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Out of all the 2 digit integers between 1 and 100 a 2 digit number has to be selected at random What is the probability that the selected number is not divisible by 7 left dfrac 13 90 right left dfrac 12 90 right left dfrac 78 90 right left dfrac 77 90 right | Factors | 55 |
Consider the field C of complex numbers with addition and multiplication Which of the following form s a subfield of C with addition and multiplication S1 the set of real numbers S2 a ib a and b are rational numbers S3 a ib a2 b2 u2264 1 S4 ia a is real only S1 S1 and S3 S2 and S3 S1 and S2 | Fields | 56 |
A FAT file allocation table based file system is being used and the total overhead of each entry in the FAT is 4 bytes in size Given a 100 imes 10 6 bytes disk on which the file system is stored and data block size is 10 3 bytes the maximum size of a file that can be stored on this disk in units of 10 6 bytes is _________ | File | 57 |
The data blocks of a very large file in the Unix file system are allocated using continuous allocation linked allocation indexed allocation an extension of indexed allocation | File System | 58 |
In the index allocation scheme of blocks to a file the maximum possible size of the file depends on the size of the blocks and the size of the address of the blocks the number of blocks used for the index and the size of the blocks the size of the blocks the number of blocks used for the index and the size of the address of the blocks None of the above | File System | 58 |
In a particular Unix OS each data block is of size 1024 bytes each node has 10 direct data block addresses and three additional addresses one for single indirect block one for double indirect block and one for triple indirect block Also each block can contain addresses for 128 blocks Which one of the following is approximately the maximum size of a file in the file system 512 MB 2 GB 8 GB 16 GB | File System | 58 |
In a file allocation system which of the following allocation scheme s can be used if no external fragmentation is allowed Contiguous Linked Indexed 1 and 3 only 2 only 3 only 2 and 3 only | File System | 58 |
A file system with a one level directory structure is implemented on a disk with disk block size of 4K bytes The disk is used as follows Disk block 0 File Allocation Table consisting of one 8 bit entry per data block representing the data block address of the next data block in the file Disk block 1 Directory with one 32 bit entry per file Disk block 2 Data block 1 Disk block 3 Data block 2 etc What is the maximum possible number of files What is the maximum possible file size in blocks | File System | 58 |
State whether the following statements are TRUE or FALSE A minimal DFA that is equiavlent to an NDFA with n nodes has always 2 n states | Finite Automata | 59 |
State True or False with one line explanation A FSM Finite State Machine can be designed to add two integers of any arbitrary length arbitrary number of digits | Finite Automata | 59 |
Let L be the set of all binary strings whose last two symbols are the same The number of states in the minimal state deterministic finite state automaton accepting L is 2 5 8 3 | Finite Automata | 59 |
Let L be the language of all binary strings in which the third symbol from the right is a 1 Give a non deterministic finite automaton that recognizes L How many states does the minimized equivalent deterministic finite automaton have Justify your answer briefly | Finite Automata | 59 |
Indicate whether the following statement is true or false providing a short explanation to substantiate your answers If a language L is accepted by an NFA with n states then there is a DFA with no more than 2 n states accepting L | Finite Automata | 59 |
Indicate whether the following statement is true or false providing a short explanation to substantiate your answers A DFA that has n states and accepts an infinite language must accept at least one string x such that 2n lt x lt 3n where x denotes the length of x | Finite Automata | 59 |
Given that L is a language accepted by a finite state machine show that L P and L R are also accepted by some finite state machines where L P left s mid ss in L ext some string s right L R left s mid s ext obtained by reversing some string in L right | Finite Automata | 59 |
