Prove: An edge e of a graph G is a cut-edge if and only if e...
Prove:
An edge e of a graph G is a cut-edge if and only if e is not part of any circuit in G.
Homework Answers
Proof: Consider that e=(x,y) is a cut edge , Suppose that there is a circuit of cycle (x, P, y, x) containing e. Then if Z = u, Z1, x, y, Z2, v is a walk from u to v using e, u, Z1, P, Z2, v is a walk from u to v that doesn’t use e. Thus e is not a cut edge .
If e is not a cut edge then G−e contains a path P from x to y (x ∼ y in G and relations are maintained after deletion of e). So (y, x, P, y) is a cycle containing e.
6. Prove that the following graphs are connected: (a) The 3 vertex cycle: (b) The following 4 vertex graph: (c) K 7. An edge e of a connected graph G is called a cut edge if the graph G obtained by d...
6. Prove that the following graphs are connected: (a) The 3 vertex cycle: (b) The following 4 vertex graph: (c) K 7. An edge e of a connected graph G is called a cut edge if the graph G obtained by deleting that edge (V(G) V(G) and E(G) E(G) \<ej) is not connected. Prove that if G1 and G2 are connected simple graphs which are isomorphic and if G1 has a cut edge, then G2 also has a cut edge....
Question 1# (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge e fa, b is...
Question 1# (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge e fa, b is removed from C then the subgraph S C G that remains is still connected. "Directly' means using only the definitions of the concepts involved, in this case connected' and 'circuit'. Hint: If z and y are vertices of G connected by path that includes e, is there an alternative path connecting x to y...
(a) Let L be a minimum edge-cut in a connected graph G with at least two vertices. Prove that G −...
(a) Let L be a minimum edge-cut in a connected graph G with at least two vertices. Prove that G − L has exactly two components. (b) Let G an eulerian graph. Prove that λ(G) is even.
Let e be the unique lightest edge in a graph G. Let T be a spanning...
Let e be the unique lightest edge in a graph G. Let T be a spanning tree of G such that e /∈ T . Prove using elementary properties of spanning trees (i.e. not the cut property) that T is not a minimum spanning tree of G.
er (a) Let G be a connected graph and C a non-trivial circuit in G. Prove...
er (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge ={a, b} is removed from then the subgraph S CG that remains is still connected. Directly' means using only the definitions of the concepts involved, in this case 'connected' and 'circuit'. Hint: If r and y are vertices of G connected by path that includes e, is there an alternative path connecting x to y that avoids e? (b)...
3. Given graph G = (V,E), prove that the following statements are equivalent. [Note: the following statements are e...
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, v EV in the graph G 2) Graph G is connected and does not contain any cycles. 3) Graph G does not contain any cycles, and a cycle is formed if any edge (u, v) E E is added to G
3. Given graph...
A bridge is an edge whose removal disconnects the graph. Prove that any bridge must be...
A bridge is an edge whose removal disconnects the graph. Prove that any bridge must be in some minimum spanning tree. You may use the cut property in the proof, if you want.
Let G be a graph, and let T, T' be spanning trees in G. Show that if e is an edge in T, then there is an edge e in T' such that the graph obtained by adding the edge e, to T-e is again a span...
Let G be a graph, and let T, T' be spanning trees in G. Show that if e is an edge in T, then there is an edge e in T' such that the graph obtained by adding the edge e, to T-e is again a spanning tree in G.
Let G be a graph, and let T, T' be spanning trees in G. Show that if e is an edge in T, then there is an edge e in...
Prove that, if even degree, then the edge conn G is a nontrivial connected graph in which every v...
Prove that, if even degree, then the edge conn G is a nontrivial connected graph in which every vertex has
Prove that, if even degree, then the edge conn G is a nontrivial connected graph in which every vertex has
3. Given graph G-(V, E), prove that the following statements are equivalent. [Note: the following statements...
3. Given graph G-(V, E), prove that the following statements are equivalent. [Note: the following statements are equivalent definitions of a "tree graph".] 4) Graph G is connected, but would become disconnected if any edge (u,v) E E is removed from G 5) Graph G is connected and has IV 1 edges 6) Graph G has no cycles and has |V| -1 edges.
6. Prove that the following graphs are connected: (a) The 3 vertex cycle: (b) The following 4 vertex graph: (c) K 7. An edge e of a connected graph G is called a cut edge if the graph G obtained by deleting that edge (V(G) V(G) and E(G) E(G) \<ej) is not connected. Prove that if G1 and G2 are connected simple graphs which are isomorphic and if G1 has a cut edge, then G2 also has a cut edge....
Question 1# (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge e fa, b is removed from C then the subgraph S C G that remains is still connected. "Directly' means using only the definitions of the concepts involved, in this case connected' and 'circuit'. Hint: If z and y are vertices of G connected by path that includes e, is there an alternative path connecting x to y...
er (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge ={a, b} is removed from then the subgraph S CG that remains is still connected. Directly' means using only the definitions of the concepts involved, in this case 'connected' and 'circuit'. Hint: If r and y are vertices of G connected by path that includes e, is there an alternative path connecting x to y that avoids e? (b)...
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, v EV in the graph G 2) Graph G is connected and does not contain any cycles. 3) Graph G does not contain any cycles, and a cycle is formed if any edge (u, v) E E is added to G
3. Given graph...
Let G be a graph, and let T, T' be spanning trees in G. Show that if e is an edge in T, then there is an edge e in T' such that the graph obtained by adding the edge e, to T-e is again a spanning tree in G.
Let G be a graph, and let T, T' be spanning trees in G. Show that if e is an edge in T, then there is an edge e in...
Prove that, if even degree, then the edge conn G is a nontrivial connected graph in which every vertex has
Prove that, if even degree, then the edge conn G is a nontrivial connected graph in which every vertex has
3. Given graph G-(V, E), prove that the following statements are equivalent. [Note: the following statements are equivalent definitions of a "tree graph".] 4) Graph G is connected, but would become disconnected if any edge (u,v) E E is removed from G 5) Graph G is connected and has IV 1 edges 6) Graph G has no cycles and has |V| -1 edges.
Most questions answered within 3 hours.
-
The contribution format income statement for Huerra Company for
last year is given below:
Total
Unit...
asked 48 seconds from now -
two identical cars approach an intersection, one is traveling east
at 18n/s. the second is traveling...
asked 5 minutes ago -
When an X-ray beam with a wavelength of 133 pm is incident on
the surface to...
asked 14 minutes ago -
Compare and contrast the internal and external organization of
Congress and the President.
"2 paragraph"
asked 22 minutes ago -
Suppose that the average waiting time for a patient at a
physician's office is just over...
asked 41 minutes ago -
C++
Can somebody help me with this
2 integer right triangles
There are right-angled triangles (i.e.,...
asked 37 minutes ago -
A researcher at Lund University is interested in studying the
biological role of a so-called type...
asked 42 minutes ago -
Consider where you currently work, where you have previously
worked, or a well-known company where you...
asked 40 minutes ago -
For college physics discussion post. Be sure to have
examples.
Respond to the following:
Explain conservation...
asked 46 minutes ago -
Develop a short argument for or against having monetary policy
conducted according to set rules instead...
asked 42 minutes ago -
Match the intermediates of the kreb cycle with their biosynthetic
products in mammals:
1) Cis-aconitate
2)...
asked 44 minutes ago -
The following info is for Johnson Company. Assume that all
balance sheet amounts represent both average...
asked 1 hour ago