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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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Hi there. My name is Sophia Meagher even though it is not the title on my beginning certificate. Since he was eighteen he's been working as an info officer but he plans on changing it. Her family members lives in Ohio. What me and my family adore is performing ballet but I've been using on new things recently.
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