Given below are the transition diagrams for two finite state machines M_1 and M_2 recognizing languages L_1 and L_2 respectively Display the transition diagram for a machine that recognizes L_1 L_2 obtained from transition diagrams for M_1 and M_2 by adding only varepsilon transitions and no new states Modify the transition diagram obtained in part a obtain a transition diagram for a machine that recognizes L_1 L_2 by adding only varepsilon transitions and no new states Final states are enclosed in double circles | Finite Automata | 59 |
Draw the state transition of a deterministic finite state automaton which accepts all strings from the alphabet a b such that no string has 3 consecutive occurrences of the letter b | Finite Automata | 59 |
Design a deterministic finite state automaton using minimum number of states that recognizes the following language L w in 0 1 mid w interpreted as binary number ignoring the leading zeros is divisible by five | Finite Automata | 59 |
Construct a finite state machine with minimum number of states accepting all strings over a b such that the number of a s is divisible by two and the number of b s is divisible by three | Finite Automata | 59 |
Consider the regular expression 0 1 0 1 dots N times The minimum state finite automaton that recognizes the language represented by this regular expression contains n states n 1 states n 2 states None of the above | Finite Automata | 59 |
Consider the following state table for a sequential machine The number of states in the minimized machine will be Input 0 1 Present State A D 0 B 1 B A 0 C 1 C A 0 B 1 D A 1 C 1 Next state Output 4 3 2 1 | Finite Automata | 59 |
A finite state machine with the following state table has a single input x and a single out z present state next state z x 1 x 0 A D 0 B 0 B B 1 C 1 C B 0 D 1 D B 1 C 0 If the initial state is unknown then the shortest input sequence to reach the final state C is 01 10 101 110 | Finite Automata | 59 |
What is the first order predicate calculus statement equivalent to the following Every teacher is liked by some student u2200 x left ext teacher left x right u2192 u2203 y left ext student left y right u2192 ext likes left y x right right right u2200 x left ext teacher left x right u2192 u2203 y left ext student left y right u2227 ext likes left y x right right right u2203 y u2200 x left ext teacher left x right u2192 left ext student left y right u2227 ext likes left y x right right right u2200 x left ext teacher left x right u2227 u2203 y left ext student left y right u2192 ext likes left y x right right right | First Order Logic | 60 |
Let P x and Q x be arbitrary predicates Which of the following statements is always TRUE left left forall x left P left x right vee Q left x right right right right implies left left forall x P left x right right vee left forall xQ left x right right right left forall x left P left x right implies Q left x right right right implies left left forall x P left x right right implies left forall xQ left x right right right left forall x left P left x right right implies forall x left Q left x right right right implies left forall x left P left x right implies Q left x right right right left forall x left P left x right right Leftrightarrow left forall x left Q left x right right right right implies left forall x left P left x right Leftrightarrow Q left x right right right | First Order Logic | 60 |
Let ext fsa and ext pda be two predicates such that ext fsa x means x is a finite state automaton and ext pda y means that y is a pushdown automaton Let ext equivalent be another predicate such that ext equivalent a b means a and b are equivalent Which of the following first order logic statements represent the following Each finite state automaton has an equivalent pushdown automaton left forall x ext fsa left x right right implies left exists y ext pda left y right wedge ext equivalent left x y right right eg forall y left exists x ext fsa left x right implies ext pda left y right wedge ext equivalent left x y right right forall x exists y left ext fsa left x right wedge ext pda left y right wedge ext equivalent left x y right right forall x exists y left ext fsa left y right wedge ext pda left x right wedge ext equivalent left x y right right | First Order Logic | 60 |
Let a x y b x y and c x y be three statements with variables x and y chosen from some universe Consider the following statement qquad exists x forall y a x y wedge b x y wedge eg c x y Which one of the following is its equivalent forall x exists y a x y vee b x y o c x y exists x forall y a x y vee b x y wedge eg c x y eg forall x exists y a x y wedge b x y o c x y eg forall x exists y a x y vee b x y o c x y | First Order Logic | 60 |
Identify the correct translation into logical notation of the following assertion Some boys in the class are taller than all the girls Note ext taller x y is true if x is taller than y exists x ext boy x rightarrow forall y ext girl y land ext taller x y exists x ext boy x land forall y ext girl y land ext taller x y exists x ext boy x rightarrow forall y ext girl y rightarrow ext taller x y exists x ext boy x land forall y ext girl y rightarrow ext taller x y | First Order Logic | 60 |
Consider the following formula and its two interpretations I_1 and I_2 alpha forall x left P_x Leftrightarrow forall y left Q_ xy Leftrightarrow eg Q_ yy right right Rightarrow forall x left eg P_x right I_1 Domain the set of natural numbers P_x x is a prime number Q_ xy y divides x I_2 same as I_1 except that P_x x is a composite number Which of the following statements is true I_1 satisfies alpha I_2 does not I_2 satisfies alpha I_1 does not Neither I_1 nor I_2 satisfies alpha Both I_1 and I_2 satisfies alpha | First Order Logic | 60 |
Consider the following first order logic formula in which R is a binary relation symbol u2200x u2200y R x y implies R y x The formula is satisfiable and valid satisfiable and so is its negation unsatisfiable but its negation is valid satisfiable but its negation is unsatisfiable | First Order Logic | 60 |
def graph ext Graph def connected ext Connected Let graph x be a predicate which denotes that x is a graph Let connected x be a predicate which denotes that x is connected Which of the following first order logic sentences DOES NOT represent the statement qquad ext Not every graph is connected lnot forall x Bigl graph x implies connected x Bigr exists x Bigl graph x land lnot connected x Bigr lnot forall x Bigl lnot graph x lor connected x Bigr forall x Bigl graph x implies lnot connected x Bigr | First Order Logic | 60 |
b Consider the following first order formula left matrix forall x exists y R x y 1em Large land 1em forall x forall y Bigl R x y implies lnot R y x Bigr 1em Large land 1em forall x forall y forall z Bigl R x y land R y z implies R x z Bigr 1em Large land 1em forall x lnot R x x right Does it have finite models Is it satisfiable If so give a countable model for it | First Order Logic | 60 |
The n bit fixed point representation of an unsigned real number X uses f bits for the fraction part Let i n f The range of decimal values for X in this representation is 2 f to 2 i 2 f to left 2 i 2 f right 0 to 2 i 0 to left 2 i 2 f right | Fixed Point Representation | 61 |
In an SR latch made by cross coupling two NAND gates if both S and R inputs are set to 0 then it will result in Q 0 Q 1 Q 1 Q 0 Q 1 Q 1 Indeterminate states | Flip Flop | 62 |
Find the maximum clock frequency at which the counter in the figure below can be operated Assume that the propagation delay through each flip flop and each AND gate is 10 ext ns Also assume that the setup time for the JK inputs of the flip flops is negligible | Flip Flop | 62 |
Design a 3 bit counter using D flip flops such that not more than one flip flop changes state between any two consecutive states | Flip Flop | 62 |
Consider a combination of T and D flip flops connected as shown below The output of the D flip flop is connected to the input of the T flip flop and the output of the T flip flop is connected to the input of the D flip flop Initially both Q_ 0 and Q_ 1 are set to 1 before the 1st clock cycle The outputs Q_ 1 Q_ 0 after the 3rd cycle are 11 and after the 4th cycle are 00 respectively Q_ 1 Q_ 0 after the 3rd cycle are 11 and after the 4th cycle are 01 respectively Q_ 1 Q_ 0 after the 3rd cycle are 00 and after the 4th cycle are 11 respectively Q_ 1 Q_ 0 after the 3rd cycle are 01 and after the 4th cycle are 01 respectively | Flip Flop | 62 |
A sequential circuit takes an input stream of 0 s and 1 s and produces an output stream of 0 s and 1 s Initially it replicates the input on its output until two consecutive 0 s are encountered on the input From then onward it produces an output stream which is the bit wise complement of input stream until it encounters two consecutive 1 s whereupon the process repeats An example input and output stream is shown below The input stream 101100 01001011 0 11 The desired output 101100 10110100 0 11 J K master slave flip flops are to be used to design the circuit Give the state transition diagram | Flip Flop | 62 |
A positive edge triggered D flip flop is connected to a positive edge triggered JK flip flop as follows The Q output of the D flip flop is connected to both the J and K inputs of the JK flip flop while the Q output of the JK flip flop is connected to the input of the D flip flop Initially the output of the D flip flop is set to logic one and the output of the JK flip flop is cleared Which one of the following is the bit sequence including the initial state generated at the Q output of the JK flip flop when the flip flops are connected to a free running common clock Assume that J K 1 is the toggle mode and J K 0 is the state holding mode of the JK flip flops Both the flip flops have non zero propagation delays 0110110 0100100 011101110 011001100 | Flip Flop | 62 |
The following is a scheme for floating point number representation using 16 bits Bit Position 15 14 9 8 0 s e m Sign Exponent Mantissa Let s e and m be the numbers represented in binary in the sign exponent and mantissa fields respectively Then the floating point number represented is begin cases 1 s left 1 m imes 2 9 right 2 e 31 amp ext if the exponent eq 111111 0 amp ext otherwise end cases What is the maximum difference between two successive real numbers representable in this system 2 40 2 9 2 22 2 31 | Floating Point Representation | 63 |
The exponent of a floating point number is represented in excess N code so that The dynamic range is large The precision is high The smallest number is represented by all zeros Overflow is avoided | Floating Point Representation | 63 |
Given the following binary number in 32 bit single precision IEEE 754 format large 00111110011011010000000000000000 The decimal value closest to this floating point number is 1 45 10 1 1 45 10 1 2 27 10 1 2 27 10 1 | Floating Point Representation | 63 |
Consider the following floating point format Mantissa is a pure fraction in sign magnitude form The normalized representation for the above format is specified as follows The mantissa has an implicit 1 preceding the binary radix point Assume that only 0 u2019s are padded in while shifting a field The normalized representation of the above number 0 239 imes 2 13 is 0A 20 11 34 49 D0 4A E8 | Floating Point Representation | 63 |
Consider the following floating point format Mantissa is a pure fraction in sign magnitude form The decimal number 0 239 imes 2 13 has the following hexadecimal representation without normalization and rounding off 0D 24 0D 4D 4D 0D 4D 3D | Floating Point Representation | 63 |
A 32 bit floating point number is represented by a 7 bit signed exponent and a 24 bit fractional mantissa The base of the scale factor is 16 The range of the exponent is ___________ | Floating Point Representation | 63 |
A process executes the following segment of code for i 1 i lt n i fork The number of new processes created is n n n 1 2 2 n 1 3 n 1 | Fork | 64 |
A process executes the following code for i 0 i lt n i fork The total number of child processes created is n 2n 1 2n 2n 1 1 | Fork | 64 |
A process executes the code fork fork fork The total number of child processes created is 3 4 7 8 | Fork | 64 |
Consider the following code fragment if fork 0 a a 5 printf d p n a amp a else a a 5 printf d p n a amp a Let u v be the values printed by the parent process and x y be the values printed by the child process Which one of the following is TRUE u x 10 and v y u x 10 and v y u 10 x and v y u 10 x and v y | Fork | 64 |
A set of Boolean connectives is functionally complete if all Boolean functions can be synthesized using those Which of the following sets of connectives is NOT functionally complete EX NOR implication negation OR negation NAND | Functional Completeness | 65 |
The implication gate shown below has two inputs x ext and y the output is 1 except when x 1 ext and y 0 ext realize f bar x y x bar y using only four implication gates Show that the implication gate is functionally complete | Functional Completeness | 65 |
