Graph
::depthFirstSearch
Makes a depth first Search in a graph.
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Graph::depthFirstSearch(G
, <StartVertex = v
>)
Graph::depthFirstSearch
traverses through
a graph via depth first search. The output shows the first time of
identification, the finishing time and the predecessor of each vertex.
If a vertex is a single vertex with no predecessor its predecessor
is infinity.
Graph::depthFirstSearch(G, StartVertex=v)
traverses
through a graph via depth first search starting from vertex v
.
The output shows the first time of identification, the finishing time
and the predecessor of each vertex. If a vertex is a single vertex
with no predecessor its predecessor is infinity.
A typical tree is created and drawn for a better understanding of the algorithm.
G := Graph([a, b, c, d, e, f, g, h, i, j, k, l], [[a, b], [a, c], [b, d], [b, e], [c, f], [c, g], [d, h], [e, i], [e, j], [f, k], [g, l]], Directed): plot( Graph::plotGridGraph(G, VerticesPerLine = [12, 12, 12, 12], VertexOrder = [ None, None, None, None, None, None, a, None, None, None, None, None, None, None, b, None, None, None, None, None, None, c, None, None, None, d, None, None, e, None, None, f, None, None, g, None, h, None, None, i, None, j, None, None, k, None, None, l ] ) )
Now we call Graph::depthFirstSearch
to find
out the starting times, the finishing times and the predecessors of
each vertex:
Graph::depthFirstSearch(G)
Vertex a
is dicovered first, then vertex
b and so on. The table in the middle shows the finishing times. h
for
example has the finishing time of 5, meaning that vertices a,
b, c, d
and h
itself were visited before
it was recognized that h
is a leaf (finishing time
= starting time + 1). The right table shows the predecessor of every
vertex. The backtacking from a single vertex is therefore really simple. a
as
the first vertex discovered in its component can not be backtracked
any further.
What happens now, if there exist a vertex that has no connection to any other vertex. The upper example is taken and a single vertex is added without changing anything else. Then a depth first search is invoked on the graph:
G := Graph([a, b, c, d, e, f, g, h, i, j, k, l], [[a, b], [a, c], [b, d], [b, e], [c, f], [c, g], [d, h], [e, i], [e, j], [f, k], [g, l]], Directed):
G2 := Graph::addVertices(G, [m]): Graph::depthFirstSearch(G2, StartVertex = [a])
The newly inserted vertex m
has no predecessor.
The predecessor holds therefore the value infinity.
If we start somewhere in the graph without knowing the root of the DAG, the results are of course different:
G := Graph([a, b, c, d, e, f, g, h, i, j, k, l], [[a, b], [a, c], [b, d], [b, e], [c, f], [c, g], [d, h], [e, i], [e, j], [f, k], [g, l]], Directed):
Graph::depthFirstSearch(G, StartVertex = [c])
The predecessor of c
is c
,
but if we look at the graph it should be a
. This
is nevertheless not quite correct. Breadth first search takes the
given vertex and uses this as the root of the graph (no invertices!).
This explains also why the next call shows a infinity as
predecessor to l:
Graph::depthFirstSearch(G, StartVertex = [l])
 

List containing one vertex. 

Defines a vertex from which to start the depth first traversal. 
List containing three tables. The first table holds the first identification timestamp of each vertex, the second the finishing timestamp and the third the predecessor vertex.