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.

- Indifference graph
- Series–parallel graph
- Threshold hypergraphs
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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).

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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.

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- ↑ Reiterman, Jan; Rödl, Vojtěch; Šiňajová, Edita; Tůma, Miroslav (1985-04-01). "Threshold hypergraphs".
*Discrete Mathematics*.**54**(2): 193–200. doi: 10.1016/0012-365X(85)90080-9 . ISSN 0012-365X.

- 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, vol. 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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