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.