Let G be connected, and let e be an edge of G. Prove that e is...

Let G be connected, and let e be an edge of G. Prove that e is a bridge if and only if it is in every spanning tree of G.

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Colby Messinger answered 1 year ago

Suppose G is a connected cubic graph (regular of degree 3) and e is an edge...

Suppose G is a connected cubic graph (regular of degree 3) and e is an edge such that G − e has two connected components G1 and G2 (a) Explain what connected means. (b) We say that e is a____________ of G (c) show that G1 has an odd number of vertices. (d) draw a connected cubic graph G with an edge e as above.

Given a connected graph G where edge costs are pair-wise distinct, prove or disprove that the...

Given a connected graph G where edge costs are pair-wise distinct, prove or disprove that the G has a unique MST. Please write Pseudo-code for the algorithms.

ii. Let G = (V, E) be a tree. Prove G has |V | − 1...

ii. Let G = (V, E) be a tree. Prove G has |V | − 1 edges using strong induction. Hint: In the inductive step, choose an edge (u, v) and partition the set vertices into two subtrees, those that are reachable from u without traversing (u, v) and those that are reachable from v without traversing (u, v). You will have to reason why these subtrees are distinct subgraphs of G. iii. What is the total degree of a...

Given a connected graph G with n vertices. We say an edge of G is a...

Given a connected graph G with n vertices. We say an edge of G is a bridge if the graph becomes a disconnected graph after removing the edge. Give an O(m + n) time algorithm that finds all the bridges. (Partial credits will be given for a polynomial time algorithm.) (Hint: Use DFS)

Let x = inf E. Prove that x is an element of the boundary of E....

Let x = inf E. Prove that x is an element of the boundary of E. Here R is regarded as a metric space with d(p,q) = |q−p| for p,q ∈R

Let (G,+) be an abelian group and U a subgroup of G. Prove that G is...

Let (G,+) be an abelian group and U a subgroup of G. Prove that G is the direct product of U and V (where V a subgroup of G) if only if there is a homomorphism f : G → U with f|U = IdU

Let G be a graph. prove G has a Eulerian trail if and only if G...

Let G be a graph. prove G has a Eulerian trail if and only if G has at most one non-trivial component and at most 2 vertices have odd degree

You are given an undirected graph G = ( V, E ) in which the edge...

You are given an undirected graph G = ( V, E ) in which the edge weights are highly restricted. In particular, each edge has a positive integer weight from 1 to W, where W is a constant (independent of the number of edges or vertices). Show that it is possible to compute the single-source shortest paths in such a graph in O(E+V) time.

Let S ⊆ R and let G be an arbitrary isometry of R . Prove that...

Let S ⊆ R and let G be an arbitrary isometry of R . Prove that the symmetry group of G(S) is isomorphic to the symmetry group of S. Hint: If F is a symmetry of S, what is the corresponding symmetry of G(S)?

1) a) Let k ≥ 2 and let G be a k-regular bipartite graph. Prove that G...

1) a) Let k ≥ 2 and let G be a k-regular bipartite graph. Prove that G has no cut-edge. (Hint: Use the bipartite version of handshaking.) b) Construct a simple, connected, nonbipartite 3-regular graph with a cut-edge. (This shows that the condition “bipartite” really is necessary in (a).) 2) Let F_n be a fan graph and Let a_n = τ(F_n) where τ(F_n) is the number of spanning trees in F_n. Use deletion/contraction to prove that a_n = 3a_n-1 - a_n-2...

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