Abstract
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.
Similar content being viewed by others
Data availability
Enquiries about data availability should be directed to the authors.
References
Angelini P, Bekos MA, Kaufmann M, Montecchiani F (2018) 3D visibility representations of 1-planar graphs. In: Graph drawing and network visualization, volume 10692 of Lecture notes in computer science. Springer, Cham, pp 102–109
Biedl T, Liotta G, Montecchiani F (2016) On visibility representations on non-planar graphs. In: 32nd International symposium on computational geometry, volume 51 of LIPIcs. Leibniz international proceedings in informatics. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, p 16
Bose P, Dean A, Hutchinson J, Shermer T (1996) On rectangle visibility graphs. In: International symposium on graph drawing. Springer, pp 25–44
Bose P, Everett H, Fekete SP, Houle ME, Lubiw A, Meijer H, Romanik K, Rote G, Shermer TC, Whitesides S, Zelle C (1998) A visibility representation for graphs in three dimensions. J Graph Algorithms Appl 2(3):16
Cahit I (1998) On \(2\)-visibility drawings of non-planar graphs. Bull Inst Combin Appl 24:24–26
Caughman J, Dunn C, Laison J, Neudauer N, Starr C (2014) Minimum representations of rectangle visibility graphs. In: Duncan C, Symvonis A (eds) Graph drawing: 22nd international symposium, GD 2014, volume 8871 of lecture notes in computer science. Springer, Heidelberg, pp 527–528
Caughman J, Dunn C, Laison J, Neudauer N, Starr C (2022) Connected 6-vertex RVGs. http://web.pdx.edu/~caughman/6-Vertex-Chart.pdf. Accessed 27 July 2022
Dean AM, Hutchinson JP (1995) Rectangle-visibility representations of bipartite graphs. In: Graph drawing (Princeton, NJ, 1994), of Lecture notes in computer science, vol 894. Springer, Berlin, pp 159–166
Dean AM, Hutchinson JP (1997) Rectangle-visibility representations of bipartite graphs. Discrete Appl Math 75(1):9–25
Dean AM, Hutchinson JP (1998) Rectangle-visibility layouts of unions and products of trees. J Graph Algorithms Appl 2(8):21
Dean AM, Hutchinson JP (2010) Representing 3-trees as unit rectangle-visibility graphs. In: Proceedings of the forty-first southeastern international conference on combinatorics, graph theory and computing, vol 203, pp 139–160
Dean AM, Ellis-Monaghan JA, Hamilton S, Pangborn G (2008) Unit rectangle visibility graphs. Electron J Combin 15(1):79, 24 (Research Paper)
Develin M, Hartke S, Moultons DP (2003) A general notion of visibility graph. Discrete Comput Geom 29(4):511–524
Di Giacomo E, Didimo W, Evans WS, Liotta G, Meijer H, Montecchiani F, Wismath SK (2018) Ortho-polygon visibility representations of embedded graphs. Algorithmica 80(8):2345–2383
Duchet P, Hamidoune Y, Las Vergnas M, Meyniel H (1983) Representing a planar graph by vertical lines joining different levels. Discrete Math 46(3):319–321
Fekete SP, Meijer H (1999) Rectangle and box visibility graphs in \(3\)D. Int J Comput Geom Appl 9(1):1–27
Garey MR, Johnson DS, So HC (1976) An application of graph coloring to printed circuit testing. IEEE Trans Circuits Syst CAS 2(10):591–599
Gethner E, Laison JD (2011) More directions in visibility graphs. Australas J Combin 50:55–65
Hutchinson JP (1993) Coloring ordinary maps, maps of empires and maps of the moon. Math Mag 66(4):211–226
Hutchinson JP, Shermer T, Vince A (1999) On representations of some thickness-two graphs. Comput Geom 13(3):161–171
Kant G, Liotta G, Tamassia R, Tollis IG (1997) Area requirement of visibility representations of trees. Inform Process Lett 62(2):81–88
Liotta G, Montecchiani F, Tappini A (2021) Ortho-polygon visibility representations of 3-connected 1-plane graphs. Theor Comput Sci 863:40–52
Keith M (2021) All small connected graphs. Accessed 20 April 2021
Streinu I, Whitesides S (2003) Rectangle visibility graphs: characterization, construction, and compaction. In: STACS 2003, volume 2607 of Lecture notes in computer science. Springer, Berlin, pp 26–37
Tamassia R, Tollis IG (1986) A unified approach to visibility representations of planar graphs. Discrete Comput Geom 1(4):321–341
Wismath SK (1985) Characterizing bar line-of-sight graphs. In: Proceedings of the first annual symposium on computational geometry, SCG ’85. Association for Computing Machinery, New York, pp 147–152
Funding
The authors have not disclosed any funding.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have not disclosed any conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Caughman, J. ., Dunn, C. ., Laison, J. . et al. Area, perimeter, height, and width of rectangle visibility graphs. J Comb Optim 46, 18 (2023). https://doi.org/10.1007/s10878-023-01084-9
Accepted:
Published:
DOI: https://doi.org/10.1007/s10878-023-01084-9