Editing Chordal graph (section)

==Minimal separators==
In any graph, a [[vertex separator]] is a set of vertices the removal of which leaves the remaining graph disconnected; a separator is minimal if it has no proper subset that is also a separator. According to a theorem of {{harvtxt|Dirac|1961}}, chordal graphs are graphs in which each minimal separator is a clique; Dirac used this characterization to prove that chordal graphs are [[perfect graph|perfect]].

The family of chordal graphs may be defined inductively as the graphs whose vertices can be divided into three nonempty subsets {{mvar|A}}, {{mvar|S}}, and {{mvar|B}}, such that {{tmath|A \cup S}} and {{tmath|S \cup B}} both form chordal [[induced subgraph]]s, {{mvar|S}} is a clique, and there are no edges from {{mvar|A}} to {{mvar|B}}. That is, they are the graphs that have a recursive decomposition by clique separators into smaller subgraphs. For this reason, chordal graphs have also sometimes been called '''decomposable graphs'''.<ref>{{cite web |url=http://www.stat.berkeley.edu/~bartlett/courses/241A-spring2007/graphnotes.pdf |title=Undirected Graphical Models: Chordal Graphs, Decomposable Graphs, Junction Trees, and Factorizations | author=Peter Bartlett}}</ref>