The following functional dependencies hold true for the relational schema R left V W X Y Z right V rightarrow W VW rightarrow X Y rightarrow VX Y rightarrow Z Which of the following is irreducible equivalent for this set of functional dependencies V rightarrow W V rightarrow X Y rightarrow V Y rightarrow Z V rightarrow W W rightarrow X Y rightarrow V Y rightarrow Z V rightarrow W V rightarrow X Y rightarrow V Y rightarrow X Y rightarrow Z V rightarrow W W rightarrow X Y rightarrow V Y rightarrow X Y rightarrow Z | Functional Dependencies | 66 |
The following functional dependencies are given AB rightarrow CD AF rightarrow D DE rightarrow F C rightarrow G F rightarrow E G rightarrow A Which one of the following options is false left CF right left ACDEFG right left BG right left ABCDG right left AF right left ACDEFG right left AB right left ABCDG right | Functional Dependencies | 66 |
Suppose we have a relation R A B C D E with the functional dependencies A rightarrow D B rightarrow C D rightarrow E CE rightarrow B If we project R and therefore its functional dependencies onto the schema ABC what will the key s for ABC be | Functional Dependencies | 66 |
State whether the following statements are TRUE or FALSE A relation r with schema X Y satisfies the function dependency X rightarrow Y The tuples langle 1 2 rangle and langle 2 2 rangle can both be in r simultaneously | Functional Dependencies | 66 |
Relation R with an associated set of functional dependencies F is decomposed into ext BCNF The redundancy arising out of functional dependencies in the resulting set of relations is Zero More than zero but less than that of an equivalent 3NF decomposition Proportional to the size of F Indeterminate | Functional Dependencies | 66 |
Relation R has eight attributes ext ABCDEFGH Fields of R contain only atomic values F ext CH u2192G A u2192BC B u2192CFH E u2192A F u2192EG is a set of functional dependencies FDs so that F is exactly the set of FDs that hold for R How many candidate keys does the relation R have 3 4 5 6 | Functional Dependencies | 66 |
Relation R has eight attributes ext ABCDEFGH Fields of R contain only atomic values F ext CH rightarrow G A rightarrow BC B rightarrow CFH E rightarrow A F rightarrow EG is a set of functional dependencies FDs so that F is exactly the set of FDs that hold for R The relation R is in 1NF but not in 2NF in 2NF but not in 3NF in 3NF but not in ext BCNF in ext BCNF | Functional Dependencies | 66 |
In a schema with attributes A B C D and E following set of functional dependencies are given A u2192 B A u2192 C CD u2192 E B u2192 D E u2192 A Which of the following functional dependencies is NOT implied by the above set CD u2192 AC BD u2192 CD BC u2192 CD AC u2192 BC | Functional Dependencies | 66 |
Given the following relation instance X Y Z 1 4 2 1 5 3 1 6 3 3 2 2 Which of the following functional dependencies are satisfied by the instance XY u2192 Z and Z u2192 Y YZ u2192 X and Y u2192 Z YZ u2192 X and X u2192 Z XZ u2192 Y and Y u2192 X | Functional Dependencies | 66 |
Form the following instance of a relation schema R A B C we can conclude that A B C 1 1 1 1 1 0 2 3 2 2 3 2 A functionally determines B and B functionally determines C A functionally determines B and B does not functionally determine C B does not functionally determine C A does not functionally determine B and B does not functionally determine C | Functional Dependencies | 66 |
Consider the relation X P Q R S T U with the following set of functional dependencies F P R rightarrow S T P S U rightarrow Q R Which of the following is the trivial functional dependency in F where F is closure to F P R rightarrow S T P R rightarrow R T P S rightarrow S P S U rightarrow Q | Functional Dependencies | 66 |
Consider the relation scheme R E F G H I J K L M N and the set of functional dependencies left E F o G F o I J E H o K L K o M L o N right on R What is the key for R E F E F H E F H K L E | Functional Dependencies | 66 |
Consider a relation R with five attributes V W X Y and Z The following functional dependencies hold VY u2192 W WX u2192 Z and ZY u2192 V Which of the following is a candidate key for R VXZ VXY VWXY VWXYZ | Functional Dependencies | 66 |
What is the maximum number of different Boolean functions involving n Boolean variables n 2 2 n 2 2 n 2 n 2 | Functions | 67 |
The number of functions from an m element set to an n element set is m n m n n m m n | Functions | 67 |
Let X Y Z be sets of sizes x y and z respectively Let W X imes Y and E be the set of all subsets of W The number of functions from Z to E is z 2 xy z imes 2 xy z 2 x y 2 xyz | Functions | 67 |
Let Sigma left a b c d e right be an alphabet We define an encoding scheme as follows g a 3 g b 5 g c 7 g d 9 g e 11 Let p_i denote the i th prime number left p_1 2 right For a non empty string s a_1 dots a_n where each a_i in Sigma define f s Pi n_ i 1 P_i g a_i For a non empty sequence left langle s_j dots s_n right rangle of stings from Sigma define h left left langle s_i dots s_n right rangle right Pi n_ i 1 P_i f left s_i right Which of the following numbers is the encoding h of a non empty sequence of strings 2 73 75 7 2 83 85 8 2 93 95 9 2 10 3 10 5 10 | Functions | 67 |
Let S 1 2 3 m m gt 3 Let X1 Xn be subsets of S each of size 3 Define a function f from S to the set of natural numbers as f i is the number of sets Xj that contain the element i That is f i left left j mid i in X_j right right then sum_ i 1 m f i is 3m 3n 2m 1 2n 1 | Functions | 67 |
Let R denote the set of real numbers Let f R imes R rightarrow R imes R be a bijective function defined by f x y x y x y The inverse function of f is given by a f 1 x y left frac 1 x y frac 1 x y right b f 1 x y x y x y c f 1 x y left frac x y 2 frac x y 2 right d f 1 x y left 2 left x y right 2 left x y right right | Functions | 67 |
Let F be the set of one to one functions from the set 1 2 dots n to the set 1 2 dots m where m geq n geq1 How many functions are members of F How many functions f in F satisfy the property f i 1 for some i 1 leq i leq n How many functions f in F satisfy the property f i lt f j for all 1 leq i leq j leq n | Functions | 67 |
Let f be a function from a set A to a set B g a function from B to C and h a function from A to C such that h a g f a for all a u2208 A Which of the following statements is always true for all such functions f and g g is onto gt h is onto h is onto gt f is onto h is onto gt g is onto h is onto gt f and g are onto | Functions | 67 |
Let f B o C and g A o B be two functions and let h f o g Given that h is an onto function which one of the following is TRUE f and g should both be onto functions f should be onto but g need not to be onto g should be onto but f need not be onto both f and g need not be onto | Functions | 67 |
Let f A rightarrow B a function and let E and F be subsets of A Consider the following statements about images S1 f E cup F f E cup f F S2 f E cap F f E cap f F Which of the following is true about S1 and S2 Only S1 is correct Only S2 is correct Both S1 and S2 are correct None of S1 and S2 is correct | Functions | 67 |
Let f A o B be an injective one to one function Define g 2 A o 2 B as g C left f x mid x in C right for all subsets C of A Define h 2 B o 2 A as h D x mid x in A f x in D for all subsets D of B Which of the following statements is always true g h D subseteq D g h D supseteq D g h D cap D phi g h D cap B D e phi | Functions | 67 |
Let A and B be sets with cardinalities m and n respectively The number of one one mappings from A to B when m lt n is m n nP_m mC_n nC_m mP_n | Functions | 67 |
Given a boolean function f x1 x2 xn which of the following equations is NOT true f x1 x2 xn x1 f x1 x2 xn x1f x1 x2 xn f x1 x2 xn x2f x1 x2 u2026 xn x2 f x1 x2 u2026 xn f x1 x2 xn xn f x1 x2 u2026 0 xnf x1 x2 u2026 1 f x1 x2 xn f 0 x2 u2026 xn f 1 x2 xn | Functions | 67 |
A function f Bbb N rightarrow Bbb N defined on the set of positive integers Bbb N satisfies the following properties f n f n 2 if n is even f n f n 5 if n is odd Let R i mid exists j f j i be the set of distinct values that f takes The maximum possible size of R is ___________ | Functions | 67 |
Suppose X and Y are sets and X ext and Y are their respective cardinalities It is given that there are exactly 97 functions from X to Y From this one can conclude that X 1 Y 97 X 97 Y 1 X 97 Y 97 None of the above | Functions | 67 |
What is the generating function G z for the sequence of Fibonacci numbers | Generating Functions | 68 |
The coefficient of x 3 in the expansion of 1 x 3 2 x 2 10 is 2 14 31 left frac 3 3 right left frac 10 1 right left frac 3 3 right 2 left frac 10 1 right left frac 3 3 right left frac 10 1 right 2 9 | Generating Functions | 68 |
The coefficient of x 12 in left x 3 x 4 x 5 x 6 dots right 3 is ___________ | Generating Functions | 68 |
If the ordinary generating function of a sequence big a_n big _ n 0 infty is large frac 1 z 1 z 3 then a_3 a_0 is equal to ___________ | Generating Functions | 68 |
When a point inside of a tetrahedron a solid with four triangular surfaces is connected by straight lines to its corners how many new internal planes are created with these lines | Geometry | 69 |
