Hasse diagramIn order theory, a Hasse diagram (ˈhæsə; ˈhasə) 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 (that is, whenever , and there is no distinct from and with ). These curves may cross each other but must not touch any vertices other than their endpoints.
Shortest path problemIn graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between two intersections on a road map may be modeled as a special case of the shortest path problem in graphs, where the vertices correspond to intersections and the edges correspond to road segments, each weighted by the length of the segment.
Feedback arc setIn graph theory and graph algorithms, a feedback arc set or feedback edge set in a directed graph is a subset of the edges of the graph that contains at least one edge out of every cycle in the graph. Removing these edges from the graph breaks all of the cycles, producing a directed acyclic graph, an acyclic subgraph of the given graph. The feedback arc set with the fewest possible edges is the minimum feedback arc set and its removal leaves the maximum acyclic subgraph; weighted versions of these optimization problems are also used.
MultitreeIn combinatorics and order theory, a multitree may describe either of two equivalent structures: a directed acyclic graph (DAG) in which there is at most one directed path between any two vertices, or equivalently in which the subgraph reachable from any vertex induces an undirected tree, or a partially ordered set (poset) that does not have four items a, b, c, and d forming a diamond suborder with a ≤ b ≤ d and a ≤ c ≤ d but with b and c incomparable to each other (also called a diamond-free poset).
Citation graphA citation graph (or citation network), in information science and bibliometrics, is a directed graph that describes the citations within a collection of documents. Each vertex (or node) in the graph represents a document in the collection, and each edge is directed from one document toward another that it cites (or vice versa depending on the specific implementation). Citation graphs have been utilised in various ways, including forms of citation analysis, academic search tools and court judgements.
Loop (graph theory)In graph theory, a loop (also called a self-loop or a buckle) is an edge that connects a vertex to itself. A simple graph contains no loops. Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow the presence of loops (often in concert with allowing or disallowing multiple edges between the same vertices): Where graphs are defined so as to allow loops and multiple edges, a graph without loops or multiple edges is often distinguished from other graphs by calling it a simple graph.
Dense graphIn mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges (where every pair of vertices is connected by one edge). The opposite, a graph with only a few edges, is a sparse graph. The distinction of what constitutes a dense or sparse graph is ill-defined, and is often represented by 'roughly equal to' statements. Due to this, the way that density is defined often depends on the context of the problem.
Linear extensionIn order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order. A partial order is a reflexive, transitive and antisymmetric relation.
Arborescence (graph theory)In graph theory, an arborescence is a directed graph in which, for a vertex u (called the root) and any other vertex v, there is exactly one directed path from u to v. An arborescence is thus the directed-graph form of a rooted tree, understood here as an undirected graph. Equivalently, an arborescence is a directed, rooted tree in which all edges point away from the root; a number of other equivalent characterizations exist. Every arborescence is a directed acyclic graph (DAG), but not every DAG is an arborescence.
Covering relationIn mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that are immediate neighbours. The covering relation is commonly used to graphically express the partial order by means of the Hasse diagram. Let be a set with a partial order . As usual, let be the relation on such that if and only if and . Let and be elements of . Then covers , written , if and there is no element such that .
