Question

Let G be a graph or order n with independence number α(G) = 2. (a) Prove...

Let G be a graph or order n with independence number α(G) = 2.

(a) Prove that if G is disconnected, then G contains K⌈ n/2 ⌉ as a subgraph.

(b) Prove that if G is connected, then G contains a path (u, v, w) such that uw /∈ E(G) and every vertex in G − {u, v, w} is adjacent to either u or w (or both).

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Prove the following bound for the independence number. If G is a n-vertex graph with e...
Prove the following bound for the independence number. If G is a n-vertex graph with e edges and maximum degree ∆ > 0, then α(G) ≤ n − e/∆.
Let G be a simple graph with n(G) > 2. Prove that G is 2-connected iff...
Let G be a simple graph with n(G) > 2. Prove that G is 2-connected iff for every set of 3 distinct vertices, a, b and c, there is an a,c-path that contains b.
let G be a connected graph such that the graph formed by removing vertex x from...
let G be a connected graph such that the graph formed by removing vertex x from G is disconnected for all but exactly 2 vertices of G. Prove that G must be a path.
Let G be an n-vertex graph with n ≥ 2 and δ(G) ≥ (n-1)/2. Prove that...
Let G be an n-vertex graph with n ≥ 2 and δ(G) ≥ (n-1)/2. Prove that G is connected and that the diameter of G is at most two.
Let G=(V,E) be a connected graph with |V|≥2 Prove that ∀e∈E the graph G∖e=(V,E∖{e}) is disconnected,...
Let G=(V,E) be a connected graph with |V|≥2 Prove that ∀e∈E the graph G∖e=(V,E∖{e}) is disconnected, then G is a tree.
Graph Theory Let v be a vertex of a non trivial graph G. prove that if...
Graph Theory Let v be a vertex of a non trivial graph G. prove that if G is connected, then v has a neighbor in every component of G-v.
Let G = (V,E) be a graph with n vertices and e edges. Show that the...
Let G = (V,E) be a graph with n vertices and e edges. Show that the following statements are equivalent: 1. G is a tree 2. G is connected and n = e + 1 3. G has no cycles and n = e + 1 4. If u and v are vertices in G, then there exists a unique path connecting u and v.
A spanning tree of connected graph G = (V, E) is an acyclic connected subgraph (V,...
A spanning tree of connected graph G = (V, E) is an acyclic connected subgraph (V, E0 ) with the same vertices as G. Show that every connected graph G = (V, E) contains a spanning tree. (It is the connected subgraph (V, E0 ) with the smallest number of edges.)
Let G be a graph with vertex set V. Define a relation R from V to...
Let G be a graph with vertex set V. Define a relation R from V to itself as follows: vertex u has this relation R with vertex v, u R v, if there is a path in G from u to v. Prove that this relation is an equivalence relation. Write your proof with complete sentences line by line in a logical order.  If you can, you may write your answer to this question directly in the space provided.Your presentation counts.
Prove or disapprove each of the following: (a) Every disconnected graph has an isolated vertex. (b)...
Prove or disapprove each of the following: (a) Every disconnected graph has an isolated vertex. (b) A graph is connected if and only if some vertex is connected to all other vertices. (c) If G is a simple, connected, Eulerian graph, with edges e, f that are incident to a common vertex, then G has an Eulerian circuit in which e and f appear consequently.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT