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Upward Planar Drawing
In graph drawing, an upward planar drawing of a directed acyclic graph is an embedding of the graph into the Euclidean plane, in which the edges are represented as non-crossing monotonic upwards curves. That is, the curve representing each edge should have the property that every horizontal line intersects it in at most one point, and no two edges may intersect except at a shared endpoint. In this sense, it is the ideal case for layered graph drawing, a style of graph drawing in which edges are monotonic curves that may cross, but in which crossings are to be minimized. Characterizations A directed acyclic graph must be planar in order to have an upward planar drawing, but not every planar acyclic graph has such a drawing. Among the planar directed acyclic graphs with a single source (vertex with no incoming edges) and sink (vertex with no outgoing edges), the graphs with upward planar drawings are the ''st''-planar graphs, planar graphs in which the source and sink both belong ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upward Planar Drawing
In graph drawing, an upward planar drawing of a directed acyclic graph is an embedding of the graph into the Euclidean plane, in which the edges are represented as non-crossing monotonic upwards curves. That is, the curve representing each edge should have the property that every horizontal line intersects it in at most one point, and no two edges may intersect except at a shared endpoint. In this sense, it is the ideal case for layered graph drawing, a style of graph drawing in which edges are monotonic curves that may cross, but in which crossings are to be minimized. Characterizations A directed acyclic graph must be planar in order to have an upward planar drawing, but not every planar acyclic graph has such a drawing. Among the planar directed acyclic graphs with a single source (vertex with no incoming edges) and sink (vertex with no outgoing edges), the graphs with upward planar drawings are the ''st''-planar graphs, planar graphs in which the source and sink both belong ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientation (graph Theory)
In graph theory, an orientation of an undirected graph is an assignment of a direction to each edge, turning the initial graph into a directed graph. Oriented graphs A directed graph is called an oriented graph if none of its pairs of vertices is linked by two symmetric edges. Among directed graphs, the oriented graphs are the ones that have no 2-cycles (that is at most one of and may be arrows of the graph). A tournament is an orientation of a complete graph. A polytree is an orientation of an undirected tree. Sumner's conjecture states that every tournament with vertices contains every polytree with vertices. The number of non-isomorphic oriented graphs with vertices (for ) is : 1, 2, 7, 42, 582, 21480, 2142288, 575016219, 415939243032, … . Tournaments are in one-to-one correspondence with complete directed graphs (graphs in which there is a directed edge in one or both directions between every pair of distinct vertices). A complete directed graph can be converted to an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reachability
In graph theory, reachability refers to the ability to get from one vertex to another within a graph. A vertex s can reach a vertex t (and t is reachable from s) if there exists a sequence of adjacent vertices (i.e. a walk) which starts with s and ends with t. In an undirected graph, reachability between all pairs of vertices can be determined by identifying the connected components of the graph. Any pair of vertices in such a graph can reach each other if and only if they belong to the same connected component; therefore, in such a graph, reachability is symmetric (s reaches t iff t reaches s). The connected components of an undirected graph can be identified in linear time. The remainder of this article focuses on the more difficult problem of determining pairwise reachability in a directed graph (which, incidentally, need not be symmetric). Definition For a directed graph G = (V, E), with vertex set V and edge set E, the reachability relation of G is the transitive closure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partially Ordered Set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of Comparability, comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hasse Diagram
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 ''(S, ≤)'' one represents each element of ''S'' as a vertex in the plane and draws a line segment or curve that goes ''upward'' from ''x'' to ''y'' whenever ''y'' ≠ ''x'' and ''y'' covers ''x'' (that is, whenever ''x'' ≤ ''y'' and there is no ''z'' such that ''x'' ≤ ''z'' ≤ ''y''). 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. The diagrams are named after Helmut Hasse (1898–1979); according to , they are so called because of the effective use Hasse made of them. However, Hasse was not the first to use these diagrams. One example that predates Hasse can be found in . Although Hasse diagrams were originally devised as a technique for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Area (graph Drawing)
