The Equitable Dominating Graph


 Miles Howard
 2 years ago
 Views:
Transcription
1 Intrnational Journal of Enginring Rsarch and Tchnology. ISSN Volum 8, Numbr 1 (015), pp Intrnational Rsarch Publication Hous Th Equitabl Dominating Graph P.N. Vinay Kumar Faculty of Mathmatics, Sri H.D.D. Govt. First Grad Collg, Hassan, Karnataka, INDIA Abstract Th quitabl dominating graph ED( G) of a graph G is a graph with V ( ED( G)) V ( G) D( G) whr DG ( ) is th st of all minimal quitabl dominating sts of G and u, v V ( ED( G)) ar adjacnt to ach othr if u V ( G ) and v is a minimal quitabl dominating st of G containing u. In this papr w charactriz th quitabl dominating graphs which ar ithr connctd or complt. Kywords: Minimum quitabl dominating st; Equitabl dominating graph; Minimum quitabl domination numbr. Mathmatics Subjct Classification (000): 05C 1. Introduction All th graphs ar simpl, undirctd without loops and multipl dgs. Lt G ( V, E ) b a graph. A subst D of V is said to b a quitabl dominating st of G if for vry v V D thr xists a vrtx u D such that uv E( G ) and d( u) d( v ) 1. Th minimum cardinality of such a dominating st D is calld th quitabl domination numbr of G and is dnotd by ( G ). An quitabl dominating st D is said to b minimal quitabl dominating st if no propr subst of D is an quitabl dominating st. Kulli and Janakiram [5] introducd a nw class of intrsction graphs. Motivatd by this w introduc a nw class of graphs in th fild of domination thory. Throughout this papr, th graph G is of p vrtics and q dgs. Th trms usd in this papr ar in th sns of Harary[4]. Dfinition 1.1[III]: A vrtx u V is said to b dgr quitabl with a vrtx v V
2 36 P.N. Vinay Kumar if d( u) d( v ) 1. A vrtx u V is said to b an quitabl isolat if d( u) d( v), v V. Dfinition 1.[III]: A minimal quitabl dominating st of maximum cardinality is calld st and its cardinality is dnotd by ( G ). Dfinition 1.3[III]: Lt u V. Th quitabl nighbourhood of u dnotd by N ( u ) is dfind as N ( u) { v V / v N( u), d( u) d( v ) 1}. Dfinition 1.4[III]: A subst S of V is calld an quitabl indpndnt st, if for any u S, v N ( u ) for all v S {} u. Th maximum cardinality of S is calld quitabl indpndnc numbr of G and is dnotd ( G ). Dfinition 1.5[III]: Th maximum ordr of a partition of V into quitabl dominating sts is calld quitabl domatic numbr of G and is dnotd by d ( G ). Dfinition 1.6: Th quitabl dominating graph ED( G) of a graph G is a graph with V ( ED( G)) V ( G) D( G) whr DG ( ) is th st of all minimal quitabl dominating sts of G and u, v V ( ED( G)) ar adjacnt to ach othr if u V ( G ) and v is a minimal quitabl dominating st of G containing u. An xampl of th quitabl dominating graph ED( G) of a graph G is givn blow:. Rsults In this sction w prov th main rsults of this papr. First w obtain th ncssary and sufficint condition for a givn graph G to b connctd and followd by som rsults on compltnss, quitabl domatic partition and th quitabl domination numbr of ED( G ). Thorm.1[III]: Lt G b a graph without quitabl isolatd vrtics. If D is a
3 Th Equitabl Dominating Graph 37 minimal quitabl dominating st, thn V D is an quitabl dominating st. Thorm.