Co-Graph

In computer science, a co- graph is an undirected graph can be constructed with certain elementary operations. In co- graphs many serious problems, such as CLIQUE and the closely related INDEPENDENT QUANTITY and KNOTS OVER COVER can be solved in linear time.

Definition

Is a graph showing a co- graph, if it can be constructed using the following three operations:

Equivalent characterizations

For a graph of the following statements are equivalent:

• Is a co- graph.
• Contains no induced, with the undirected path with four nodes respectively.
• The Komplementgraph each contiguous induced subgraphs of at least two nodes is incoherent.
• Can be constructed with the following three rules:

Co - tree

In order to efficiently solve hard problems on co- graphs they can be using co- trees represent. A co- tree is a binary tree whose leaves are labeled with and whose internal nodes with or.

A co- tree is defined as follows:

Example

The following example outlines the construction of a co- graph with associated co- tree:

Other examples of co- graphs are complete graphs and completely unconnected graph.

Properties of the co- graphene

It is easy to see that co- graphs are closed under complementation. To produce the Komplementgraphen must be exchanged and only the operations in the accompanying co- tree.

Furthermore, the amount of co- graph is completed with the formation of induced subgraph.

It is also known that each co- graph is a perfect graph.

Application in algorithmics

Some heavy graph problems can be solved on co- graphs in linear time. These include among other things a problem INDEPENDENT QUANTITY, CLIQUE and NODE OVERLAP.

Using dynamic programming to the associated co- trees can be simple and elegant solutions to the above problems.