Question

We define a forest to be a graph with no cycles. It can be though of as a collection (or union) of individual trees. (A connected forest is also itself a tree.) a. Suppose F is a forest consisting of m trees and v vertices. How many edges does F have? (Hint: Drawing some sample forests (i.e., groups of trees which are disconnected from each other) can help you derive this formula.) e = v - m Box 1: Enter your answer as an expression. Example: 3x^2+1, x/5, (a+b)/c Be sure your variables match those in the question b. Prove, by contrapositive, that any graph G with v vertices and e edges that satisfies v < e + 1 must contain a cycle (i.e., not be a forest) by matching the correct steps with their order in the desired proof. - Let G be an acyclic graph (i.e., a forest). - Therefore, a graph that satisfies v < e + 1 must contain a cycle. - ? v ? e + 1 for the acyclic graph G. - e = (vT1 + vT2 + vT3 + ... + vTm) - (1 ? m) - e = (vT1 - 1) + (vT2 - 1) + (vT3 - 1) + ... + (vTm - 1), where m is the number of trees contained within the forest G. - e = v - m ? v = e + m - m ? 1, since there is at least 1 tree in a forest G. - The sum of all eT in the forest G produces e, while the sum of all vT in the forest G produces v. - Let T be a tree which exists as a connected component of the forest G, and let eT be the number of edges of T and vT be the number of vertices of T. - ?T, eT = vT - 1 Box 2: In each pull-down, select the item that matches with the displayed item

          We define a forest to be a graph with no cycles. It can be though of as a collection (or union) of individual trees. (A connected forest is also itself a tree.)

a. Suppose F is a forest consisting of m trees and v vertices. How many edges does F have?
(Hint: Drawing some sample forests (i.e., groups of trees which are disconnected from each other) can help you derive this formula.)
e = v - m
Box 1: Enter your answer as an expression. Example: 3x^2+1, x/5, (a+b)/c
Be sure your variables match those in the question

b. Prove, by contrapositive, that any graph G with v vertices and e edges that satisfies v < e + 1 must contain a cycle (i.e., not be a forest) by matching the correct steps with their order in the desired proof.
- Let G be an acyclic graph (i.e., a forest).
- Therefore, a graph that satisfies v < e + 1 must contain a cycle.
- ? v ? e + 1 for the acyclic graph G.
- e = (vT1 + vT2 + vT3 + ... + vTm) - (1 ? m)
- e = (vT1 - 1) + (vT2 - 1) + (vT3 - 1) + ... + (vTm - 1), where m is the number of trees contained within the forest G.
- e = v - m ? v = e + m
- m ? 1, since there is at least 1 tree in a forest G.
- The sum of all eT in the forest G produces e, while the sum of all vT in the forest G produces v.
- Let T be a tree which exists as a connected component of the forest G, and let eT be the number of edges of T and vT be the number of vertices of T.
- ?T, eT = vT - 1
Box 2: In each pull-down, select the item that matches with the displayed item

We define a forest to be a graph with no cycles. It can be though of as a collection (or union) of individual trees. (A connected forest is also itself a tree.)

a. Suppose F is a forest consisting of m trees and v vertices. How many edges does F have?
(Hint: Drawing some sample forests (i.e., groups of trees which are disconnected from each other) can help you derive this formula.)
e = v - m
Box 1: Enter your answer as an expression. Example: 3x^2+1, x/5, (a+b)/c
Be sure your variables match those in the question

b. Prove, by contrapositive, that any graph G with v vertices and e edges that satisfies v < e + 1 must contain a cycle (i.e., not be a forest) by matching the correct steps with their order in the desired proof.
- Let G be an acyclic graph (i.e., a forest).
- Therefore, a graph that satisfies v < e + 1 must contain a cycle.
- ? v ? e + 1 for the acyclic graph G.
- e = (vT1 + vT2 + vT3 + ... + vTm) - (1 ? m)
- e = (vT1 - 1) + (vT2 - 1) + (vT3 - 1) + ... + (vTm - 1), where m is the number of trees contained within the forest G.
- e = v - m ? v = e + m
- m ? 1, since there is at least 1 tree in a forest G.
- The sum of all eT in the forest G produces e, while the sum of all vT in the forest G produces v.
- Let T be a tree which exists as a connected component of the forest G, and let eT be the number of edges of T and vT be the number of vertices of T.
- ?T, eT = vT - 1
Box 2: In each pull-down, select the item that matches with the displayed item

Added by Cheryl V.

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