[I]: A graph G is Eulrian if and only vry of vrtx of G is of vn dgr. Thorm.3: For any graph G with p and without quitabl isolatd vrtics, th quitabl dominating graph ED( G ) of G is connctd if and only if ( G) p 1. Proof: Lt ( G) p 1. Lt D1 and D b two minimal quitabl dominating sts of G. W considr th following cass: Cas i): Suppos thr xists two vrtics u D 1 and v D such that u and v ar not adjacnt to ach othr. Thn, thr xists a maximal quitabl indpndnt st D 3 containing u and v. Sinc vry maximal quitabl indpndnt st is a minimal quitabl dominating st, D3 is a minimal quitabl dominating st joining D1 and D. Hnc thr is a path in ED( G ) joining th vrtics of VG ( ) togthr with th minimal quitabl dominating sts of G. Thus, ED( G ) is connctd. Cas ii): Suppos for any two vrtics u D 1 and v D, thr xists a vrtx w D1 D such that w is adjacnt to nithr u not v. Thn, thr xists two maximal quitabl indpndnt sts D 3 and D4 containing u,w and w,v rspctivly. Thus, th vrtics u,v,w and th minimal quitabl dominating sts D1, D, D3, D 4 ar connctd by th path D1 u D3 w D4 v D. Thus, ED( G ) is connctd. Convrsly, suppos that ED( G ) is connctd. Lt us assum that ( G) p 1 and lt {} u b a vrtx of dgr p 1. Thn, { u} is a minimal quitabl dominating st of G and by thorm.1, V D has a minimal quitabl dominating st say D. This implis that ED( G) has at last two componnts, a contradiction. Hnc, ( G) p 1. Hnc th rsult. Rmark.4: In ED( G ), any two vrtics u and v of VG ( ) ar connctd by a path of lngth at most four. Thorm.5: For any graph G with ( G) p 1 and without quitabl isolatd vrtics, diam( ED( G )) 5. Proof: As ( G) p 1, by thorm.4, G is connctd. Lt ED( G) V Y, E, whr Y is th st of all minimal quitabl dominating sts of G. Lt u, v V Y. Thn, by abov thorm.4, diam( ED( G )) 4 if u, v V or u, v Y. On th othr hand, if u V and v Y thn v D is a minimal quitabl dominating st of G. If
4 38 P.N. Vinay Kumar u D, thn d( u, v ) 4; othrwis, thr xists a vrtx w D such that d( u, v) d( u, w) d( w, v ) This provs th rsult. Thorm.6: For any graph G without quitabl isolatd vrtics, ED( G ) is a complt bipartit graph if and only if K p. Proof: Suppos that ED( G ) is not a complt bipartit graph with G K p. As G K p th minimal quitabl dominating st of G is VG, ( ) vry isolatd vrtx in ED( G ) is adjacnt to th vrtx VG. ( ) This implis that ED( G ) is K 1, p, which is a contradiction. Thus, ED( G) is complt bipartit graph. Convrsly, suppos that ED( G ) is complt bipartit graph and G K. Thus G contains a nontrivial subgraph G 1. Thn, by thorm.1, for som vrtx u G 1, thr xists a minimal quitabl dominating sts D and D with u D and u D, which is a contradiction to th fact that G is complt bipartit graph with u G 1. Hnc G K p. This complts th proof. Thorm.7: For any graph G without quitabl isolatd vrtics, d ( G) ( ED( G )). Furthr, th quality holds if and only if VG ( ) can b partitiond into union of disjoint minimal quitabl dominating sts of cardinality on. Proof: Lt S b th maximum ordr of quitabl domatic partition of VG. ( ) If vry quitabl dominating st is minimal and S consists of all minimal quitabl dominating sts of G, thn S is a maximum quitabl indpndnt sts of ED( G ). Hnc d ( G) ( ED( G )). Othrwis, lt D b a maximum quitabl indpndnt st with D S. Hnc, D is a minimal quitabl dominating st of G. Lt u D. Thn, thr ar two following cass: Cas i): If u D', whr D' S. Thn, clarly S {} u is a quitabl indpndnt st in ED( G ). Hnc th rsult holds. Cas ii): If u D ', whr D' S. Thn, thr xists a vrtx w V ( G) such that S { u, w} is an quitabl indpndnt st. Hnc th rsult. Clarly, th quality condition follows as vry componnt of ED( G ) is K as VG ( ) is th union of disjoint minimal quitabl dominating sts of cardinality on. This complts th proof. Corollary.8: For any graph G, V ( ED( G) d ( G ). p
