In graph theory, a **threshold graph** is a graph that can be constructed from a one-vertex graph by repeated applications of the following two operations:

- Alternative definitions
- Decomposition
- Related classes of graphs and recognition
- See also
- References
- External links

- Addition of a single isolated vertex to the graph.
- Addition of a single dominating vertex to the graph, i.e. a single vertex that is connected to all other vertices.

For example, the graph of the figure is a threshold graph. It can be constructed by beginning with a single-vertex graph (vertex 1), and then adding black vertices as isolated vertices and red vertices as dominating vertices, in the order in which they are numbered.

Threshold graphs were first introduced by Chvátal & Hammer (1977). A chapter on threshold graphs appears in Golumbic (1980), and the book Mahadev & Peled (1995) is devoted to them.

An equivalent definition is the following: a graph is a threshold graph if there are a real number and for each vertex a real vertex weight such that for any two vertices , is an edge if and only if .

Another equivalent definition is this: a graph is a threshold graph if there are a real number and for each vertex a real vertex weight such that for any vertex set , is independent if and only if

The name "threshold graph" comes from these definitions: *S* is the "threshold" for the property of being an edge, or equivalently *T* is the threshold for being independent.

Threshold graphs also have a forbidden graph characterization: A graph is a threshold graph if and only if it no four of its vertices form an induced subgraph that is a three-edge path graph, a four-edge cycle graph, or a two-edge matching.

From the definition which uses repeated addition of vertices, one can derive an alternative way of uniquely describing a threshold graph, by means of a string of symbols. is always the first character of the string, and represents the first vertex of the graph. Every subsequent character is either , which denotes the addition of an isolated vertex (or *union* vertex), or , which denotes the addition of a dominating vertex (or *join* vertex). For example, the string represents a star graph with three leaves, while represents a path on three vertices. The graph of the figure can be represented as

Threshold graphs are a special case of cographs, split graphs, and trivially perfect graphs. A graph is a threshold graph if and only if it is both a cograph and a split graph. Every graph that is both a trivially perfect graph and the complementary graph of a trivially perfect graph is a threshold graph. Threshold graphs are also a special case of interval graphs. All these relations can be explained in terms of their characterisation by forbidden induced subgraphs. A cograph is a graph with no induced path on four vertices, P_{4}, and a threshold graph is a graph with no induced P_{4}, C_{4} nor 2K_{2}. C_{4} is a cycle of four vertices and 2K_{2} is its complement, that is, two disjoint edges. This also explains why threshold graphs are closed under taking complements; the P_{4} is self-complementary, hence if a graph is P_{4}-, C_{4}- and 2K_{2}-free, its complement is as well.

Heggernes & Kratsch (2007) showed that threshold graphs can be recognized in linear time; if a graph is not threshold, an obstruction (one of P_{4}, C_{4}, or 2K_{2}) will be output.

This is a **glossary of graph theory terms**. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by edges.

In graph theory, a **perfect graph** is a graph in which the chromatic number of every induced subgraph equals the order of the largest clique of that subgraph. Equivalently stated in symbolic terms an arbitrary graph is perfect if and only if for all we have .

In graph theory, the **perfect graph theorem** of László Lovász states that an undirected graph is perfect if and only if its complement graph is also perfect. This result had been conjectured by Berge, and it is sometimes called the weak perfect graph theorem to distinguish it from the strong perfect graph theorem characterizing perfect graphs by their forbidden induced subgraphs.

In the mathematical area of graph theory, a **chordal graph** is one in which all cycles of four or more vertices have a *chord*, which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree. They are sometimes also called **rigid circuit graphs** or **triangulated graphs**.

In 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 nor odd antiholes. 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.

In graph theory, the **complement** or **inverse** of a graph G is a graph H on the same vertices such that two distinct vertices of H are adjacent if and only if they are not adjacent in G. That is, to generate the complement of a graph, one fills in all the missing edges required to form a complete graph, and removes all the edges that were previously there. It is not, however, the set complement of the graph; only the edges are complemented.

In graph theory, a **cograph**, or **complement-reducible graph**, or ** P_{4}-free graph**, is a graph that can be generated from the single-vertex graph

In graph theory, a **comparability graph** is an undirected graph that connects pairs of elements that are comparable to each other in a partial order. Comparability graphs have also been called **transitively orientable graphs**, **partially orderable graphs**, **containment graphs**, and **divisor graphs**. An **incomparability graph** is an undirected graph that connects pairs of elements that are not comparable to each other in a partial order.

In the mathematical field of graph theory, the **Grötzsch graph** is a triangle-free graph with 11 vertices, 20 edges, chromatic number 4, and crossing number 5. It is named after German mathematician Herbert Grötzsch.

In the mathematical area of graph theory, an **induced path** in an undirected graph *G* is a path that is an induced subgraph of *G*. That is, it is a sequence of vertices in *G* such that each two adjacent vertices in the sequence are connected by an edge in *G*, and each two nonadjacent vertices in the sequence are not connected by any edge in *G*. An induced path is sometimes called a **snake**, and the problem of finding long induced paths in hypercube graphs is known as the snake-in-the-box problem.

In graph theory, a branch of mathematics, a **split graph** is a graph in which the vertices can be partitioned into a clique and an independent set. Split graphs were first studied by Földes and Hammer, and independently introduced by Tyshkevich and Chernyak (1979).

In graph theory, a **clique cover** or **partition into cliques** of a given undirected graph is a partition of the vertices of the graph into cliques, subsets of vertices within which every two vertices are adjacent. A **minimum clique cover** is a clique cover that uses as few cliques as possible. The minimum *k* for which a clique cover exists is called the **clique cover number** of the given graph.

In graph theory, a branch of discrete mathematics, a **distance-hereditary graph** is a graph in which the distances in any connected induced subgraph are the same as they are in the original graph. Thus, any induced subgraph inherits the distances of the larger graph.

In graph theory, a **trivially perfect graph** is a graph with the property that in each of its induced subgraphs the size of the maximum independent set equals the number of maximal cliques. Trivially perfect graphs were first studied by but were named by Golumbic (1978); Golumbic writes that "the name was chosen since it is trivial to show that such a graph is perfect." Trivially perfect graphs are also known as **comparability graphs of trees**, **arborescent comparability graphs**, and **quasi-threshold graphs**.

In graph theory, a **perfectly orderable graph** is a graph whose vertices can be ordered in such a way that a greedy coloring algorithm with that ordering optimally colors every induced subgraph of the given graph. Perfectly orderable graphs form a special case of the perfect graphs, and they include the chordal graphs, comparability graphs, and distance-hereditary graphs. However, testing whether a graph is perfectly orderable is NP-complete.

In graph theory, the **modular decomposition** is a decomposition of a graph into subsets of vertices called **modules.** A module is a generalization of a connected component of a graph. Unlike connected components, however, one module can be a proper subset of another. Modules therefore lead to a recursive (hierarchical) decomposition of the graph, instead of just a partition.

In graph theory, a **skew partition** of a graph is a partition of its vertices into two subsets, such that the induced subgraph formed by one of the two subsets is disconnected and the induced subgraph formed by the other subset is the complement of a disconnected graph. Skew partitions play an important role in the theory of perfect graphs.

In graph theory, a branch of mathematics, the **Erdős–Hajnal conjecture** states that families of graphs defined by forbidden induced subgraphs have either large cliques or large independent sets. It is named for Paul Erdős and András Hajnal.

In graph theory, a **universal vertex** is a vertex of an undirected graph that is adjacent to all other vertices of the graph. It may also be called a **dominating vertex**, as it forms a one-element dominating set in the graph.

In graph theory, the **graph removal lemma** states that when a graph contains few copies of a given subgraph, then all of the copies can be eliminated by removing a small number of edges. The special case in which the subgraph is a triangle is known as the **triangle removal lemma**.

- Chvátal, Václav; Hammer, Peter L. (1977), "Aggregation of inequalities in integer programming", in Hammer, P. L.; Johnson, E. L.; Korte, B. H.; et al. (eds.),
*Studies in Integer Programming (Proc. Worksh. Bonn 1975)*, Annals of Discrete Mathematics,**1**, Amsterdam: North-Holland, pp. 145–162. - Golumbic, Martin Charles (1980),
*Algorithmic Graph Theory and Perfect Graphs*, New York: Academic Press. 2nd edition, Annals of Discrete Mathematics,**57**, Elsevier, 2004. - Heggernes, Pinar; Kratsch, Dieter (2007), "Linear-time certifying recognition algorithms and forbidden induced subgraphs" (PDF),
*Nordic Journal of Computing*,**14**(1–2): 87–108 (2008), MR 2460558 . - Mahadev, N. V. R.; Peled, Uri N. (1995),
*Threshold Graphs and Related Topics*, Elsevier.

- Threshold graphs, Information System on Graph Classes and their Inclusions.

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