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| [[Image:threshold graph.png|thumb|240px|An example of a threshold graph.]]
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| 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:
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| #Addition of a single isolated vertex to the graph.
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| #Addition of a single dominating vertex to the graph, i.e. a single vertex that is connected to all other vertices.
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| 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.
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| Threshold graphs were first introduced by {{harvtxt|Chvátal|Hammer|1977}}. A chapter on threshold graphs appears in {{harvtxt|Golumbic|1980}}, and the book {{harvtxt|Mahadev|Peled|1995}} is devoted to them.
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| == Alternative definitions ==
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| An equivalent definition is the following: a graph is a threshold graph if there are a real number <math>S</math> and for each vertex <math>v</math> a real vertex weight <math>w(v)</math> such that for any two vertices <math>v,u</math>, <math>uv</math> is an edge if and only if <math>w(u)+w(v)\ge S</math>.
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| <!-- To see why the two definition are equivalent, observe that a graph constructed using the first definition is also a graph of the second definition by setting <math>S=1</math>, <math>w(u)=1</math> for vertices that were added as dominating vertices and <math>w(u)=0</math> for all other vertices. For the other direction, given <math>S</math> and <math>w</math> we can construct the same graph by starting with the graph whose only vertex is the vertex with the minimum weight and then examining all remaining vertices in order of non-decreasing weight. Each one is added as a -->
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| Another equivalent definition is this: a graph is a threshold graph if there are a real number <math>T</math> and for each vertex <math>v</math> a real vertex weight <math>a(v)</math> such that for any vertex set <math>X\subseteq V</math>, <math>X</math> is independent if and only if <math>\sum_{v \in X} a(v) \ge T.</math>
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| 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.
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| ==Decomposition==
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| 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. <math>\epsilon</math> is always the first character of the string, and represents the first vertex of the graph. Every subsequent character is either <math>u</math>, which denotes the addition of an isolated vertex (or ''union'' vertex), or <math>j</math>, which denotes the addition of a dominating vertex (or ''join'' vertex). For example, the string <math>\epsilon u u j</math> represents a star graph with three leaves, while <math>\epsilon u j</math> represents a path on three vertices. The graph of the figure can be represented as <math>\epsilon uuujuuj </math>
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| ==Related classes of graphs==
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| Threshold graphs are a special case of [[cograph]]s, [[split graph]]s, and [[trivially perfect graph]]s. Every graph that is both a cograph and a split graph is a threshold graph. Every graph that is both a trivially perfect graph and the [[Complement graph|complementary graph]] of a trivially perfect graph is a threshold graph. Threshold graphs are also a special case of [[interval graph]]s.
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| ==See also==
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| *[[Series-parallel graph]]
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| ==References==
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| *{{citation
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| | last1 = Chvátal | first1 = Václav | author1-link = Václav Chvátal
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| | last2 = Hammer | first2 = Peter L. | author2-link = Peter Hammer
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| | contribution = Aggregation of inequalities in integer programming
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| | editor1-last = Hammer | editor1-first = P. L.
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| | editor2-last = Johnson | editor2-first = E. L.
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| | editor3-last = Korte | editor3-first = B. H.
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| | editor4-last = Nemhauser | editor4-first = G. L.
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| | location = Amsterdam
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| | pages = 145–162
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| | publisher = North-Holland
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| | series = Annals of Discrete Mathematics
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| | title = Studies in Integer Programming (Proc. Worksh. Bonn 1975)
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| | volume = 1
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| | year = 1977}}.
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| *{{citation
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| | last = Golumbic | first = Martin Charles | author-link = Martin Charles Golumbic
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| | location = New York
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| | publisher = Academic Press
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| | title = Algorithmic Graph Theory and Perfect Graphs
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| | year = 1980}}. 2nd edition, Annals of Discrete Mathematics, '''57''', Elsevier, 2004.
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| *{{citation
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| | last1 = Mahadev | first1 = N. V. R.
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| | last2 = Peled | first2 = Uri N.
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| | publisher = Elsevier
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| | title = Threshold Graphs and Related Topics
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| | year = 1995}}.
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| ==External links==
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| *[http://wwwteo.informatik.uni-rostock.de/isgci/classes/gc_328.html Threshold graphs], Information System on Graph Class Inclusions, Univ. of Rostock.
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| {{DEFAULTSORT:Threshold Graph}}
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| [[Category:Graph families]]
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| [[Category:Perfect graphs]]
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