5 Th Equitabl Dominating Graph 39 Proof: Follows from thorm.8 and th fact that for any graph G, V ( G) ( G ). Thorm.9: For any graph G without quitabl isolatd vrtics p d ( G) p ' p( ( G ) 1), whr p ' is th numbr of vrtics of ED( G ). Furthr th lowr bound is attaind if and only if vry minimal quitabl dominating st of G is indpndnt and th uppr bound is attaind if and only if vry maximum quitabl indpndnt st is of cardinality on. Proof: Th graph ED( G) has th vrtx st V ( G) D( G) and it has at last d ( G) numbr of minimal quitabl dominating sts, hnc th lowr bound follows. Clarly uppr bound follows as vry maximal quitabl indpndnt st is a minimal quitabl dominating st and vry vrtx is prsnt in at most ( p 1) minimal quitabl dominating sts. Furthr, suppos that p d ( G) p '. As thr ar d ( G) numbr of minimal quitabl dominating sts and ach vrtx is prsnt in xactly on of th minimal quitabl dominating st and hnc ths minimal quitabl dominating sts ar indpndnt. Also, suppos that vry maximum quitabl indpndnt st is of cardinality on thn, ths ar minimal quitabl dominating sts of G and ar indpndnt and as vry vrtx is prsnt in at most ( p 1) minimal quitabl dominating st, th quality holds. This implis th ncssary condition. Convrs of th rsult trivially holds. Thorm.10: For any graph G without quitabl isolatd vrtics p d ( G) q ' p( p 1), whr q ' is th numbr of dgs of ED( G ). Furthr, th lowr bound is attaind if and only if vry minimal quitabl dominating st is indpndnt and th uppr bound is attaind if and only if G is ( p ) rgular. Proof: Th proof of th lowr bound follows by th sam lins of thorm.10. Suppos th lowr bound is attaind. As vry vrtx must b in xactly on of th dominating st, Clary vry minimal quitabl dominating st is indpndnt. As vry vrtx is in at most ( p 1) minimal quitabl dominating st, uppr bound follows. Suppos th uppr bound is attaind. Thn, ach vrtx is in xactly ( p 1) minimal quitabl dominating sts and hnc G is ( p ) rgular. This complts th proof. Thorm.11: For any graph G with p 3, d ( ED( G)) 1 if and only if G K p,
6 40 P.N. Vinay Kumar whr K p is th complmnt of K p or ED( G ) has an quitabl isolatd vrtx. Proof: Suppos that d ( ED( G )) 1. Thn, ED( G) has a vrtx D with D V ( G ). Thus ED( G ) is K 1, p and hnc G K p. Othrwis, suppos assum that ED( G) has p' no quitabl isolatd vrtx and V ( ED( G)) p '. Thn, ( ED( G )). If D is an quitabl dominating st, thn V D is an quitabl dominating st and hnc d ( ED( G )), a contradiction. Hnc ED( G) has an quitabl isolatd vrtx. Th convrs is obvious. Thorm.1: If a graph G is connctd, ( p 1) isolatd vrtics thn, ( ED( G)) p. rgular and without quitabl Proof: As G is connctd and ( G) p 1, by thorm.4, ED( G ) is disconnctd. Also, w know that vry vrtx is prsnt in at most ( p 1) minimal quitabl dominating sts. Thus, ED( G) is a disconnctd graph with ach of th componnt bing K, thr ar p numbr of componnts. Hnc ( ED( G)) p. Thorm.13: For any graph G of ordr p, without quitabl isolatd vrtics and ( G) p 1, th quitabl dominating graph ED( G) of a graph G is a tr if and only if G K p. Proof: As G is a graph of ordr p,without quitabl isolatd vrtics and ( G) p 1, by thorm.4, ED( G) is connctd. Suppos assum that ED( G) of G is a tr. Thn, clarly G has no cycl. On th contrary assum that G K p. Thn, by thorm.1, d ( ED( G )) 1. Hnc thr xists at last two minimal quitabl dominating sts containing whr u and v ar any two vrtics in G. If u and v ar in th sam minimal quitabl dominating st D thn, u D v u is a cycl in ED( G ), a contradiction. On th othr hand, if u and v ar in diffrnt minimal quitabl dominating st. Thn, thr xists vrtics u 1, v1 and th minimal quitabl dominating sts D 1, D and D3 such that uu1 D 1, u1v1 D and v1v D 3. Thus, u and v ar connctd by two paths in ED( G ), a contradiction. Hnc Convrsly, suppos that G K p and ( G) p 1. Thn, by thorm.4, ED( G) is connctd. Also, by thorm.1, d ( ED( G )) 1. i.., thr xists a minimal quitabl dominating st D with D V ( G ). Thus, ED( G) is connctd, K and has no cycl. Hnc ED( G ) is a tr. This complts th proof. 1, p Thorm.15: For any graph G, ED( G) is ithr connctd or has at most on G K p
7 Th Equitabl Dominating Graph 41 componnt that is not K. Proof: W considr th following cass: Cas i): If ( G) p 1, thn by thorm.4, ED( G ) is connctd. Cas ii): If ( G) ( G) p 1, thn G K p. Hnc ach of th vrtx v V ( G) is a minimal quitabl dominating st of G and hnc ach of th componnt of ED( G ) is K. Cas iii): If ( G) ( G) p 1. Lt v1, v,..., vn b n vrtics of G of dgr p 1. Lt H G thn ( H) V ( H ) 1. Hnc by thorm.4, ED( H ) is { v, v,..., v } 1 n connctd. Sinc ED( G) ( V ( ED( H) V ( G1) V ( G)... V ( Gn)) whr G1, G,..., Gnar th graphs joining v1, v,..., vnwith { v1},{ v},...,{ v n } rspctivly. Thn, xactly on of th componnt of ED( G) is not K. Hnc th proof. Thorm.17: If G is a r rgular graph with ( G) and vry vrtx is in xactly vn numbr of minimal quitabl dominating sts thn ED( G) is ulrian. Proof: Lt G is a r rgular graph. Sinc ach of th vrtx of G is in vn numbr of minimal quitabl dominating sts, ach of thm contributs vn numbr to th dgr of th vrtx in ED( G ) and as ( G ), ach of th minimal quitabl dominating st of G is a vrtx of dgr two in ED( G ). Thus, by thorm., ED( G) is ulrian. Thorm.18: Lt G b a graph with ( G) p 1and ( G ). If vry vrtx is prsnt in xactly two minimal quitabl dominating sts thn, ED( G) is Hamiltonian. Proof: As ( G) p 1, G is connctd by thorm.4. Also, sinc vry vrtx is prsnt in xactly two minimal quitabl dominating sts, ( G) ( G) and also dg( u) dg( D ) in ED( G ), whr D is a minimal quitabl dominating st in G. Thus, ED( G) is connctd and rgular. Hnc ED( G) is Hamiltonian. REFERENCES: I. Chartrand G. and Zhang P., Introduction to graph thory, Tata McGrawHill Inc., Nwyork (006). II. Cockayn E.J. and Hdtnimi S.T., Towards a thory of domination in
8 4 P.N. Vinay Kumar graphs, Ntworks, 7, (1977). III. Dharmalingam K.M.,Studis in graph thory quitabl domination and bottlnck domination, Ph.D. Thsis (006). IV. Harary F., Graph Thory, AddisonWsly Publ. Comp. Inc., Rading, Mass (1969). V. Kulli V.R. and Janakiram B., Th Minimal Dominating Graph, Graph thory nots of Nw York XXVIII, 115 (1995).
SOME PARAMETERS ON EQUITABLE COLORING OF PRISM AND CIRCULANT GRAPH.
SOME PARAMETERS ON EQUITABLE COLORING OF PRISM AND CIRCULANT GRAPH. K VASUDEVAN, K. SWATHY AND K. MANIKANDAN 1 Dpartmnt of Mathmatics, Prsidncy Collg, Chnnai05, India. EMail:vasu k dvan@yahoo.com. 2,
More informationWeek 3: Connected Subgraphs
Wk 3: Connctd Subgraphs Sptmbr 19, 2016 1 Connctd Graphs Path, Distanc: A path from a vrtx x to a vrtx y in a graph G is rfrrd to an xypath. Lt X, Y V (G). An (X, Y )path is an xypath with x X and y
More informationInternational Journal of Scientific & Engineering Research, Volume 6, Issue 7, July ISSN
Intrnational Journal of Scintific & Enginring Rsarch, Volum 6, Issu 7, July25 64 ISSN 2229558 HARATERISTIS OF EDGE UTSET MATRIX OF PETERSON GRAPH WITH ALGEBRAI GRAPH THEORY Dr. G. Nirmala M. Murugan
More informationCombinatorial Networks Week 1, March 1112
1 Nots on March 11 Combinatorial Ntwors W 1, March 111 11 Th Pigonhol Principl Th Pigonhol Principl If n objcts ar placd in hols, whr n >, thr xists a box with mor than on objcts 11 Thorm Givn a simpl
More information(UpsideDown o Direct Rotation) β  Numbers
Amrican Journal of Mathmatics and Statistics 014, 4(): 5864 DOI: 10593/jajms0140400 (UpsidDown o Dirct Rotation) β  Numbrs Ammar Sddiq Mahmood 1, Shukriyah Sabir Ali,* 1 Dpartmnt of Mathmatics, Collg
More informationcycle that does not cross any edges (including its own), then it has at least
W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th
More informationSection 6.1. Question: 2. Let H be a subgroup of a group G. Then H operates on G by left multiplication. Describe the orbits for this operation.
