Concept
Théorème du séparateur planaire
Concepts associés (9)
Branch-decomposition
In graph theory, a branch-decomposition of an undirected graph G is a hierarchical clustering of the edges of G, represented by an unrooted binary tree T with the edges of G as its leaves. Removing any edge from T partitions the edges of G into two subgraphs, and the width of the decomposition is the maximum number of shared vertices of any pair of subgraphs formed in this way. The branchwidth of G is the minimum width of any branch-decomposition of G.
Shallow minor
In graph theory, a shallow minor or limited-depth minor is a restricted form of a graph minor in which the subgraphs that are contracted to form the minor have small diameter. Shallow minors were introduced by , who attributed their invention to Charles E. Leiserson and Sivan Toledo. One way of defining a minor of an undirected graph G is by specifying a subgraph H of G, and a collection of disjoint subsets Si of the vertices of G, each of which forms a connected induced subgraph Hi of H.
Bounded expansion
In graph theory, a family of graphs is said to have bounded expansion if all of its shallow minors are sparse graphs. Many natural families of sparse graphs have bounded expansion. A closely related but stronger property, polynomial expansion, is equivalent to the existence of separator theorems for these families. Families with these properties have efficient algorithms for problems including the subgraph isomorphism problem and model checking for the first order theory of graphs.
Pathwidth
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.
Graph embedding
In topological graph theory, an embedding (also spelled imbedding) of a graph on a surface is a representation of on in which points of are associated with vertices and simple arcs (homeomorphic images of ) are associated with edges in such a way that: the endpoints of the arc associated with an edge are the points associated with the end vertices of no arcs include points associated with other vertices, two arcs never intersect at a point which is interior to either of the arcs. Here a surface is a compact, connected -manifold.
Largeur arborescente
En théorie des graphes et en informatique théorique, la largeur arborescente ou largeur d'arbre d'un graphe (treewidth en anglais) est un nombre qui, intuitivement, mesure s'il est proche d'un arbre. Elle peut être définie de plusieurs manières, notamment en utilisant la décomposition arborescente. Souvent, un problème algorithmique facile sur les arbres est en fait facile pour les graphes qui ressemblent à des arbres. Ainsi, ce paramètre est souvent utilisé en algorithmique de graphes, notamment pour les schémas d'approximation polynomiaux et complexité paramétrée.
Book embedding
In graph theory, a book embedding is a generalization of planar embedding of a graph to embeddings in a book, a collection of half-planes all having the same line as their boundary. Usually, the vertices of the graph are required to lie on this boundary line, called the spine, and the edges are required to stay within a single half-plane. The book thickness of a graph is the smallest possible number of half-planes for any book embedding of the graph. Book thickness is also called pagenumber, stacknumber or fixed outerthickness.
Mineur (théorie des graphes)
La notion de mineur d'un graphe est un concept de théorie des graphes. Il a été défini et étudié par Robertson et Seymour dans une série d'articles intitulée Graph minors (I à XXIII), publiée dans le Journal of Combinatorial Theory entre 1983 et 2011. Soit un graphe non orienté fini. Un graphe est un mineur de s'il peut être obtenu en contractant des arêtes d'un sous-graphe de .
Graphe planaire
Dans la théorie des graphes, un graphe planaire est un graphe qui a la particularité de pouvoir se représenter sur un plan sans qu'aucune arête (ou arc pour un graphe orienté) n'en croise une autre. Autrement dit, ces graphes sont précisément ceux que l'on peut plonger dans le plan, ou encore les graphes dont le nombre de croisements est nul. Les méthodes associées à ces graphes permettent de résoudre des problèmes comme l'énigme des trois maisons et d'autres plus difficiles comme le théorème des quatre couleurs.