In a triangle PQR PS is the angle bisector of angle QPR ext and angle QPS 60 circ What is the length of PS left dfrac q r qr right left dfrac qr q r right large sqrt q 2 r 2 left dfrac q r 2 qr right | Geometry | 69 |
A cube is built using 64 cubic blocks of side one unit After it is built one cubic block is removed from every corner of the cube The resulting surface area of the body in square units after the removal is ________ 56 64 72 96 | Geometry | 69 |
An unrestricted use of the goto statement is harmful because it makes it more difficult to verify programs it increases the running time of the programs it increases the memory required for the programs it results in the compiler generating longer machine code | Goto | 70 |
What is the chromatic number of the following graph 2 3 4 5 | Graph Coloring | 71 |
What is the chromatic number of an n vertex simple connected graph which does not contain any odd length cycle Assume n gt 2 2 3 n 1 n | Graph Coloring | 71 |
The minimum number of colours that is sufficient to vertex colour any planar graph is ________ | Graph Coloring | 71 |
The minimum number of colours required to colour the vertices of a cycle with n nodes in such a way that no two adjacent nodes have the same colour is 2 3 4 n 2 left lfloor frac n 2 right rfloor 2 | Graph Coloring | 71 |
The minimum number of colours required to colour the following graph such that no two adjacent vertices are assigned the same color is 2 3 4 5 | Graph Coloring | 71 |
Let G V E be a simple undirected graph on n vertices A colouring of G is an assignment of colours to each vertex such that endpoints of every edge are given different colours Let chi G denote the chromatic number of G i e the minimum numbers of colours needed for a valid colouring of G A set B subseteq V is an independent set if no pair of vertices in B is connected by an edge Let a G be the number of vertices in a largest possible independent set in G In the absence of any further information about G we can conclude chi left G right geq a left G right chi left G right leq a left G right a left G right geq frac n chi left G right a left G right leq frac n chi left G right None of the above | Graph Coloring | 71 |
How many ways are there to assign colours from range left 1 2 r right to vertices of the following graph so that adjacent vertices receive distinct colours r 4 r 4 4r 3 r 4 5r 3 8r 2 4r r 4 4r 3 9r 2 3r r 4 5r 3 10r 2 15r | Graph Coloring | 71 |
Consider the undirected graph G defined as follows The vertices of G are bit strings of length n We have an edge between vertex u and vertex v if and only if u and v differ in exactly one bit position in other words v can be obtained from u by flipping a single bit The ratio of the chromatic number of G to the diameter of G is frac 1 left 2 n 1 right left frac 1 n right left frac 2 n right left frac 3 n right | Graph Coloring | 71 |
A vertex colouring with three colours of a graph G V E is a mapping c V rightarrow R G B so that adjacent vertices receive distinct colours Consider the following undirected graph How many vertex colouring with three colours does this graph have 3 9 6 3 3 imes 2 8 27 24 | Graph Coloring | 71 |
A vertex colouring of a graph G V E with k coulours is a mapping c V rightarrow 1 dots k such that c u eq c v for every u v in E Consider the following statements If every vertex in G has degree at most d then G admits a vertex coulouring using d 1 colours Every cycle admits a vertex colouring using 2 colours Every tree admits a vertex colouring using 2 colours Which of the above statements is are TRUE Choose from the following options only i only i and ii only i and iii only ii and iii i ii and iii | Graph Coloring | 71 |
What is the number of vertices in an undirected connected graph with 27 edges 6 vertices of degree 2 3 vertices of degree 4 and remaining of degree 3 10 11 18 19 | Graph Connectivity | 72 |
What is the maximum number of edges in an acyclic undirected graph with n vertices n 1 n n 1 2n 1 | Graph Connectivity | 72 |
The minimum number of edges in a connected cyclic graph on n vertices is a n 1 b n c n 1 d None of the above | Graph Connectivity | 72 |
The maximum number of possible edges in an undirected graph with n vertices and k components is ______ | Graph Connectivity | 72 |
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