User:Vernon1016

In graph theory, a decomposable graph is a type of graph which is used to model various mathematical concepts. These include modeling probabilities in [[Bayesian statistics, and forming junction trees. Junction trees are a specialized type of a mathematical structure known as a tree. These junction trees can be used to solve problems such as graph coloring and  constraint satisfaction.

Definitions
We say that $$A$$, $$B$$, and $$S$$, subsets of a graph $$G$$, form a proper decomposition of $$G$$, written as the ordered 3-tuple $$(A,B,S)$$ if the following statements all hold:
 * 1) $$A$$, $$B$$, and $$S$$ are subsets of $$G$$ such that $$V = A$$&cup;$$B$$&cup;$$S$$, where $$V$$ is the vertex set of $$G$$.
 * 2) The elements of $$A$$,$$B$$, and $$S$$ are distinct and share no common elements.
 * 3) The set $$S$$ is a clique, which is a  complete subset of vertices in $$G$$.
 * 4) $$S$$ is a vertex separator in $$G$$.

A graph $$G$$ is said to be decomposable if either of the following holds:
 * 1) $$G$$ is complete.
 * 2) $$G$$ possesses a proper decomposition $$(A,B,S)$$ such that the induced subgraphs $$G$$($A$&cup;$S$) and $$G$$($B$&cup;$S$) are decomposable.

Properties and Facts

 * All decomposable graphs are triangulated, or chordal. The converse is also true. This means that every cycle in a graph with length of at least 4 contains a chord, which is an edge connecting non-adjacent vertices.
 * The removal of an edge (x,y), from a graph G, results in a decomposable graph if and only if the two vertices x and y are in exactly one clique.
 * The addition of an edge (x,y), into a graph G, results in a decomposable graph if and only if x and y are unconnected, and contained in adjacent cliques in some junction tree of the graph G.