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Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graphs arising from applications such as social network analysis, cartography, linguistics, and bioinformatics.
In order theory, a Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set one represents each element of as a vertex in the plane and draws a line segment or curve that goes upward from one vertex to another vertex whenever covers . These curves may cross each other but must not touch any vertices other than their endpoints. Such a diagram, with labeled vertices, uniquely determines its partial order.
In computational complexity, problems that are in the complexity class NP but are neither in the class P nor NP-complete are called NP-intermediate, and the class of such problems is called NPI. Ladner's theorem, shown in 1975 by Richard E. Ladner, is a result asserting that, if P ≠ NP, then NPI is not empty; that is, NP contains problems that are neither in P nor NP-complete. Since it is also true that if NPI problems exist, then P ≠ NP, it follows that P = NP if and only if NPI is empty.
In graph theory, a path decomposition of a graph G is, informally, a representation of G as a "thickened" path graph, and the pathwidth of G is a number that measures how much the path was thickened to form G. More formally, a path-decomposition is a sequence of subsets of vertices of G such that the endpoints of each edge appear in one of the subsets and such that each vertex appears in a contiguous subsequence of the subsets, and the pathwidth is one less than the size of the largest set in such a decomposition. Pathwidth is also known as interval thickness, vertex separation number, or node searching number.
In topological graph theory, an embedding of a graph on a surface is a representation of on in which points of are associated with vertices and simple arcs are associated with edges in such a way that:
In graph theory, a branch of mathematics, the triconnected components of a biconnected graph are a system of smaller graphs that describe all of the 2-vertex cuts in the graph. An SPQR tree is a tree data structure used in computer science, and more specifically graph algorithms, to represent the triconnected components of a graph. The SPQR tree of a graph may be constructed in linear time and has several applications in dynamic graph algorithms and graph drawing.
A program structure tree (PST) is a hierarchical diagram that displays the nesting relationship of single-entry single-exit (SESE) fragments/regions, showing the organization of a computer program. Nodes in this tree represent SESE regions of the program, while edges represent nesting regions. The PST is defined for all control flow graphs.
In graph theory, the planarity testing problem is the algorithmic problem of testing whether a given graph is a planar graph (that is, whether it can be drawn in the plane without edge intersections). This is a well-studied problem in computer science for which many practical algorithms have emerged, many taking advantage of novel data structures. Most of these methods operate in O(n) time (linear time), where n is the number of edges (or vertices) in the graph, which is asymptotically optimal. Rather than just being a single Boolean value, the output of a planarity testing algorithm may be a planar graph embedding, if the graph is planar, or an obstacle to planarity such as a Kuratowski subgraph if it is not.
In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph. A path is called simple if it does not have any repeated vertices; the length of a path may either be measured by its number of edges, or by the sum of the weights of its edges. In contrast to the shortest path problem, which can be solved in polynomial time in graphs without negative-weight cycles, the longest path problem is NP-hard and the decision version of the problem, which asks whether a path exists of at least some given length, is NP-complete. This means that the decision problem cannot be solved in polynomial time for arbitrary graphs unless P = NP. Stronger hardness results are also known showing that it is difficult to approximate. However, it has a linear time solution for directed acyclic graphs, which has important applications in finding the critical path in scheduling problems.
Layered graph drawing or hierarchical graph drawing is a type of graph drawing in which the vertices of a directed graph are drawn in horizontal rows or layers with the edges generally directed downwards. It is also known as Sugiyama-style graph drawing after Kozo Sugiyama, who first developed this drawing style.
In graph drawing, the angular resolution of a drawing of a graph is the sharpest angle formed by any two edges that meet at a common vertex of the drawing.
Dorothea Wagner is a German computer scientist, known for her research in graph drawing, route planning, and social network analysis. She heads the Institute of Theoretical Informatics at the Karlsruhe Institute of Technology.
In graph drawing, a universal point set of order n is a set S of points in the Euclidean plane with the property that every n-vertex planar graph has a straight-line drawing in which the vertices are all placed at points of S.
In topological graph theory, the Hanani–Tutte theorem is a result on the parity of edge crossings in a graph drawing. It states that every drawing in the plane of a non-planar graph contains a pair of independent edges that cross each other an odd number of times. Equivalently, it can be phrased as a planarity criterion: a graph is planar if and only if it has a drawing in which every pair of independent edges crosses evenly.
In the mathematical field of graph theory, planarization is a method of extending graph drawing methods from planar graphs to graphs that are not planar, by embedding the non-planar graphs within a larger planar graph.
In graph drawing styles that represent the edges of a graph by polylines, it is desirable to minimize the number of bends per edge or the total number of bends in a drawing. Bend minimization is the algorithmic problem of finding a drawing that minimizes these quantities.
In graph theory, a branch of mathematics, a radio coloring of an undirected graph is a form of graph coloring in which one assigns positive integer labels to the graphs such that the labels of adjacent vertices differ by at least two, and the labels of vertices at distance two from each other differ by at least one. Radio coloring was first studied by Griggs & Yeh (1992), under a different name, L(2,1)-labeling. It was called radio coloring by Frank Harary because it models the problem of channel assignment in radio broadcasting, while avoiding electromagnetic interference between radio stations that are near each other both in the graph and in their assigned channel frequencies.
Simultaneous embedding is a technique in graph drawing and information visualization for visualizing two or more different graphs on the same or overlapping sets of labeled vertices, while avoiding crossings within both graphs. Crossings between an edge of one graph and an edge of the other graph are allowed.
In graph theory, a sum coloring of a graph is a labeling of its vertices by positive integers, with no two adjacent vertices having equal labels, that minimizes the sum of the labels. The minimum sum that can be achieved is called the chromatic sum of the graph. Chromatic sums and sum coloring were introduced by Supowit in 1987 using non-graph-theoretic terminology, and first studied in graph theoretic terms by Ewa Kubicka in her 1989 doctoral thesis.
References
- ↑ Mutzel, Petra. "Petra Mutzel's academic website".
- ↑ Faculty profile, TU Dortmund, retrieved 2014-07-04.
- ↑ Petra Mutzel at the Mathematics Genealogy Project
- ↑ ""Education and Positions" section on Petra Mutzel's academic homepage".
- ↑ Jünger & Mutzel (1996).
- ↑ Jünger & Mutzel (1997).
- ↑ Gutwenger & Mutzel (2001).
- ↑ Jünger & Mutzel (2004).
- ↑ GD 2001 Archived 2013-04-30 at the Wayback Machine web site, retrieved 2014-07-04.
- ↑ De Simone et al. (1995).
- ↑ Hetzl & Mutzel (2005).
- ↑ Ljubić et al. (2006).
- ↑ ALENEX 2012 web site, retrieved 2014-07-04.
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