Cluster graphIn graph theory, a branch of mathematics, a cluster graph is a graph formed from the disjoint union of complete graphs. Equivalently, a graph is a cluster graph if and only if it has no three-vertex induced path; for this reason, the cluster graphs are also called P_3-free graphs. They are the complement graphs of the complete multipartite graphs and the 2-leaf powers. The cluster graphs are transitively closed, and every transitively closed undirected graph is a cluster graph.
Degeneracy (graph theory)In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has a vertex of degree at most k: that is, some vertex in the subgraph touches k or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of k for which it is k-degenerate. The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph.
Strong perfect graph theoremIn graph theory, the strong perfect graph theorem is a forbidden graph characterization of the perfect graphs as being exactly the graphs that have neither odd holes (odd-length induced cycles of length at least 5) nor odd antiholes (complements of odd holes). It was conjectured by Claude Berge in 1961. A proof by Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomas was announced in 2002 and published by them in 2006.
K-treeIn graph theory, a k-tree is an undirected graph formed by starting with a (k + 1)-vertex complete graph and then repeatedly adding vertices in such a way that each added vertex v has exactly k neighbors U such that, together, the k + 1 vertices formed by v and U form a clique. The k-trees are exactly the maximal graphs with a treewidth of k ("maximal" means that no more edges can be added without increasing their treewidth).
Hereditary propertyIn mathematics, a hereditary property is a property of an object that is inherited by all of its subobjects, where the meaning of subobject depends on the context. These properties are particularly considered in topology and graph theory, but also in set theory. In topology, a topological property is said to be hereditary if whenever a topological space has that property, then so does every subspace of it. If the latter is true only for closed subspaces, then the property is called weakly hereditary or closed-hereditary.
Vertex separatorIn graph theory, a vertex subset S \subset V is a vertex separator (or vertex cut, separating set) for nonadjacent vertices a and b if the removal of S from the graph separates a and b into distinct connected components. Consider a grid graph with r rows and c columns; the total number n of vertices is r × c. For instance, in the illustration, r = 5, c = 8, and n = 40. If r is odd, there is a single central row, and otherwise there are two rows equally close to the center; similarly, if c is odd, there is a single central column, and otherwise there are two columns equally close to the center.
Clique-widthIn graph theory, the clique-width of a graph G is a parameter that describes the structural complexity of the graph; it is closely related to treewidth, but unlike treewidth it can be small for dense graphs. It is defined as the minimum number of labels needed to construct G by means of the following 4 operations : Creation of a new vertex v with label i (denoted by i(v)) Disjoint union of two labeled graphs G and H (denoted by ) Joining by an edge every vertex labeled i to every vertex labeled j (denoted by η(i,j)), where i ≠ j Renaming label i to label j (denoted by ρ(i,j)) Graphs of bounded clique-width include the cographs and distance-hereditary graphs.
Graph structure theoremIn mathematics, the graph structure theorem is a major result in the area of graph theory. The result establishes a deep and fundamental connection between the theory of graph minors and topological embeddings. The theorem is stated in the seventeenth of a series of 23 papers by Neil Robertson and Paul Seymour. Its proof is very long and involved. and are surveys accessible to nonspecialists, describing the theorem and its consequences. A minor of a graph G is any graph H that is isomorphic to a graph that can be obtained from a subgraph of G by contracting some edges.
Strangulated graphIn graph theoretic mathematics, a strangulated graph is a graph in which deleting the edges of any induced cycle of length greater than three would disconnect the remaining graph. That is, they are the graphs in which every peripheral cycle is a triangle. In a maximal planar graph, or more generally in every polyhedral graph, the peripheral cycles are exactly the faces of a planar embedding of the graph, so a polyhedral graph is strangulated if and only if all the faces are triangles, or equivalently it is maximal planar.
Parameterized complexityIn computer science, parameterized complexity is a branch of computational complexity theory that focuses on classifying computational problems according to their inherent difficulty with respect to multiple parameters of the input or output. The complexity of a problem is then measured as a function of those parameters. This allows the classification of NP-hard problems on a finer scale than in the classical setting, where the complexity of a problem is only measured as a function of the number of bits in the input.
