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isConnectedGraph -- determines if a graph is connected

Synopsis

Description

This function checks if the given graph G is connected. A graph is said to be connected if it has exactly one connected component.

Isolated vertices form their own connected components and will cause this method return false. This is in contrast to isConnected in which isolated vertices are not in any connected components. See the Connected Components Tutorial for more information.

i1 : S = QQ[a..e];
 
i2 : G = graph {a*b,b*c,c*d,d*e,a*e} -- the 5-cycle (connected)

o2 = Graph{"edges" => {{a, b}, {b, c}, {c, d}, {a, e}, {d, e}}}
           "ring" => S
           "vertices" => {a, b, c, d, e}

o2 : Graph
 
i3 : H = graph {a*b,b*c,c*a,d*e} -- a 3-cycle and a disjoint edge (not connected)

o3 = Graph{"edges" => {{a, b}, {a, c}, {b, c}, {d, e}}}
           "ring" => S
           "vertices" => {a, b, c, d, e}

o3 : Graph
 
i4 : isConnectedGraph G

o4 = true
 
i5 : isConnectedGraph H

o5 = false

In the following example, the graph G has the isolated vertex e. As d forms its own connected component, this graph is not connected.

i6 : S = QQ[a..e];
 
i7 : G = graph {a*b,b*c,c*d,a*d} -- 4-cycle with isolated vertex (not connected)

o7 = Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, d}}}
           "ring" => S
           "vertices" => {a, b, c, d, e}

o7 : Graph
 
i8 : isolatedVertices G

o8 = {e}

o8 : List
 
i9 : isConnectedGraph G

o9 = false

See also

Ways to use isConnectedGraph :

For the programmer

The object isConnectedGraph is a method function.