Theoretical and Experimental DNA Computation (Natural ...

2.6 Computational Complexity 39
Another problem may be concerned with cliques. A clique is a set of vertices
in an undirected graph in which each vertex is connected to every other
vertex. Cliques are complete graphs on n vertices, where n is the number of
vertices in the clique. The maximum clique problem is concerned with finding
the largest clique in a graph.
For example, consider the graph depicted in Fig. 2.13. It contains a subgraph
that is a clique of size 4.
Fig. 2.13. Graph containing a clique (shown in bold)
The problem of identifying “clusters” of related objects is often equivalent to
finding cliques in graphs. One “real world” application of this was developed
by the US Internal Revenue Service to detect organized tax fraud, where
groups of “phony” tax forms are submitted. A graph is constructed, where
vertices correspond to tax return forms and edges link any forms that look
similar. The existence of a large clique suggests large-scale organized fraud
[146].
As we can see, our problem is to either decide if a graph has certain properties
or construct from it a path or subgraph with certain properties. We show in
a later chapter how this may be achieved in the context of models of DNA
computation.
Another type of graph is the tree. These are connected, undirected, acyclic
graphs. This structure is accessed at the root vertex (usually drawn, perhaps
confusingly, at the top of any diagrammatic representation). Each vertex in a
tree is either a leaf node or an internal node (the terms vertex and node are
interchangeable). An internal node has one or more child nodes, and is called
the parent of the child node. An example tree is depicted in Fig. 2.14.
2.6 Computational Complexity
It should be clear that different algorithms require different resources. A program
to sort one hundred integers into ascending order will “obviously” run