MAT 444 H Barclo Spring 004 Homwork 6 Solutions Sction 6 Lt H b a subgroup of a group G Thn H oprats on G by lft multiplication Dscrib th orbits for this opration Th orbits of G ar th right costs of H
More informationFinding low cost TSP and 2matching solutions using certain half integer subtour vertices
Finding low cost TSP and 2matching solutions using crtain half intgr subtour vrtics Sylvia Boyd and Robrt Carr Novmbr 996 Introduction Givn th complt graph K n = (V, E) on n nods with dg costs c R E,
More informationOn spanning trees and cycles of multicolored point sets with few intersections
On spanning trs and cycls of multicolord point sts with fw intrsctions M. Kano, C. Mrino, and J. Urrutia April, 00 Abstract Lt P 1,..., P k b a collction of disjoint point sts in R in gnral position. W
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013
18.782 Introduction to Arithmtic Gomtry Fall 2013 Lctur #20 11/14/2013 20.1 Dgr thorm for morphisms of curvs Lt us rstat th thorm givn at th nd of th last lctur, which w will now prov. Thorm 20.1. Lt φ:
More informationMutually Independent Hamiltonian Cycles of Pancake Networks
Mutually Indpndnt Hamiltonian Cycls of Pancak Ntworks ChngKuan Lin Dpartmnt of Mathmatics National Cntral Univrsity, ChungLi, Taiwan 00, R O C discipl@ms0urlcomtw HuaMin Huang Dpartmnt of Mathmatics
More informationStrongly Connected Components
Strongly Connctd Componnts Lt G = (V, E) b a dirctd graph Writ if thr is a path from to in G Writ if and is an quivalnc rlation: implis and implis s quivalnc classs ar calld th strongly connctd componnts
More informationSCHUR S THEOREM REU SUMMER 2005
SCHUR S THEOREM REU SUMMER 2005 1. Combinatorial aroach Prhas th first rsult in th subjct blongs to I. Schur and dats back to 1916. On of his motivation was to study th local vrsion of th famous quation
More information1 Minimum Cut Problem
CS 6 Lctur 6 Min Cut and argr s Algorithm Scribs: Png Hui How (05), Virginia Dat: May 4, 06 Minimum Cut Problm Today, w introduc th minimum cut problm. This problm has many motivations, on of which coms
More informationConstruction of asymmetric orthogonal arrays of strength three via a replacement method
isid/ms/26/2 Fbruary, 26 http://www.isid.ac.in/ statmath/indx.php?modul=prprint Construction of asymmtric orthogonal arrays of strngth thr via a rplacmnt mthod Tianfang Zhang, Qiaoling Dng and Alok Dy
More informationSome Results on E  Cordial Graphs
Intrnational Journal of Mathmatics Trnds and Tchnology Volum 7 Numbr 2 March 24 Som Rsults on E  Cordial Graphs S.Vnkatsh, Jamal Salah 2, G.Sthuraman 3 Corrsponding author, Dpartmnt of Basic Scincs, Collg
More informationSquare of Hamilton cycle in a random graph
Squar of Hamilton cycl in a random graph Andrzj Dudk Alan Friz Jun 28, 2016 Abstract W show that p = n is a sharp thrshold for th random graph G n,p to contain th squar of a Hamilton cycl. This improvs
More informationDerangements and Applications
2 3 47 6 23 Journal of Intgr Squncs, Vol. 6 (2003), Articl 03..2 Drangmnts and Applications Mhdi Hassani Dpartmnt of Mathmatics Institut for Advancd Studis in Basic Scincs Zanjan, Iran mhassani@iasbs.ac.ir
More informationBasic Polyhedral theory
Basic Polyhdral thory Th st P = { A b} is calld a polyhdron. Lmma 1. Eithr th systm A = b, b 0, 0 has a solution or thr is a vctorπ such that π A 0, πb < 0 Thr cass, if solution in top row dos not ist
More informationPROOF OF FIRST STANDARD FORM OF NONELEMENTARY FUNCTIONS
Intrnational Journal Of Advanc Rsarch In Scinc And Enginring http://www.ijars.com IJARSE, Vol. No., Issu No., Fbruary, 013 ISSN3198354(E) PROOF OF FIRST STANDARD FORM OF NONELEMENTARY FUNCTIONS 1 Dharmndra
More informationSearching Linked Lists. Perfect Skip List. Building a Skip List. Skip List Analysis (1) Assume the list is sorted, but is stored in a linked list.
