Question: If G = (V, E) is an undirected graph, a subset K of V is called a covering of G if for every edge {a,

If G = (V, E) is an undirected graph, a subset K of V is called a covering of G if for every edge {a, b} of G either a or b is in K. The set K is a minimal covering if K - {x} fails to cover G for each x e K. The number of vertices in a smallest covering is called the covering number of G.
(a) Prove that if I c V, then / is an independent set in G if and only if V - I is a covering of G.
(b) Verify that | V | is the sum of the independence number of G (as defined in Exercise 25 for Section 11.5) and its covering number.

Step by Step Solution

★★★★★

3.61 Rating (155 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a Let I be independent and a b E If neither a nor b is in V 1 then a b I and sinc... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

954-M-L-A-L-S (8244).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

(a) If G = (V, E) is an undirected graph with |V| = v, |E| = e, and no loops, prove that 2e v2 - v. (b) State the corresponding inequality for the case when G is directed.

If G is an undirected graph with n vertices and e edges, let = minvV{deg(v)} and let = maxvV{deg(v)}. Prove that 2(e/n)

If G = (V, E) is an undirected graph with |V|-n and |E| = k, the following matrices are used to represent G. Let V = {v1, v2, . . . , Define the adjacency matrix A = (aI j)nÃn where aIJ = 1 if...

If G = (V, E) is an undirected graph, a subset D of V is called a dominating set if for all v V, either v D or v is adjacent to a vertex in D. If D is a dominating set and no proper subset of D has...

If G = (V, E) is an undirected graph, a subset I of V is called independent if no two vertices in I are adjacent. An independent set I is called maximal if no vertex v can be added to I with I ª...

ANSI-SPARC6 Programming Language Compilation Write notes on each of the following topics: (a) the implementation of labels and jumps in a recursive, block structured programming language [7 marks]...

2. Let G (V, E) be an undirected graph with V-[1,2, .. ,n]. Given a vertex r, let A subset S S V is called independent if for every pair of distinct vertices x,y from S, the following holds: A set...

Let G (V, E) be an undirected graph with n nodes. Recall that a subset of the nodes is called an independent set if no two of them are joined by an edge. Finding large independent sets is difficult...

Let G = (V, E) be an undirected graph with no parallel edges. A 2-edge-colouring of G is an assignment of a colour 'red' or 'blue' (for short r and b) to each edge of G. Let c : E {r, E be a...

Let G = (V. E) be an undirected graph with all edge weights equal to I. Let d(i, j) denote the length of the shortest path between i and j. Now, suppose we wish to answer queries from the user of the...

When MCI Communications Corporation (now part of Verizon Communications) purchased Satellite Business Systems (SBS) from International Business Machines Corporation (IBM), it issued common stock to...

tungsten mass: 1.48 kg (density of tungsten = 19.25 g/cm 3 ) convert to cm^3

Tasha is 58 years old with an adjusted gross income (AGI) of $70,000. Tasha had medical expenses of $6,000 in the current year and withdrew $6,000 from her traditional IRA to cover the expenses. What...

What factors would a potential sponsor look for when considering whether to sponsor the festival identified in question 1?

With a specific line of fit, the data point (6, 47) has a residual of 2.8. The slope of the line of fit is 2.4. What is the equation of the line of fit?

a. The following readings were taken from a display outside the First River Bank. The display alternated between °F and °C. However, there was an error within the system that calculated the...

Alex says, &8220; The formula for the root mean square error is long. Why do you have to square and then take the square root? Isn't that just doing a lot of work for nothing? Can't you just make...

Prepare the journal entries for each of company A and Company B, assuming that the trade is one without commercial substance Company A Cost $1,100 Accumulated amortization $350 Book value $750...

Assuming the market is arbitrage-free, if a six-month pure discount bond yields 1.9%, a one year-pure discount bond yields 2.3%, an eighteen-month bond yields 3.05%, what should be the price of a...

Type your question here Suppose the real risk-free rate is 4.20%, the average expected future inflation rate is 6.50%, and a maturity risk premium of 0.10% per year to maturity applies, i.e., MRP =...