In graph drawing, the area used by a drawing is a commonly used way of measuring its quality. Definition For a drawing style in which the vertices are placed on the integer lattice, the area of the drawing may be defined as the area of the smallest axis-aligned bounding box of the drawing: that is, it the product of the largest difference in ''x''-coordinates of two vertices with the largest difference in ''y''-coordinates. For other drawing styles, in which vertices are placed more freely, the drawing may be scaled so that the closest pair of vertices have distance one from each other, after which the area can again be defined as the area of a smallest bounding box of a drawing. Alternatively, the area can be defined as the area of the convex hull of the drawing, again after appropriate scaling.. Polynomial bounds For straight-line drawings of planar graphs with ''n'' vertices, the optimal worst-case bound on the area of a drawing is Θ(''n''2). The nested triangles graph requi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dominance Drawing
Dominance drawing is a style of graph drawing of directed acyclic graphs that makes the reachability relations between vertices visually apparent. In dominance drawing, vertices are placed at distinct points of the Euclidean plane and a vertex ''v'' is reachable from another vertex ''u'' if and only if both Cartesian coordinates of ''v'' are greater than or equal to the coordinates of ''u''. The edges of a dominance drawing may be drawn either as straight line segments, or, in some cases, as polygonal chains. Planar graphs Every transitively reduced ''st''-planar graph, a directed acyclic planar graph with a single source and a single sink, both on the outer face of some embedding of the graph, has a dominance drawing. The left–right algorithm for finding these drawings sets the ''x'' coordinate of every vertex to be its position in a depth-first search ordering of the graph, starting with ''s'' and prioritizing edges in right-to-left order, and by setting the ''y'' coordi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transitive Reduction
In the mathematical field of graph theory, a transitive reduction of a directed graph is another directed graph with the same vertices and as few edges as possible, such that for all pairs of vertices , a (directed) path from to in exists if and only if such a path exists in the reduction. Transitive reductions were introduced by , who provided tight bounds on the computational complexity of constructing them. More technically, the reduction is a directed graph that has the same reachability relation as . Equivalently, and its transitive reduction should have the same transitive closure as each other, and the transitive reduction of should have as few edges as possible among all graphs with that property. The transitive reduction of a finite directed acyclic graph (a directed graph without directed cycles) is unique and is a subgraph of the given graph. However, uniqueness fails for graphs with (directed) cycles, and for infinite graphs not even existence is guaranteed. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fáry's Theorem
In the mathematical field of graph theory, Fáry's theorem states that any simple, planar graph can be drawn without crossings so that its edges are straight line segments. That is, the ability to draw graph edges as curves instead of as straight line segments does not allow a larger class of graphs to be drawn. The theorem is named after István Fáry, although it was proved independently by , , and . Proof One way of proving Fáry's theorem is to use mathematical induction. Let be a simple plane graph with vertices; we may add edges if necessary so that is a maximally plane graph. If < 3, the result is trivial. If ≥ 3, then all faces of must be triangles, as we could add an edge into any face with more sides while preserving planarity, contradicting the assumption of maximal planarity. Choose some three vertices forming a triangular face of . We prove by induction on that there exists a straight-line combinatorially isomorphic re-embedding of in which triangle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pair where * ''V'' is a set whose elements are called '' vertices'', ''nodes'', or ''points''; * ''A'' is a set of ordered pairs of vertices, called ''arcs'', ''directed edges'' (sometimes simply ''edges'' with the corresponding set named ''E'' instead of ''A''), ''arrows'', or ''directed lines''. It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called ''edges'', ''links'' or ''lines''. The aforementioned definition does not allow a directed graph to have multiple arrows with the same source and target nodes, but some authors consider a broader definition that allows directed graphs to have such multiple arcs (namely, they allow the arc set to be a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclomatic Number
In graph theory, a branch of mathematics, the circuit rank, cyclomatic number, cycle rank, or nullity of an undirected graph is the minimum number of edges that must be removed from the graph to break all its cycles, making it into a tree or forest. It is equal to the number of independent cycles in the graph (the size of a cycle basis). Unlike the corresponding feedback arc set problem for directed graphs, the circuit rank is easily computed using the formula :r = m - n + c, where is the number of edges in the given graph, is the number of vertices, and is the number of connected components. . It is also possible to construct a minimum-size set of edges that breaks all cycles efficiently, either using a greedy algorithm or by complementing a spanning forest. The circuit rank can be explained in terms of algebraic graph theory as the dimension of the cycle space of a graph, in terms of matroid theory as the corank of a graphic matroid, and in terms of topology as one of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
