What is Chordal Graphs?

The Fundamental Theory of Chordal Graphs was developed in the 1960s and 1970s. Most of us are now familiar with the term “chordal” as well as other names. Chordal graphs have also been called rigid-circuit, triangulated, and decomposable graphs. Chordal graphs have many practical applications, especially in biology. i.e., modeling and analysis of molecular networks and genetic interactions. In particular, several classes of chordal graphs have been used to model different types of biological networks and analyze their structural and functional properties.

A chordal graph is a special type of undirected simple graph. A graph is chordal if and only if every induced cycle of length 4 or more is such that any two non-adjacent vertices of the cycle are connected to each other as a chord or edge. That chord is not part of the graph cycle. A graph cycle without a chord is called a “graph hole” or “chordless cycle.”

A chord divides a cycle into two parts. If at least one of the cycles is not a cycle, then it has another chord.

Subclasses of Chordal Graphs 

There are several important subclasses of graphs, and all have their own properties and applications. Some of the most important are mentioned below.

Complete Graph 

A complete graph is a chordal graph, in which every vertex is connected to adjacent vertices. Every vertex has the same degree. A complete graph is a superset of a chordal graph. because every induced subgraph of a graph is also a chordal graph.

Interval Graph 

An interval graph is a chordal graph that can be represented by a set of intervals on a line such that two intervals have an intersection if and only if the corresponding vertices in the graph are adjacent.

Block Graph 

A block graph is a chordal graph where every block (maximal 2 connected subgraphs) is a complete graph.

Clique-Sum Graph 

A clique-sum graph is a chordal graph that can be constructed by sequences of operations called clique-sum. where two graphs are merged by identifying a common clique.

Perfect Elimination Graph  

A Perfect Elimination Graph is a chordal graph in which each induced subgraph has a perfect elimination ordering, which is an ordering of vertices in which every vertex is adjacent to every later vertex in the ordering that is also a neighbor of the vertex. 

Perfect elimination ordering — (B,A,C,D)

Strongly Chordal Graph  

A strongly chordal graph is a chordal graph in which every induced subgraph has a perfect elimination ordering. That is, there exists an ordering of the vertices such that, for each vertex, its neighbors that come later in the ordering form a clique.