3 3 4 8 6 3 3 4 8 6 3 3 4 8 6 () (d) 3 Sarching Linkd Lists Sarching Linkd Lists Sarching Linkd Lists ssum th list is sortd, but is stord in a linkd list. an w us binary sarch? omparisons? Work? What if
More informationarxiv: v2 [cs.dm] 17 May 2018
Inhritanc of Convxity for th P min Rstrictd Gam arxiv:708.0675v [cs.dm] 7 May 08 A. Skoda July 6, 08 Abstract W considr rstrictd gams on wightd graphs associatd with minimum partitions. W rplac in th
More informationRecall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1
Chaptr 11 Th singular sris Rcall that by Thorms 10 and 104 togthr provid us th stimat 9 4 n 2 111 Rn = SnΓ 2 + on2, whr th singular sris Sn was dfind in Chaptr 10 as Sn = q=1 Sq q 9, with Sq = 1 a q gcda,q=1
More informationON THE COMPLEXITY OF KSTEP AND KHOP DOMINATING SETS IN GRAPHS
MATEMATICA MONTISNIRI Vol XL (2017) MATEMATICS ON TE COMPLEXITY OF KSTEP AN KOP OMINATIN SETS IN RAPS M FARAI JALALVAN AN N JAFARI RA partmnt of Mathmatcs Shahrood Unrsty of Tchnology Shahrood Iran Emals:
More informationNEW APPLICATIONS OF THE ABELLIOUVILLE FORMULA
NE APPLICATIONS OF THE ABELLIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals
More informationON RIGHT(LEFT) DUO POSEMIGROUPS. S. K. Lee and K. Y. Park
KangwonKyungki Math. Jour. 11 (2003), No. 2, pp. 147 153 ON RIGHT(LEFT) DUO POSEMIGROUPS S. K. L and K. Y. Park Abstract. W invstigat som proprtis on right(rsp. lft) duo posmigroups. 1. Introduction
More informationCOUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM
COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM Jim Brown Dpartmnt of Mathmatical Scincs, Clmson Univrsity, Clmson, SC 9634, USA jimlb@g.clmson.du Robrt Cass Dpartmnt of Mathmatics,
More informationSpectral Synthesis in the Heisenberg Group
Intrnational Journal of Mathmatical Analysis Vol. 13, 19, no. 1, 15 HIKARI Ltd, www.mhikari.com https://doi.org/1.1988/ijma.19.81179 Spctral Synthsis in th Hisnbrg Group Yitzhak Wit Dpartmnt of Mathmatics,
More informationCPSC 665 : An Algorithmist s Toolkit Lecture 4 : 21 Jan Linear Programming
CPSC 665 : An Algorithmist s Toolkit Lctur 4 : 21 Jan 2015 Lcturr: Sushant Sachdva Linar Programming Scrib: Rasmus Kyng 1. Introduction An optimization problm rquirs us to find th minimum or maximum) of
More informationCLONES IN 3CONNECTED FRAME MATROIDS
CLONES IN 3CONNECTED FRAME MATROIDS JAKAYLA ROBBINS, DANIEL SLILATY, AND XIANGQIAN ZHOU Abstract. W dtrmin th structur o clonal classs o 3connctd ram matroids in trms o th structur o biasd graphs. Robbins
More information10. EXTENDING TRACTABILITY
Coping with NPcompltnss 0. EXTENDING TRACTABILITY ining small vrtx covrs solving NPhar problms on trs circular arc covrings vrtx covr in bipartit graphs Q. Suppos I n to solv an NPcomplt problm. What
More informationGRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS
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
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim (implicit in notation and n a positiv intgr, lt ν(n dnot th xponnt of p in n, and U(n n/p ν(n, th unit
More informationEngineering 323 Beautiful HW #13 Page 1 of 6 Brown Problem 512
Enginring Bautiful HW #1 Pag 1 of 6 5.1 Two componnts of a minicomputr hav th following joint pdf for thir usful liftims X and Y: = x(1+ x and y othrwis a. What is th probability that th liftim X of th
More informationLimiting value of higher Mahler measure
Limiting valu of highr Mahlr masur Arunabha Biswas a, Chris Monico a, a Dpartmnt of Mathmatics & Statistics, Txas Tch Univrsity, Lubbock, TX 7949, USA Abstract W considr th khighr Mahlr masur m k P )
More informationu 3 = u 3 (x 1, x 2, x 3 )
Lctur 23: Curvilinar Coordinats (RHB 8.0 It is oftn convnint to work with variabls othr than th Cartsian coordinats x i ( = x, y, z. For xampl in Lctur 5 w mt sphrical polar and cylindrical polar coordinats.
