Theoretical and Experimental DNA Computation (Natural ...

2.5 Data Structures 37
tions and perhaps redesign the network such that their removal does not lead
to the graph becoming disconnected (at the very least).
Another category of graphs is made up of those that can have their vertices
“colored” using one of three colors (say, red, green, and blue). These graphs
fall into the “3-vertex-colorable” category. Graphs that do and do not fall
into this category are depicted in Fig. 2.10. The problem is to decide whether
three colors are sufficient to achieve such a coloring for an arbitrary graph
[67]. If we consider a graph with n vertices, there are clearly 3 n possible ways
of assigning colors to vertices, but only a fraction of them will encode proper
colorings.
(a) (b)
Fig. 2.10. (a) 3-colorable. (b) Non-3-colorable graph
In order to clarify this, consider Fig. 2.11a. All possible three-colorings of G
(a) are depicted graphically in (b), with all proper colorings framed. A proper
coloring is highlighted in (c), and an illegal coloring in (d) (note how v2 and
v3 are colored the same).
Coloring could be applied to a situation where, for example, a company
manufactures several chemicals, certain pairs of which could explode if brought
into close contact. The company wishes to partition its warehouse into sealed
compartments and store incompatible chemicals in different compartments.
Obviously, the company wishes to minimize the amount of building work required,
and so needs to know the least number of compartments into which
the warehouse should be partitioned. We can construct a graph with vertices
representing chemicals, and edges between vertices representing incompatibilities.
The answer to the question of how many partitions are required is
therefore equal to the smallest number of colors required to obtain a proper
(i.e., legal) coloring of the graph.
Other questions involve asking whether or not a graph contains a subgraph
with its own particular properties. For example, “Does it contain a cycle?”
The cycle shown in the example graph G depicted in Fig. 2.9 is considered a
subgraph of G, as we see in Fig. 2.12.