View source for Strength of a graph

{{Short description|Graph-theoretic connectivity parameter}}
{{infobox graph
 | name     = Strength of a graph: example
 | image    = [[Image:Force-wiki.jpg|360px]]
 | image_caption = A graph with strength 2: the graph is here decomposed into three parts, with 4 edges between the parts, giving a ratio of 4/(3-1)=2.
}}

In [[graph theory]], the '''strength''' of an undirected [[Graph (discrete mathematics)|graph]] corresponds to the minimum ratio of ''edges removed''/''components created'' in a decomposition of the graph in question. It is a method to compute [[Partition of a set|partitions]] of the set of vertices and detect zones of high concentration of edges, and is analogous to [[graph toughness]] which is defined similarly for vertex removal.

== Definitions ==

The '''strength''' <math>\sigma(G)</math> of an undirected [[simple graph]] ''G''&nbsp;=&nbsp;(''V'',&nbsp;''E'') admits the three following definitions:

* Let <math>\Pi</math> be the set of all [[Partition of a set|partitions]] of <math>V</math>, and <math>\partial \pi</math> be the set of edges crossing over the sets of the partition <math>\pi\in\Pi</math>, then  <math>\displaystyle\sigma(G)=\min_{\pi\in\Pi}\frac{|\partial \pi|}{|\pi|-1}</math>.
* Also if <math> \mathcal T</math> is the set of all spanning trees of ''G'', then
:: <math>\sigma(G)=\max\left\{\sum_{T\in\mathcal T}\lambda_T\ :\ \forall T\in {\mathcal T}\ \lambda_T\geq 0\mbox{ and }\forall e\in E\ \sum_{T\ni e}\lambda_T\leq1\right\}.</math>
* And by linear programming duality,
:: <math>\sigma(G)=\min\left\{\sum_{e\in E}y_e\ :\ \forall e\in E\ y_e\geq0\mbox{ and }\forall T\in {\mathcal T}\ \sum_{e\in E}y_e\geq1\right\}.</math>

== Complexity ==

Computing the strength of a graph can be done in polynomial time, and the first such algorithm
was discovered by Cunningham (1985). The algorithm with best complexity for computing exactly the strength is due to Trubin (1993), uses the flow decomposition of Goldberg and Rao (1998), in time <math>O(\min(\sqrt{m},n^ {2/3})mn\log(n^2/m+2))</math>.

== Properties ==

* If <math>\pi=\{V_1,\dots,V_k\}</math> is one partition that maximizes, and for <math> i\in\{1,\dots,k\}</math>, <math>G_i=G/V_i</math> is the restriction of ''G'' to the set <math>V_i</math>, then <math>\sigma(G_k)\geq\sigma(G)</math>.
* The Tutte-Nash-Williams theorem:  <math>\lfloor\sigma(G)\rfloor</math> is the maximum number of edge-disjoint spanning trees that can be contained in ''G''.
* Contrary to the [[graph partition]] problem, the partitions output by computing the strength are not necessarily balanced (i.e. of almost equal size).

== References ==
*W. H. Cunningham. [http://portal.acm.org/citation.cfm?id=3829 ''Optimal attack and reinforcement of a network,''] J of ACM, 32:549&ndash;561, 1985.
*[[Alexander Schrijver|A. Schrijver]]. Chapter 51. [https://www.springer.com/math/applications/book/978-3-540-44389-6 ''Combinatorial Optimization,''] Springer, 2003.
*V. A. Trubin. [https://doi.org/10.1007%2FBF01125543 ''Strength of a graph and packing of trees and branchings,''], Cybernetics and Systems Analysis, 29:379&ndash;384, 1993.

[[Category:Graph connectivity]]
[[Category:Graph invariants]]