AffiliationUniv Arizona, Dept Comp Sci
MetadataShow full item record
CitationDe Luca, F., Hossain, M. I., Kobourov, S., Lubiw, A., & Mondal, D. (2019). Recognition and drawing of stick graphs. Theoretical Computer Science, 796, 22-33.
JournalTHEORETICAL COMPUTER SCIENCE
Rights© 2019 Elsevier B.V. All rights reserved.
Collection InformationThis item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at email@example.com.
AbstractA Stick graph is an intersection graph of axis-aligned segments such that the left endpoints of the horizontal segments and the bottom end-points of the vertical segments lie on a "ground line," a line with slope - 1. It is an open question to decide in polynomial time whether a given bipartite graph G with bipartition A boolean OR B has a Stick representation where the vertices in A and B correspond to horizontal and vertical segments, respectively. We prove that G has a Stick representation if and only if there are orderings of A and B such that G's bipartite adjacency matrix with rows A and columns B excludes three small 'forbidden' submatrices. This is similar to characterizations for other classes of bipartite intersection graphs. We present an algorithm to test whether given orderings of A and B permit a Stick representation respecting those orderings, and to find such a representation if it exists. The algorithm runs in time linear in the size of the adjacency matrix. For the case when only the ordering of A is given, or neither ordering is given, we present some partial results about graphs that are, or are not, Stick representable. (C) 2019 Elsevier B.V. All rights reserved.
Note24 month embargo; available online 19 August 2019.
VersionFinal accepted manuscript
SponsorsNatural Sciences and Engineering Research Council of Canada (NSERC); National Science Foundation (NSF) [CCF-1740858, CCF-1712119, DMS-1839274, DMS-1839307]
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SAME STATS, DIFFERENT GRAPHS (GRAPH STATISTICS AND WHY WE NEED GRAPH DRAWING)Kobourov, Stephen; Chen, Hang (The University of Arizona., 2018)Data analysts commonly utilize statistics to summarize large datasets. While it is often sufficient to explore only the summary statistics of a dataset (e.g., min/mean/max), Anscombe’s Quartet demonstrates how such statistics can be misleading. We consider a similar problem in the context of graph mining. To study the relationships between different graph properties and statistics, we examine all low-order ( 10) non-isomorphic graphs and provide a simple visual analytics system to explore correlations across multiple graph properties. However, for graphs with more than ten nodes, generating the entire space of graphs becomes quickly intractable. We use different random graph generation methods to further look into the distribution of graph statistics for higher order graphs and investigate the impact of various sampling methodologies. We also describe a method for generating many graphs that are identical over a number of graph properties and statistics yet are clearly different and identifiably distinct.
Interactive graph layout: The exploration of large graphs.Hudson, Scott; Schlichting, Richard; Henry, Tyson Rombauer.; Peterson, Larry L.; Snodgrass, Richard T. (The University of Arizona., 1992)Directed and undirected graphs provide a natural notation for describing many fundamental structures of computer science. Unfortunately graphs are hard to draw in an easy to read fashion. Traditional graph layout algorithms have focused on creating good layouts for the entire graph. This approach works well with smaller graphs, but often cannot produce readable layouts for large graphs. This dissertation presents a novel methodology for viewing large graphs. The basic concept is to allow the user to interactively navigate through large graphs, learning about them in appropriately small and concise pieces. The motivation of this approach is that large graphs contain too much information to be conveyed by a single canonical layout. For a user to be able to understand the data encoded in the graph she must be able to carve up the graph into manageable pieces and then create custom layouts that match her current interests. An architecture is presented that supports graph exploration. It contains three new concepts for supporting interactive graph layout: interactive decomposition of large graphs, end-user specified layout algorithms, and parameterized layout algorithms. The mechanism for creating custom layout algorithms provides the non-programming end-user with the power to create custom layouts that are well suited for the graph at hand. New layout algorithms are created by combining existing algorithms in a hierarchical structure. This method allows the user to create layouts that accurately reflect the current data set and her current interests. In order to explore a large graph, the user must be able to break the graph into small, more manageable pieces. A methodology is presented that allows the user to apply graph traversal algorithms to large graphs to carve out reasonably sized pieces. Graph traversal algorithms can be combined using a visual programming language. This provides the user with the control to select subgraphs that are of particular interest to her. The ability to Parameterize layout algorithms provides the user with control over the layout process. The user can customize the generated layout by changing parameters to the layout algorithm. Layout algorithm parameterization is placed into an interactive framework that allows the user to iteratively fine tune the generated layout. As a proof of concept, examples are drawn from a working prototype that incorporates this methodology.
The Perception of Graph Properties in Graph LayoutsSoni, Utkarsh; Lu, Yafeng; Hansen, Brett; Purchase, Helen C.; Kobourov, Stephen; Maciejewski, Ross; Univ Arizona, Dept Comp Sci (WILEY, 2018-06)When looking at drawings of graphs, questions about graph density, community structures, local clustering and other graph properties may be of critical importance for analysis. While graph layout algorithms have focused on minimizing edge crossing, symmetry, and other such layout properties, there is not much known about how these algorithms relate to a user's ability to perceive graph properties for a given graph layout. In this study, we apply previously established methodologies for perceptual analysis to identify which graph drawing layout will help the user best perceive a particular graph property. We conduct a large scale (n = 588) crowdsourced experiment to investigate whether the perception of two graph properties (graph density and average local clustering coefficient) can be modeled using Weber's law. We study three graph layout algorithms from three representative classes (Force Directed - FD, Circular, and Multi-Dimensional Scaling - MDS), and the results of this experiment establish the precision of judgment for these graph layouts and properties. Our findings demonstrate that the perception of graph density can be modeled with Weber's law. Furthermore, the perception of the average clustering coefficient can be modeled as an inverse of Weber's law, and the MDS layout showed a significantly different precision of judgment than the FD layout.