Graph Theory - Chordal Graphs

Chordal Graphs

A chordal graph, also known as a "perfectly triangulated graph," is a special type of graph where every cycle of length greater than 3 contains a chord. A chord is defined as an edge that connects two non-adjacent vertices within the cycle.

Chordal graphs are important in graph theory due to their structural properties and the fact that they allow efficient algorithms for various graph problems.

The following image shows an example of a chordal graph −

Properties of Chordal Graphs

Chordal graphs have several unique properties that distinguish them from other types of graphs. Some of these properties are −

• Perfect Elimination Ordering: Chordal graphs allow for a perfect elimination ordering, which is a sequence of vertices such that each vertex has all its neighbors from earlier in the ordering.
• Cliques: A maximal clique in a chordal graph is a subset of vertices that forms a clique (every vertex is connected to every other vertex in the subset) and cannot be extended by adding an adjacent vertex.
• Cycle Condition: Any cycle in the graph with more than 3 vertices must have a chord, which ensures that the graph has a more rigid structure than non-chordal graphs.
• Planarity: Chordal graphs may or may not be planar, meaning they can either be drawn on a flat plane without any edges crossing or require the use of additional dimensions to be drawn without crossings. However, many chordal graphs used in practical problems are planar.
• Graph Decomposition: Chordal graphs can be decomposed into cliques using algorithms like clique tree decomposition. This makes them useful in optimization problems.

Chordal graphs are useful in computer science, especially for solving problems like graph coloring, clique problems, and optimization tasks.

Classification of Chordal Graphs

Chordal graphs can be classified into various subtypes based on their structure and properties. These classifications are −

• Triangulated Graph: A triangulated graph is another name for a chordal graph, where each cycle of length greater than 3 contains a chord.
• Interval Graph: An interval graph is a type of chordal graph where each vertex corresponds to an interval on a line, and edges represent overlap between intervals.
• Split Graph: A split graph is a chordal graph that can be divided into two sets of vertices: one set that forms a clique (i.e., all vertices in this set are connected to each other) and the other set that forms an independent set (i.e., no vertices in this set are connected to each other).
• Packing Graph: A packing graph is a chordal graph where the vertices can be assigned to "packing" regions, such as in geometric problems.

Applications of Chordal Graphs

Chordal graphs have various applications in computer science and optimization problems, such as −

• Graph Coloring: Chordal graphs are used in graph coloring algorithms. A coloring problem involves assigning colors to the vertices of a graph such that no two adjacent vertices share the same color. The structure of chordal graphs makes this task easier, especially in scheduling problems and resource allocation.
• Perfect Elimination Orderings: The concept of perfect elimination orderings in chordal graphs is useful in various algorithms, such as those used for finding cliques in large networks, decomposing graphs, and solving constraint satisfaction problems.
• Optimization Problems: Chordal graphs are used in solving linear programming problems, as well as in various optimization tasks.
• Biological Networks: Chordal graphs are used to model interactions within biological networks, such as protein-protein interaction networks. Their ability to identify cliques and manage complex interactions makes them valuable in the study of biological systems and diseases.

Testing Chordality in Graphs

There are various methods to test whether a graph is chordal, such as −

• Perfect Elimination Ordering: If a graph allows for a perfect elimination ordering, it is chordal.
• Cycle Condition: Testing whether every cycle of length greater than 3 contains a chord is another way to check for chordality.

Various Types of Chordal Graphs

There are several special types of chordal graphs that exhibit unique properties. The following table summarizes some of the most common types of chordal graphs −