Geographic Information Systems

graph graph and geography the geographical problem Figure 3.11: The Koenigsberg bridge problem and its graphical representation. The problem and its associated graph is shown in Figure 3.11 and this allows us to prove it is impossible to visit all the land masses, crossing each bridge only once. Graphs are a topological construct, and do not have a notion of distance associated with them in general. Rather that spend a long time reviewing the theory of graphs, we give a number of definitions: graph: a set of nodes and edges; edge: e = {x, y} is the edge joining nodes x and y; directed graph: each edge of the graph has a direction attached; labelled graph: each edge has an associated label; connected graph: there is a path between all pairs of nodes in the graph; isomorphic graphs: two graphs which have the same connectedness are said to be isomorphic; 14 CS321 Geographic Information Systems cycle: a path in the graph from a node back to itself; acyclic graph: a graph with no cycles (easily analysed); tree: a connected acyclic graph, frequently used as a data structure in CS in general and GIS. Figure 3.12: A graph and two of its possible planar embeddings. Planar graphs are graphs which can be represented as planar embeddings of the abstract graph with edges only crossing at nodes. An example of a planar graph and two possible embeddings are shown in Figure 3.12. Note that the two planar embeddings shown are not homeomorphic to each other, so there is some ambiguity as to which is appropriate. We will return to some of these concepts as we need them. In particular we will explore network data structures further later in the course.