# 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: In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices, nodes, or points which are connected by edges, arcs, or lines. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another; see Graph for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics.

## Contents

1. Addition of a single isolated vertex to the graph.
2. 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.

## Alternative definitions

An equivalent definition is the following: a graph is a threshold graph if there are a real number $S$ and for each vertex $v$ a real vertex weight $w(v)$ such that for any two vertices $v,u$ , $uv$ is an edge if and only if $w(u)+w(v)>S$ . In mathematics, and more specifically in graph theory, a vertex or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges, while a directed graph consists of a set of vertices and a set of arcs. In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another.

Another equivalent definition is this: a graph is a threshold graph if there are a real number $T$ and for each vertex $v$ a real vertex weight $a(v)$ such that for any vertex set $X\subseteq V$ , $X$ is independent if and only if $\sum _{v\in X}a(v)\leq T.$  In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set S of vertices such that for every two vertices in S, there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in S. The size of an independent set is the number of vertices it contains. Independent sets have also been called internally stable sets.

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.

## Decomposition

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. $\epsilon$ is always the first character of the string, and represents the first vertex of the graph. Every subsequent character is either $u$ , which denotes the addition of an isolated vertex (or union vertex), or $j$ , which denotes the addition of a dominating vertex (or join vertex). For example, the string $\epsilon uuj$ represents a star graph with three leaves, while $\epsilon uj$ represents a path on three vertices. The graph of the figure can be represented as $\epsilon uuujuuj$ Threshold graphs are a special case of cographs, split graphs, and trivially perfect graphs. 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 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, P4, and a threshold graph is a graph with no induced P4, C4 nor 2K2. C4 is a cycle of four vertices and 2K2 is its complement, that is, two disjoint edges. This also explains why threshold graphs are closed under taking complements; the P4 is self-complementary, hence if a graph is P4-, C4- and 2K2-free, its complement is as well. In graph theory, a cograph, or complement-reducible graph, or P4-free graph, is a graph that can be generated from the single-vertex graph K1 by complementation and disjoint union. That is, the family of cographs is the smallest class of graphs that includes K1 and is closed under complementation and disjoint union. 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 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.

Heggernes & Kratsch (2007) showed that threshold graphs can be recognized in linear time; if a graph is not threshold, an obstruction (one of P4, C4, or 2K2) will be output.

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• Chvátal, Václav; Hammer, Peter L. (1977), "Aggregation of inequalities in integer programming", in Hammer, P. L.; Johnson, E. L.; Korte, B. H.; et al., 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.