ann Neudauer, Colin L. Starr\",\"doi\":\"10.1007/s10878-023-01084-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A rectangle visibility graph (RVG) is represented by assigning to each vertex a rectangle in the plane with horizontal and vertical sides in such a way that edges in the graph correspond to unobstructed horizontal and vertical lines of sight between their corresponding rectangles. To discretize, we consider only rectangles whose corners have integer coordinates. For any given RVG, we seek a representation with smallest bounding box as measured by its area, perimeter, height, or width (height is assumed not to exceed width). We derive a number of results regarding these parameters. Using these results, we show that these four measures are distinct, in the sense that there exist graphs <span>\\\\(G_1\\\\)</span> and <span>\\\\(G_2\\\\)</span> with <span>\\\\({{\\\\,\\\\textrm{area}\\\\,}}(G_1) < {{\\\\,\\\\textrm{area}\\\\,}}(G_2)\\\\)</span> but <span>\\\\({{\\\\,\\\\textrm{perim}\\\\,}}(G_2) < {{\\\\,\\\\textrm{perim}\\\\,}}(G_1)\\\\)</span>, and analogously for all other pairs of these parameters. We further show that there exists a graph <span>\\\\(G_3\\\\)</span> with representations <span>\\\\(S_1\\\\)</span> and <span>\\\\(S_2\\\\)</span> such that <span>\\\\({{\\\\,\\\\textrm{area}\\\\,}}(G_3)={{\\\\,\\\\textrm{area}\\\\,}}(S_1)<{{\\\\,\\\\textrm{area}\\\\,}}(S_2)\\\\)</span> but <span>\\\\({{\\\\,\\\\textrm{perim}\\\\,}}(G_3)={{\\\\,\\\\textrm{perim}\\\\,}}(S_2)<{{\\\\,\\\\textrm{perim}\\\\,}}(S_1)\\\\)</span>. In other words, <span>\\\\(G_3\\\\)</span> requires distinct representations to minimize area and perimeter. Similarly, such graphs exist to demonstrate the independence of all other pairs of these parameters. Among graphs with <span>\\\\(n \\\\le 6\\\\)</span> vertices, the empty graph <span>\\\\(E_n\\\\)</span> requires largest area. But for graphs with <span>\\\\(n=7\\\\)</span> and <span>\\\\(n=8\\\\)</span> vertices, we show that the complete graphs <span>\\\\(K_7\\\\)</span> and <span>\\\\(K_8\\\\)</span> require larger area than <span>\\\\(E_7\\\\)</span> and <span>\\\\(E_8\\\\)</span>, respectively. Using this, we show that for all <span>\\\\(n \\\\ge 8\\\\)</span>, the empty graph <span>\\\\(E_n\\\\)</span> does not have largest area, perimeter, height, or width among all RVGs on <i>n</i> vertices.</p>\",\"PeriodicalId\":50231,\"journal\":{\"name\":\"Journal of Combinatorial Optimization\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-09-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Optimization\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10878-023-01084-9\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10878-023-01084-9","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
Area, perimeter, height, and width of rectangle visibility graphs
A rectangle visibility graph (RVG) is represented by assigning to each vertex a rectangle in the plane with horizontal and vertical sides in such a way that edges in the graph correspond to unobstructed horizontal and vertical lines of sight between their corresponding rectangles. To discretize, we consider only rectangles whose corners have integer coordinates. For any given RVG, we seek a representation with smallest bounding box as measured by its area, perimeter, height, or width (height is assumed not to exceed width). We derive a number of results regarding these parameters. Using these results, we show that these four measures are distinct, in the sense that there exist graphs \(G_1\) and \(G_2\) with \({{\,\textrm{area}\,}}(G_1) < {{\,\textrm{area}\,}}(G_2)\) but \({{\,\textrm{perim}\,}}(G_2) < {{\,\textrm{perim}\,}}(G_1)\), and analogously for all other pairs of these parameters. We further show that there exists a graph \(G_3\) with representations \(S_1\) and \(S_2\) such that \({{\,\textrm{area}\,}}(G_3)={{\,\textrm{area}\,}}(S_1)<{{\,\textrm{area}\,}}(S_2)\) but \({{\,\textrm{perim}\,}}(G_3)={{\,\textrm{perim}\,}}(S_2)<{{\,\textrm{perim}\,}}(S_1)\). In other words, \(G_3\) requires distinct representations to minimize area and perimeter. Similarly, such graphs exist to demonstrate the independence of all other pairs of these parameters. Among graphs with \(n \le 6\) vertices, the empty graph \(E_n\) requires largest area. But for graphs with \(n=7\) and \(n=8\) vertices, we show that the complete graphs \(K_7\) and \(K_8\) require larger area than \(E_7\) and \(E_8\), respectively. Using this, we show that for all \(n \ge 8\), the empty graph \(E_n\) does not have largest area, perimeter, height, or width among all RVGs on n vertices.
期刊介绍:
The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering.
The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.