Warm Up – 4.2 - Wednesday

Trees A tree is a network without any circuits. Note: Our definition of circuit here is different than an Euler Circuit or a Hamilton Circuit. The graph to the left has a general circuit starting at C and following C, B, E, D, C. A circuit simply starts and ends at the same vertex.

Not a tree! Because this graph has a circuit, this network is not a tree!

The following are trees! They have no circuits within the graph.

Key Properties of Trees The single path property: In a tree, there is only one path connecting two vertices. Notice if we have two paths then we also have a circuit and thus a non-tree.

Key Properties of Trees All Bridges Property: In a tree, every edge is a bridge (Remember that a bridge is an edge that if removed, makes the graph disconnected). The N-1 edges property: A tree with N vertices has N-1 Edges.

Key Properties of Trees If a network satisfies one of those three properties then it satisfies all three, and is a tree! WOOT! The single path property: In a tree, there is only one path connecting two vertices. All Bridges Property: In a tree, every edge is a bridge (Remember that a bridge is an edge that if removed, makes the graph disconnected). The N-1 edges property: A tree with N vertices has N-1 Edges.

Trees #1

Subtrees A subtree is a tree within a network. Consider the network above. Do you see any trees within the graph (meaning if I ignore certain edges)?

Spanning Trees A spanning tree is a subtree that uses all the vertices.

Spanning Trees Thus we have three different spanning trees! If I remove one of the three edges in the triangular path, I get a tree because I have removed the circuit. Thus we have three different spanning trees!

Redundancy The redundancy is a number that helps us when counting spanning trees in a graph. The redundancy is the number of Edges (M) minus one less than the number of vertices (N – 1). 𝑹=𝑴− 𝑵−𝟏 We define redundancy this way so a tree has a redundancy of 0!

Redundancy The redundancy tells you the number of edges I need to remove in order to find a spanning tree. I cannot remove Bridges! (Edges that would disconnect the graph)

Example #1 Calculate the Redundancy and find a spanning tree in the graph below.

Example #1 There are 12 vertices and 14 edges. Thus my redundancy is: 14 – 12 – 1 =3 This means I need to remove 3 edges to get a spanning tree.

Example #1 If I remove edges BE, HI, and IJ, a spanning tree is created. Notice we still have a network and there are no circuits.