In mathematical graph theory, a chordal graph is one in which all cycles of four or more vertices have an edge called a chord. One of the most central structured representations used for network simplification is chordal or triangulated graphs.
These graphs are subgraphs of complete graphs - and they have many uses in applications such as database theory, perfect phylogeny, or sparse matrix computations.
Important Points to Remember
- A chordal graph is a subset of the complete graph.
- A graph containing no chord is called a hole or chordless cycle.
- A complete graph is a chordal graph itself.
1) Consider the below graph ABCD. An edge CB is drawn from vertex C to vertex B which acts as a chord in this graph. Hence the below graph is said to be a chordal graph.
2) Consider the below example of the complete graph. A graph consisting of n*(n-1)/2 edges is said to be a complete graph, where n is the number of vertices. In the complete graph, the edges connecting all vertices are already present and act as a chord. Hence, a complete graph is a superset of a chordal graph.
Minimum-coloring Algorithm
In this article, we will discuss coloring the chordal graphs by using the greedy coloring algorithm.
Greedy Coloring Algorithm:
- First, we need to find the perfect elimination ordering (PEO) of the given chordal graph.
- Then, traverse the vertices in the reverse order of that PEO that we have already found.
- Gives the smallest coloring to each vertex such that the current vertex color is not used in its neighboring vertices.
- If the color of the current vertex is already given to its neighbor then increment the color and assign it to the current vertex.
- Repeat steps 3 and 4 until all the vertices are colored.
Example: Consider the below chordal graph ABCD. Let the perfect elimination ordering of this graph be [D, C, B, A]. We will traverse the graph PEO in the reverse direction. Hence, [A, B, C, D] is the reverse of PEO.
1. First, we will color the vertex A with 1 as we need to color with the smallest present color.
2. Now for vertex B, we will check whether its neighbors contain color 1 or not. As vertex 'A' contains color 1, therefore we will increment the color by 1 to color vertex B.
3. Similarly, for vertex C, its neighbor A contains color 1, and B contains color 2 so we will color vertex C to 3.
4. Now, for vertex D, as its neighbors do not contain the color 1 so we will color 1 to vertex D.
Solved Examples on Coloring of Chordal Graphs
Example 1: Coloring a Simple Chordal Graph
Consider the following chordal graph:
Vertices: A, B, C, D
Edges: AB, AC, AD, BC, BD
Perfect Elimination Ordering (PEO): [D, B, C, A]
Solution:
Traverse in reverse PEO: [A, C, B, D]
Color vertex A with color 1.
Vertex C: neighbor A has color 1, so color C with color 2.
Vertex B: neighbor A has color 1, C has color 2, so color B with color 3.
Vertex D: no neighbors with color 1, so color D with color 1.
Final Coloring: A - 1, C - 2, B - 3, D - 1
Example 2: Coloring a Complete Graph
Consider the following complete graph:
Vertices: A, B, C, D
Edges: AB, AC, AD, BC, BD, CD
Perfect Elimination Ordering (PEO): [D, C, B, A]
Solution:
Traverse in reverse PEO: [A, B, C, D]
Color vertex A with color 1.
Vertex B: neighbor A has color 1, so color B with color 2.
Vertex C: neighbors A and B have colors 1 and 2, so color C with color 3.
Vertex D: neighbors A, B, and C have colors 1, 2, and 3, so color D with color 4.
Final Coloring: A - 1, B - 2, C - 3, D - 4
Practice Problems - Coloring of Chordal Graphs
Problem 1: Vertices: A, B, C, D, E
Edges: AB, AC, AD, BE, CE
Find the perfect elimination ordering (PEO) and color the graph using the greedy coloring algorithm.
Problem 2: Vertices: P, Q, R, S, T
Edges: PQ, PR, PS, QR, RS
Determine the PEO and apply the greedy coloring algorithm.
Problem 3: Vertices: M, N, O, P, Q
Edges: MN, MO, MP, NQ, OQ
Identify the PEO and color the graph using the greedy coloring algorithm.
Problem 4: Vertices: X, Y, Z, W
Edges: XY, XZ, YZ, YW
Find the PEO and color the graph using the greedy coloring algorithm.
Problem 5: Vertices: A, B, C, D, E, F
Edges: AB, AC, BD, BE, CF, DF
Determine the PEO and use the greedy coloring algorithm to color the graph.
Problem 6: Vertices: L, M, N, O, P, Q
Edges: LM, LN, MO, NP, OQ, PQ
Identify the PEO and apply the greedy coloring algorithm.
Problem 7: Vertices: U, V, W, X, Y, Z
Edges: UV, UW, VX, WX, XY, YZ
Find the PEO and color the graph using the greedy coloring algorithm.
Problem 8: Vertices: G, H, I, J, K
Edges: GH, GI, HJ, IJ, JK
Determine the PEO and apply the greedy coloring algorithm.
Problem 9: Vertices: A, B, C, D, E, F, G
Edges: AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE, DF, EF
Find the PEO and use the greedy coloring algorithm to color the graph.
Problem 10: Vertices: P, Q, R, S, T, U
Edges: PQ, PR, PS, PT, QU, RU, SU, TU
Find the PEO and color the graph using the greedy coloring algorithm.