Crossing Minimization meets Simultaneous Drawing

Abstract

We define the concept of crossing numbers for simultaneous graphs by extending the crossing number problem of traditional graphs. We discuss differences to the traditional crossing number problem, and give an NP-completeness proof and lower and upper bounds for the new problem. Furthermore, we show how existing heuristic and exact algorithms for the traditional problem can be adapted to the new task of simultaneous crossing minimization, and report on a brief experimental study of their implementations.

A Graph Drawing Application to Web Site Traffic Analysis

“…Kobourov and Pitta describe an interactive multi-user environment for simultaneous embedding where the number of edge crossings is kept under control by means of heuristic primitives [23]. Chimani et al further investigate the problem of minimizing the number of crossings in a simultaneous embedding; they show the NP-hardness of the question and experimentally compare heuristics and exact drawing algorithms [7].…”

“…However, since G pag and G usr share the whole vertex set, the centrality of the shared vertices (considered in [16]) is not a meaningful aesthetic requirement in our case. Also, motivated by recent cognitive experiments, we do not insist on minimizing the number of edge crossings (as done in [7,23]) but rather we try to control the visual quality of the edge crossings. More precisely, human-computer interaction experiments have shown that orthogonal crossings do not inhibit human task performance when reading a drawing and that users have a geodesic tendency when discovering relations between pairs of vertices [20,21,22].…”

“…The interaction with COWA makes it possible to measure the interest in a pair of concepts; it would be interesting to extend this facility and measure the interest for groups or clusters of concepts. The graph drawing heuristic draws edges having a bend for every crossing; it would be interesting to reduce the curve complexity of the edges by integrating our heuristic with crossing reduction techniques (see, e.g., [7]). …”

Graph visualization techniques for conceptual Web site traffic analysis

“…Kobourov and Pitta describe an interactive multi-user environment for simultaneous embedding where the number of edge crossings is kept under control by means of heuristic primitives [17]. Chimani et al further investigate the problem of minimizing the number of crossings in a simultaneous embedding; they both show the NP-Hardness of the question and experimentally compare heuristics and exact drawing algorithms [5].…”

“…The interaction with COWA makes it possible to measure the interest in a pair of concepts; it would be interesting to extend this facility and measure the interest for groups or clusters of concepts. The graph drawing heuristic draws edges having a bend for every crossing; it would be interesting to reduce the curve complexity of the edges by integrating our heuristic with crossing reduction techniques (see, e.g., [5]). The logical connection between the two concepts Series A and School is not reliable.…”