More informationMulticoloured Hamilton cycles and perfect matchings in pseudorandom graphs
Multicolourd Hamilton cycls and prfct matchings in psudorandom graphs Danila Kühn Dryk Osthus Abstract Givn 0 < p < 1, w prov that a psudorandom graph G with dg dnsity p and sufficintly larg ordr has
More informationPropositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric RodríguezCarbonell. May 17, 2018
Propositional Logic Combinatorial Problm Solving (CPS) Albrt Olivras Enric RodríguzCarbonll May 17, 2018 Ovrviw of th sssion Dfinition of Propositional Logic Gnral Concpts in Logic Rduction to SAT CNFs
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES. 1. Statement of results
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim and n a positiv intgr, lt ν p (n dnot th xponnt of p in n, and u p (n n/p νp(n th unit part of n. If α
More informationSome remarks on Kurepa s left factorial
Som rmarks on Kurpa s lft factorial arxiv:math/0410477v1 [math.nt] 21 Oct 2004 Brnd C. Kllnr Abstract W stablish a connction btwn th subfactorial function S(n) and th lft factorial function of Kurpa K(n).
More information1 Isoparametric Concept
UNIVERSITY OF CALIFORNIA BERKELEY Dpartmnt of Civil Enginring Spring 06 Structural Enginring, Mchanics and Matrials Profssor: S. Govindj Nots on D isoparamtric lmnts Isoparamtric Concpt Th isoparamtric
More informationOn Grids in Topological Graphs
On Grids in Topological Graphs Eyal Ackrman Dpartmnt of Computr Scinc Fri Univrsität Brlin Takustr. 9, 14195 Brlin, Grmany yal@inf.fubrlin.d Jacob Fox Dpartmnt of Mathmatics Princton Univrsity Princton,
More informationa g f 8 e 11 Also: Minimum degree, maximum degree, vertex of degree d 1 adjacent to vertex of degree d 2,...
Warmup: Lt b 2 c 3 d 1 12 6 4 5 10 9 7 a 8 11 (i) Vriy tat G is connctd by ivin an xampl o a walk rom vrtx a to ac o t vrtics b. (ii) Wat is t sortst pat rom a to c? to? (iii) Wat is t lonst pat rom a
More informationarxiv: v1 [cs.cg] 28 Feb 2017
On th Rlationship btwn kplanar and kquasi Planar Graphs Patrizio Anglini 1, Michal A. Bkos 1, Franz J. Brandnburg 2, Giordano Da Lozzo 3, Giuspp Di Battista 4, Waltr Didimo 5, Giuspp Liotta 5, Fabrizio
More informationSuperposition. Thinning
Suprposition STAT253/317 Wintr 213 Lctur 11 Yibi Huang Fbruary 1, 213 5.3 Th Poisson Procsss 5.4 Gnralizations of th Poisson Procsss Th sum of two indpndnt Poisson procsss with rspctiv rats λ 1 and λ 2,
More informationsets and continuity in fuzzy topological spaces
Journal of Linar and Topological Algbra Vol. 06, No. 02, 2017, 125134 Fuzzy sts and continuity in fuzzy topological spacs A. Vadivl a, B. Vijayalakshmi b a Dpartmnt of Mathmatics, