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.
Boucle (théorie des graphes)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.
Connectivity (graph theory)In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network. In an undirected graph G, two vertices u and v are called connected if G contains a path from u to v.
PolyarbreEn mathématiques, et notamment en théorie des graphes, un polyarbre (aussi appelé arbre dirigé, arbre orienté ou singly connected network) est graphe orienté acyclique dont le graphe non orienté sous-jacent est un arbre (théorie des graphes). En d'autres termes, si on remplace les arcs par des arêtes, on obtient un graphe non orienté qui est à la fois connexe et sans cycle. Une polyforêt (ou forêt dirigée ou forêt orientée) est un graphe orienté dont le graphe non orienté sous-jacent est une forêt.
MultigraphIn mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called parallel edges), that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge. There are 2 distinct notions of multiple edges: Edges without own identity: The identity of an edge is defined solely by the two nodes it connects. In this case, the term "multiple edges" means that the same edge can occur several times between these two nodes.
Densité d'un grapheEn mathématiques, et plus particulièrement en théorie des graphes, on peut associer à tout graphe un entier appelé densité du graphe. Ce paramètre mesure si le graphe a beaucoup d'arêtes ou peu. Un graphe dense (dense graph) est un graphe dans lequel le nombre d'arêtes (ou d'arcs) est proche du nombre maximal, par exemple un nombre quadratique par rapport au nombre de sommets. Un graphe creux (sparse graph) a au contraire peu d'arêtes, par exemple un nombre linéaire. La distinction entre graphe creux et dense est plutôt vague et dépend du contexte.
ReachabilityIn graph theory, reachability refers to the ability to get from one vertex to another within a graph. A vertex can reach a vertex (and is reachable from ) if there exists a sequence of adjacent vertices (i.e. a walk) which starts with and ends with . 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 ( reaches iff reaches ).
Graphe (mathématiques discrètes)Dans le domaine des mathématiques discrètes, la théorie des graphes définit le graphe, une structure composée d'objets et de relations entre deux de ces objets. Abstraitement, lesdits objets sont appelés sommets (ou nœuds ou points), et les relations entre eux sont nommées arêtes (ou liens ou lignes). On distingue les graphes non orientés, où les arêtes relient deux sommets de manière symétrique, et les graphes orientés, où les arêtes, alors appelées arcs (ou flèches), relient deux sommets de manière asymétrique.
Carquois (théorie des catégories)Un carquois est une collection d'arcs joignant des couples de points. En ce sens, il s'agit d'un graphe orienté, mais la notion intervient en physique théorique ainsi qu'en théorie des représentations, des groupes et des catégories de manière naturelle. En effet, une catégorie est un carquois doté d'une structure supplémentaire : nommément la présence d'identités et de compositions. On parle donc de carquois lorsque l'on souhaite évoquer ce contexte catégorique (ou de représentation), plutôt que de (multi-di-)graphe orienté.
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. 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 (x, y) and (y, x) may be arrows of the graph). A tournament is an orientation of a complete graph. A polytree is an orientation of an undirected tree.
