a) Suppose is maximally acyclic. Let ; we will show that there is a path in connecting .
If then of course this itself is a path in . Suppose that . Then contains a cycle, and this cycle must contain the edge because otherwise it will be a cycle in which is acyclic. But then, this cycle must be of the form
,
showing that contains the path connecting .
This proves that is connected. To show that it is minimally connected, consider any edge , say . If is connected, then there is a path
in , showing that
is a cycle in . Again, this is impossible since is acyclic. Thus, is not connected. This shows that is minimally connected.
Conversely, suppose is minimally connected. We will show that it is acyclic.
Assume, if possible, that there is a cycle
in . Then is connected because every vertex in is connected to via a path in and if this path contains we can replace it with
This contradicts that is minimally connected. Thus, must be acyclic.
Now consider vertices such that . Since connected, there is a path
in , showing that contains the cycle
This proves that is maximally acyclic.
This proves the equivalence.
b) Suppose that is connected acyclic graph. We will use induction on to prove that .
If then , and , showing if .
Suppose that is connected acyclic, such that for some . Suppose that holds for all such that . Let . Since is connected acyclic, can not have any path between . Thus, is not connected, and since this is obtained by removing just one vertex from a connected graph, it has two connected components, say
Being connected components, these are connected; being subgraph of acyclic graph, these are acyclic. Thus, by induction hypothesis, we know
This shows
By induction, we have for all connected acyclic.
By Euler's formula, we have . If is connected acyclic, then
so that
e:= 3. (i) Let T = (V, E) be a graph. Prove that the following are...
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Let G= (V, E) be a connected undirected graph and let v be a vertex in G. Let T be the depth-first search tree of G starting from v, and let U be the breadth-first search tree of G starting from v. Prove that the height of T is at least as great as the height of U
1) Professor Sabatier conjectures the following converse of Theorem 23.1. Let G=(V,E) be a connected, undirected graph with a real-valued weight function w defined on E. Let A be a subset of E that is included in some minimum spanning tree for G, let (S,V−S) be any cut of G that respects A, and let (u,v) be a safe edge for A crossing (S,V−S). Then, (u,v) is a light edge for the cut. Show that the professor's conjecture is incorrect...
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Let G = (V, E) be a finite graph. We will use a few definitions for the statement of this problem. The Tutte polynomial is defined as the polynomial in 2 variables, 2 and y, given by: Definition 1 Tg(x,y) = (x - 1)*(A)-k(E)(y - 1)*(A)+|A1-1V1 ACE where for A CE, k(A) is the number of connected components of the graph (V, A). For this problem we will need the following definition: Definition 2 (Acyclic Graph) A graph is called...
3. Given graph G = (V,E), prove that the following statements are equivalent. [Note: the following statements are equivalent definitions of a tree graph" 1) There exist exactly one path between any of two vertices u,vEV in the graph G 3. Given graph G = (V,E), prove that the following statements are equivalent. [Note: the following statements are equivalent definitions of a tree graph" 1) There exist exactly one path between any of two vertices u,vEV in the graph G
1. Prove the following properties of scalar multiplication and addition for vectors. Let s,t e R and v,w E Rn (a) (st)v s(tv) (b) (st)(vw) = sv + tv + sw + tw) (c) Use a special form of w and part (b) to instantly prove (s + t)v = sv + tv. 1. Prove the following properties of scalar multiplication and addition for vectors. Let s,t e R and v,w E Rn (a) (st)v s(tv) (b) (st)(vw) = sv...
For any n ≥ 1 let Kn,n be the complete bipartite graph (V, E) where V = {xi : 1 ≤ i ≤ n} ∪ {yi : 1 ≤ i ≤ n} E = {{xi , yj} : 1 ≤ i ≤ n, 1 ≤ j ≤ n} (a) Prove that Kn,n is connected for all n ≤ 1. (b) For any n ≥ 3 find two subsets of edges E 0 ⊆ E and E 00 ⊆ E such...