Concept

Forbidden graph characterization

Résumé
In graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to the family and further exclude all graphs from the family which contain any of these forbidden graphs as (induced) subgraph or minor. A prototypical example of this phenomenon is Kuratowski's theorem, which states that a graph is planar (can be drawn without crossings in the plane) if and only if it does not contain either of two forbidden graphs, the complete graph K_5 and the complete bipartite graph K_3,3. For Kuratowski's theorem, the notion of containment is that of graph homeomorphism, in which a subdivision of one graph appears as a subgraph of the other. Thus, every graph either has a planar drawing (in which case it belongs to the family of planar graphs) or it has a subdivision of at least one of these two graphs as a subgraph (in which case it does not belong to the planar graphs). More generally, a forbidden graph characterization is a method of specifying a family of graph, or hypergraph, structures, by specifying substructures that are forbidden to exist within any graph in the family. Different families vary in the nature of what is forbidden. In general, a structure G is a member of a family if and only if a forbidden substructure is not contained in G. The forbidden substructure might be one of: subgraphs, smaller graphs obtained from subsets of the vertices and edges of a larger graph, induced subgraphs, smaller graphs obtained by selecting a subset of the vertices and using all edges with both endpoints in that subset, homeomorphic subgraphs (also called topological minors), smaller graphs obtained from subgraphs by collapsing paths of degree-two vertices to single edges, or graph minors, smaller graphs obtained from subgraphs by arbitrary edge contractions. The set of structures that are forbidden from belonging to a given graph family can also be called an obstruction set for that family.
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Publications associées (71)
Concepts associés (46)
Graphe à seuil
vignette| Un graphe à seuil. En théorie des graphes, un graphe à seuil est un graphe qui peut être construit, en partant d'un graphe à un seul sommet, par application répétée d'une des deux opérations suivantes : Ajout d'un sommet isolé au graphe. Ajout d'un sommet dominant au graphe, c'est-à-dire d'un sommet connecté à tous les autres sommets. Par exemple, le graphe de la figure ci-contre est un graphe de seuil : il peut être construit en commençant par un graphe à un seul sommet (sommet 1), puis en ajoutant les sept autres dans l'ordre dans lequel ils sont numérotés, les sommets noirs comme sommets isolés et les sommets rouges comme sommets dominants.
Cographe
Un cographe est, en théorie des graphes, un graphe qui peut être généré par complémentation et union disjointe à partir du graphe à un nœud. La plupart des problèmes algorithmiques peuvent être résolus sur cette classe en temps polynomial, et même linaire, du fait de ses propriétés structurelles. Cette famille de graphe a été introduite par plusieurs auteurs indépendamment dans les années 1970 sous divers noms, notamment D*-graphes, hereditary Dacey graphs et 2-parity graphs.
Claw-free graph
In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph. A claw is another name for the complete bipartite graph K1,3 (that is, a star graph comprising three edges, three leaves, and a central vertex). A claw-free graph is a graph in which no induced subgraph is a claw; i.e., any subset of four vertices has other than only three edges connecting them in this pattern. Equivalently, a claw-free graph is a graph in which the neighborhood of any vertex is the complement of a triangle-free graph.
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