GRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS

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1 GRAPH THEORY AND APPLICATIONS MARKS QUESTIONS AND ANSWERS UNIT I INTRODUCTION 1. Dfin Graph. A graph G = (V, E) consists of a st of objcts V={1,, 3, } calld rtics (also calld points or nods) and othr st E = {1,, 3,...} whos lmnts ar calld dgs (also calld lins or arcs). Th st V(G) is calld th rtx st of G and E(G) is th dg st of G. For xampl : A graph G is dfind by th sts V(G) = {u,, w, x, y, z} and E(G) = {u, uw, wx, xy, xz}. Graph G: u w x y z A graph with p-rtics and q-dgs is calld a (P, Q) graph. Th (1, 0) graph is calld triial graph.. Dfin Simpl graph. An dg haing th sam rtx as its nd rtics is calld a slf-loop. Mor than on dg associatd a gin pair of rtics calld paralll dgs. A graph that has nithr slf-loops nor paralll dgs is calld simpl graph. Graph G: Graph H: u u w x y w x y Simpl Graph 3. Writ fw problms sold by th applications of graph thory. Konigsbrg bridg problm Utilitis problm Elctrical ntwork problms Sating problms Psudo Graph 4. Dfin incidnc, adjacnt and dgr. Whn a rtx i is an nd rtx of som dg j, i and j ar said to b incidnt with ach othr. Two non paralll dgs ar said to b adjacnt if thy ar incidnt on a common rtx. Th numbr of dgs incidnt on a rtx i, with slf-loops countd twic, is calld th dgr (also calld alncy), d(i), of th rtx i. A graph in which all rtics ar of qual dgr is calld rgular graph. Graph G:

2 Th dgs, 6 and 7 ar incidnt with rtx 4. Th dgs and 7 ar adjacnt. Th dgs and 4 ar not adjacnt. Th rtics 4 and 5 ar adjacnt. Th rtics 1 and 5 ar not adjacnt. d(1) = d(3) = d(4) = 3. d() = 4. d(5) = What ar finit and infinit graphs? A graph with a finit numbr off rtics as wll as a finit numbr of dgs is calld a finit graph; othrwis, it is an infinit graph. Finit Graphs Infinit Graphs 6. Dfin Isolatd and pndnt rtx. A rtx haing no incidnt dg is calld an isolatd rtx. In othr words, isolatd rtics ar rtics with zro dgr. A rtx of dgr on is calld a pndant rtx or an nd rtx. Graph G: Th rtics 6 and 7 ar isolatd rtics. Th rtx 5 is a pndant rtx. 7. Dfin null graph. In a graph G=(V, E), If E is mpty (Graph without any dgs) Thn G is calld a null graph. Graph G:

3 8. Dfin Multigraph In a multigraph, no loops ar allowd but mor than on dg can join two rtics, ths dgs ar calld multipl dgs or paralll dgs and a graph is calld multigraph. Graph G: Th dgs 5 and 4 ar multipl (paralll) dgs. 9. Dfin complt graph A simpl graph G is said to b complt if ry rtx in G is connctd with ry othr rtx. i.., if G contains xactly on dg btwn ach pair of distinct rtics. A complt graph is usually dnotd by K n. It should b notd that Kn has xactly n(n-1)/ dgs. Th complt graphs K n for n = 1,, 3, 4, 5 ar show in th following Figur. 10. Dfin Rgular graph A graph in which all rtics ar of qual dgr, is calld a rgular graph. If th dgr of ach rtx is r, thn th graph is calld a rgular graph of dgr R. 11. Dfin Cycls Th cycl C n, n 3, consists of n rtics 1,,..., n and dgs { 1, }, {, 3 },..., { n 1, n}, and {n, 1}. Th cyls c 3, c 4 and c 5 ar shown in th following Figurs Dfin Isomorphism. Two graphs G and G' ar said to b isomorphic to ach othr if thr is a on-to-on corrspondnc btwn thir rtics and btwn thir dgs such that th incidnc rlationship is prsrd

4 Graph G: b 1 a Graph G': c 3 d 1 4 Corrspondnc of rtics Corrspondnc of dgs f(a) = 1 f(1) = 1 f(b) = f() = f(c) = 3 f(3) = 3 f(d) = 4 f(4) = 4 f() = 5 f(5) = 5 Adjacncy also prsrd. Thrfor G and G' ar said to b isomorphic Dfin Eulr graph. A path in a graph G is calld Eulr path if it includs ry dgs xactly onc. Sinc th path contains ry dg xactly onc, it is also calld Eulr trail / Eulr lin. A closd Eulr path is calld Eulr circuit. A graph which contains an Eulrian circuit is calld an Eulrian graph is an Eulr circuit. So th abo graph is Eulr graph. 14. Dfin Hamiltonian circuits and paths A Hamiltonian circuit in a connctd graph is dfind as a closd walk that trarss ry rtx of graph G xactly onc xcpt starting and trminal rtx. Rmoal of any on dg from a Hamiltonian circuit gnrats a path. This path is calld Hamiltonian path. 15. Dfin Tr A tr is a connctd graph without any circuits. Trs with 1,, 3, and 4 rtics ar shown in figur. 16.List out fw Proprtis of trs. 1. Thr is on and only on path btwn ry pair of rtics in a tr T.. In a graph G thr is on and only on path btwn ry pair of rtics, G is a tr. 3. A tr with n rtics has n-1 dgs. 4. Any connctd graph with n rtics has n-1 dgs is a tr. 5. A graph is a tr if and only if it is minimally connctd. 6. A graph G with n rtics has n-1 dgs and no circuits ar connctd.

5 17. What ar th Radius and Diamtr in a tr. Th ccntricity of a cntr in a tr is dfind as th radius of tr. Th lngth of th longst path in a tr is calld th diamtr of tr. 18. Dfin Rootd tr A tr in which on rtx (calld th root) is distinguishd from all th othrs is calld a rootd tr. In gnral tr mans without any root. Thy ar somtims calld as fr trs (non rootd trs). Th root is nclosd in a small triangl. All rootd trs with four rtics ar shown blow. 19.What is Distanc in a tr? In a connctd graph G, th distanc d(i, j) btwn two of its rtics i and j is th lngth of th shortst path. Graph G: d 1 a c f h j k b 3 g 6 4 i 5

6 Rank of G= = numbr UNIT II TREES, CONNECTIVITY & PLANARITY 1. Dfin Spanning trs. A tr T is said to b a spanning tr of a connctd graph G if T is a subgraph of G and T contains all rtics (maximal tr subgraph). Graph G: Spanning Tr T: Dfin Branch and chord. An dg in a spanning tr T is calld a branch of T. An dg of G is not in a gin spanning tr T is calld a chord (ti or link). Graph G: 1 4 Spanning Tr T: Edg 1 is a branch of T Edg 5 is a chord of T 4 3. Dfin complmnt of tr. If T is a spanning tr of graph G, thn th complmnt of T of G dnotd by of chords. It also calld as chord st (ti st or cotr) of T Graph G: Spanning Tr T: :Complmnt of Tr T is th collction Rank + 4. Dfin Rank and Nullity: = A graph G with n numbr of rtics, numbr of dgs, and k numbr of componnts with th following constraints and. Nullity Nullity of = G = numbr of chords in G + Rank + Nullity = = numbr of dgs in G (Nullity also calld as Cyclomatic numbr or first btti numbr) of branchs in any spanning tr of G 5. How Fundamntal circuits cratd? Addition of an dg btwn any two rtics of a tr crats a circuit. This is bcaus thr alrady xists a path btwn any two rtics of a tr.

7 6. Dfin Spanning trs in a wightd graph A spanning tr in a graph G is a minimal subgraph conncting all th rtics of G. If G is a wightd graph, thn th wight of a spanning tr T of G is dfind as th sum of th wights of all th branchs in T. A spanning tr with th smallst wight in a wightd graph is calld a shortst spanning tr (shortst-distanc spanning tr or minimal spanning tr). 7. Dfin dgr-constraind shortst spanning tr. A shortst spanning tr T for a wightd connctd graph G with a constraint rtics in T. for k=, th tr will b Hamiltonian path. i for all 8. Dfin cut sts and gi xampl. In a connctd graph G, a cut-st is a st of dgs whos rmoal from G la th graph G disconnctd. Graph G: 3 3 k 1 a 4 b c b g h g 6 6 d k 1 4 f 5 5 Disconnctd graph G with componnts aftr rmoing cut st {a, c, d, f} Possibl cut sts ar {a, c, d, f}, {a, b,, f}, {a, b, g}, {d, h, f}, {k}, and so on. {a, c, h, d} is not a cut st, bcaus its propr subst {a, c, h} is a cut st. {g, h} is not a cut st. A minimal st of dgs in a connctd graph whos rmoal rducs th rank by on is calld minimal cut st (simpl cut-st or cocycl). Ery dg of a tr is a cut st. 9. Writ th Proprtis of cut st Ery cut-st in a connctd graph G must contain at last on branch of ry spanning tr of G. In a connctd graph G, any minimal st of dgs containing at last on branch of ry spanning tr of G is a cut-st. Ery circuit has an n numbr of dgs in common with any cut st. 10. Dfin Fundamntal circuits Adding just on dg to a spanning tr will crat a cycl; such a cycl is calld a fundamntal cycl (Fundamntal circuits). Thr is a distinct fundamntal cycl for ach dg; thus, thr is a on-to-on corrspondnc btwn fundamntal cycls and dgs not in th spanning tr. For a connctd graph with V rtics, any spanning tr will ha V 1 dgs, and thus, a graph of E dgs and on of its spanning trs will ha E V + 1 fundamntal cycls. 11. Dfin Fundamntal cut sts Dual to th notion of a fundamntal cycl is th notion of a fundamntal cutst. By dlting just on dg of th spanning tr, th rtics ar partitiond into two disjoint sts. Th fundamntal cutst is dfind as th st of dgs that must b rmod from th graph G to accomplish th sam partition. Thus, ach spanning tr dfins a st of V 1 fundamntal cutsts, on for ach dg of th spanning tr. h

8 1. Dfin dg Connctiity. Each cut-st of a connctd graph G consists of crtain numbr of dgs. Th numbr of dgs in th smallst cut-st is dfind as th dg Connctiity of G. Th dg Connctiity of a connctd graph G is dfind as th minimum numbr of dgs whos rmoal rducs th rank of graph by on. Th dg Connctiity of a tr is on. 1 Th dg Connctiity of th abo graph G is thr. 13. Dfin rtx Connctiity Th rtx Connctiity of a connctd graph G is dfind as th minimum numbr of rtics whos rmoal from G las th rmaining graph disconnctd. Th rtx Connctiity of a tr is on. 1 Th rtx Connctiity of th abo graph G is on. 14. Dfin sparabl and non-sparabl graph. A connctd graph is said to b sparabl graph if its rtx connctiity is on. All othr connctd graphs ar calld non-sparabl graph. Sparabl Graph G: Non-Sparabl Graph H: Dfin articulation point. In a sparabl graph a rtx whos rmoal disconncts th graph is calld a cut-rtx, a cut-nod, or an articulation point. 1 1 is an articulation point. 16. What is Ntwork flows A flow ntwork (also known as a transportation ntwork) is a graph whr ach dg has a capacity and ach dg rcis a flow. Th amount of flow on an dg cannot xcd th capacity of th dg. 17. Dfin max-flow and min-cut thorm (quation). Th maximum flow btwn two rtics a and b in a flow ntwork is qual to th minimum of th capacitis of all cut-sts with rspct to a and b.

9 UNIT III MATRICES, COLOURING AND DIRECTED GRAPH 1. What is propr coloring? Painting all th rtics of a graph with colors such that no two adjacnt rtics ha th sam color is calld th propr coloring (simply coloring) of a graph. A graph in which ry rtx has bn assignd a color according to a propr coloring is calld a proprly colord graph.. Dfin Chromatic numbr A graph G that rquirs k diffrnt colors for its propr coloring, and no lss, is calld k- chromatic graph, and th numbr k is calld th chromatic numbr of G. Th minimum numbr of colors rquird for th propr coloring of a graph is calld Chromatic numbr. Th abo graph initially colord with 5 diffrnt colors, thn 4, and finally 3. So th chromatic numbr is 3. i.., Th graph is 3-chromatic. 3. Writ th proprtis of chromatic numbrs (obsrations). A graph consisting of only isolatd rtics is 1-chromatic. Ery tr with two or mor rtics is -chromatic. A graph with on or mor rtics is at last -chromatic. A graph consisting of simply on circuit with n 3 rtics is -chromatic if n is n and 3-chromatic if n is odd. A complt graph consisting of n rtics is n-chromatic. 4. Dfin Chromatic partitioning A propr coloring of a graph naturally inducs a partitioning of th rtics into diffrnt substs basd on colors. For xampl, th coloring of th abo graph producs th portioning { 1, 4 }, { }, and { 3, 5 }.

10 5. Dfin indpndnt st and maximal indpndnt st. A st of rtics in a graph is said to b an indpndnt st of rtics or simply indpndnt st (or an intrnally stabl st) if two rtics in th st ar adjacnt. c f a b d g For xampl, in th abo graph producs {a, c, d} is an indpndnt st. A singl rtx in any graph constituts an indpndnt st. A maximal indpndnt st is an indpndnt st to which no othr rtx can b addd without dstroying its indpndnc proprty. {a, c, d, f} is on of th maximal indpndnt st. {b, f} is on of th maximal indpndnt st. Th numbr of rtics in th largst indpndnt st of a graph G is calld th indpndnc numbr ( or cofficints of intrnal stability), dnotd by β(g). For a K- chromatic graph of n rtics, th indpndnc numbr β(g). 6. Dfin uniquly colorabl graph. A graph that has only on chromatic partition is calld a uniquly colorabl graph. For xampl, Uniquly colorabl graph G: Not uniquly colorabl graph H: 1 5 a b c f 4 3 d g 7. Dfin dominating st. A dominating st (or an xtrnally stabl st) in a graph G is a st of rtics that dominats ry rtx in G in th following sns: Eithr is includd in th dominating st or is adjacnt to on or mor rtics includd in th dominating st. c f a b d g {b, g} is a dominating st, {a, b, c, d, f} is a dominating st. A is a dominating st nd not b indpndnt st. St of all rtics is a dominating st. A minimal dominating st is a dominating st from which no rtx can b rmod without dstroying its dominanc proprty. {b, } is a minimal dominating st.

11 8. Dfin Chromatic polynomial. A graph G of n rtics can b proprly colord in many diffrnt ways using a sufficintly larg numbr of colors. This proprty of a graph is xprssd lgantly by mans of polynomial. This polynomial is calld th Chromatic polynomial of G. Th alu of th Chromatic polynomial P n (λ) of a graph with n rtics th numbr of ways of proprly coloring th graph, using λ or fwr colors. 9. Dfin Matching (Assignmnt). A matching in a graph is a subst of dgs in which no two dgs ar adjacnt. A singl dg in a graph is a matching. A maximal matching is a matching to which no dg in th graph can b addd. Th maximal matching with th largst numbr of dgs ar calld th largst maximal matching Graph G Matching Maximal matching 10. What is Coring? A st g of dgs in a graph G is said to b cor og G if ry rtx in G is incidnt on at last on dg in g. A st of dgs that cors a graph G is said to b a coring ( or an dg coring, or a corring subgraph) of G. Ery graph is its own coring. A spanning tr in a connctd graph is a coring. A Hamiltonian circuit in a graph is also a coring. 11. Dfin minimal cor. A minimal coring is a coring from which no dg can b rmod without dstroying it ability to cor th graph G Graph G Minimal cor 1. What is dimr coring? A coring in which ry rtx is of dgr on is calld a dimmr coring is a maximal matching bcaus no two dgs in Appasami, Assistant profssor, Dr. pauls Enginring Collg. dimr coring or a 1-factor. A it ar adjacnt. Prpard by G.

12 13. Dfin four color problm / conjctur. Ery planar graph has a chromatic numbr of four or lss. Ery triangular planar graph has a chromatic numbr of four or lss. Th rgions of ry planar, rgular graph of dgr thr can b colord proprly with four colors. 14. Stat fi color thorm Ery planar map can b proprly colord with fi colors. i.., th rtics of ry plannar graph can b proprly colord with fi colors. 15. Writ about rtx coloring and rgion coloring. A graph has a dual if and only if it is planar. Thrfor, coloring th rgions of a planar graph G is quialnt to coloring th rtics of its dual G* and ic rsa. UNIT IV PERMUTATIONS & COMBINATIONS 1. Dfin Fundamntal principls of counting Th Fundamntal Counting Principl is a way to figur out th total numbr of ways diffrnt nts can occur. If th first task can b prformd in m ways, whil a scond task can b prformd in n ways, and th two tasks cannot b prformd simultanously, thn prforming ithr task can b accomplishd in any on of m + n ways. If a procdur can b brokn into first and scond stags, and if thr ar m possibl outcoms for th first stag and if, for ach of ths outcoms, thr ar n possibl outcoms for th scond stag, thn th total procdur can b carrid out, in th dsignd ordr, in mn ways.. Dfin rul of sum. If th first task can b prformd in m ways, whil a scond task can b prformd in n ways, and th two tasks cannot b prformd simultanously, thn prforming ithr task can b accomplishd in any on of m + n ways. Exampl: A collg library has 40 books on C++ and 50 books on Jaa. A studnt at this collg can slct 40+50=90 books to larn programming languag. 3. Dfin rul of Product If a procdur can b brokn into first and scond stags, and if thr ar m possibl outcoms for th first stag and if, for ach of ths outcoms, thr ar n possibl outcoms for th scond stag, thn th total procdur can b carrid out, in th dsignd ordr, in mn ways. Exampl: A drama club with six mn and ight can slct mal and fmal rol in 6 x 8 = 48 ways. 4. Dfin Catalan numbrs Th Catalan numbrs form a squnc of natural numbrs that occur in arious counting problms, oftn inoling rcursily-dfind objcts. Thy ar namd aftr th Blgian mathmatician Eugèn Charls Catalan. th nth Catalan numbr is gin dirctly in trms of binomial cofficints by Prpard by G. Appasami, Assistant profssor, Dr. pauls Enginring Collg.

13 5.Writ th Principl of inclusion and xclusion formula. For any sts, C 1 and C, For any 3 sts, C 1, C and C 3, For any 4 sts, C 1, C, C and C 4,

14 UNIT V GENERATING FUNCTIONS 1. Dfin Gnrating function. A gnrating function dscribs an infinit squnc of numbrs (an) by trating thm lik th cofficints of a sris xpansion. Th sum of this infinit sris is th gnrating function. Unlik an ordinary sris, this formal sris is allowd to dirg, maning that th gnrating function is not always a tru function and th "ariabl" is actually an indtrminat.. What is Partitions of intgr? Partitioning a positi n into positi summands and sking th numbr of such partitions without rgard to ordr is calld Partitions of intgr. This numbr is dnotd by p(n). For xampl P(1) = 1: 1 P() = : = P(3) = 3: 3 = +1 = P(4) = 5: 4 = = + = = P(5) = 7: 5 = = 3 + = = = = What is Rcurrnc rlation? A rcurrnc rlation is an quation that rcursily dfins a squnc or multidimnsional array of alus, onc on or mor initial trms ar gin: ach furthr trm of th squnc or array is dfind as a function of th prcding trms. Th trm diffrnc quation somtims (and for th purposs of this articl) rfrs to a spcific typ of rcurrnc rlation. Howr, "diffrnc quation" is frquntly usd to rfr to any rcurrnc rlation. 4.Writ Fibonacci numbrs and rlation Th rcurrnc satisfid by th Fibonacci numbrs is th archtyp of a homognous linar rcurrnc rlation with constant cofficints (s blow). Th Fibonacci squnc is dfind using th rcurrnc Fn = Fn-1 + Fn- with sd alus F 0 = 0 and F 1 = 1 W obtain th squnc of Fibonacci numbrs, which bgins 0, 1, 1,, 3, 5, 8, 13, 1, 34, 55, 89, Dfin First ordr linar rcurrnc rlation Th gnral form of First ordr linar homognous rcurrnc rlation can b writtn as an+1 = d an, n 0, whr d is a constant. Th rlation is first ordr sinc an+1 dpnds on an. a0 or a1 ar calld boundary conditions. Dfin Scond ordr rcurrnc rlation

15 6.Brifly xplain Non-homognous rcurrnc rlation.

16 PART B UNIT I INTRODUCTION 1. Explain arious applications of graph.. Dfin th following kn, cn, kn,n, dn, trail, walk, path, circuit with an xampl. 3. Show that a connctd graph G is an Eulr graph iff all rtics ar n dgr. 4. Pro that a simpl graph with n rtics and k componnts can ha at most (n-k)(n-k+1)/ dgs. 5. Ar thy isomorphic? 6. Pro that in a complt graph with n rtics thr ar (n-1)/ dgs-disjoint Hamiltonian circuits, if n is odd numbr Pro that, thr is on and only on path btwn ry pair of rtics in a tr T. 8. Pro th gin statmnt, A tr with n rtics has n-1 dgs. 9. Pro that, any connctd graph with n rtics has n-1 dgs is a tr. 10. Show that a graph is a tr if and only if it is minimally connctd. 11. Pro that, a graph G with n rtics has n-1 dgs and no circuits ar connctd. UNIT II TREES, CONNECTIVITY & PLANARITY 1. Find th shortst spanning tr for th following graph.. Explain 1 - isomarphism and - isomarphism of graphs with your own xampl. 3. Pro that a connctd graph G with n rtics and dgs has -n+ rgions. 4. Writ all possibl spanning tr for K5. 5. Pro that ry cut-st in a connctd graph G must contain at last on branch of ry spanning tr of G. 6. Pro that th ry circuit which has n numbr of dgs in common with any cut-st. 7. Show that th ring sum of any two cut-sts in a graph is ithr a third cut st or n dg disjoint union of cut sts. 8. Explain ntwork flow problm in dtail. 9. If G 1 and G ar two 1-isomorphic graphs, th rank of G 1 quals th rank of G and th nullity of G 1 quals th nullity of G, pro this. 10. Pro that any two graphs ar -isomorphic if and only if thy ha circuit corrspondnc.

17 UNIT III MATRICES, COLOURING AND DIRECTED GRAPH 1. Pro that any simpl planar graph can b mbddd in a plan such that ry dg is drawn as a straight lin.. Show that a connctd planar graph with n rtics and dgs has -n+ rgions. 3. Dfin chromatic polynomial. Find th chromatic polynomial for th following graph. 4. Explain matching and bipartit graph in dtail. 5. Writ th obsrations of minimal coring of a graph. 6. Pro that th rtics of ry planar graph can b proprly colord with fi colors. 7. Explain matching in dtail. 8. Pro that a coring g of graph G is minimal iff g contains no path of lngth thr or mor. 9. Illustrat four-color problm. 10. Explain Eulr digraphs in dtail. UNIT IV PERMUTATIONS & COMBINATIONS PART B 1. Explain th Fundamntal principls of counting.. Find th numbr of ways of ways of arranging th word APPASAMIAP and out of it how many arrangmnts ha all A s togthr. 3. Discuss th ruls of sum and product with xampl. 4. Dtrmin th numbr of (staircas) paths in th xy-plan from (, 1) to (7, 4), whr ach path is mad up of indiidual stps going 1 unit to th right (R) or on unit upward (U). i Stat and pro binomial thorm. 6. How many tims th print statmnt xcutd in this program sgmnt? 7. Discuss th Principl of inclusion and xclusion. 8. How many intgrs btwn 1 and 300 (inc.) ar not diisibl by at last on of 5, 6, 8? 9. How 3 bit procssors addrss th contnt? How many addrss ar possibl? 10. Explain th Arrangmnts with forbiddn positions.

18 UNIT V GENERATING FUNCTIONS PART B 1. Explain Gnrating functions. Find th conolution of th squncs 1, 1, 1, 1,.. and 1,-1,1,-1,1, Find th numbr of non ngati & positi intgr solutions of for x1+x+x3+x4=5. 4. Find th cofficint of x5 in(1-x)7. 5. Th numbr of irus affctd fils in a systm is 1000 and incrass 50% ry hours. 6. Explain Partitions of intgrs 7. Us a rcurrnc rlation to find th numbr of iruss aftr on day. 8. Explain First ordr homognous rcurrnc rlations. 9. Sol th rcurrnc rlation an+-4an+1+3an=-00 with a0=3000 and a1= Sol th Fibonacci rlation Fn = Fn-1+Fn Find th rcurrnc rlation from th squnc 0,, 6, 1, 0, 30, 4,. 1. Dtrmin (1+ 3i) Discuss Mthod of gnrating functions.

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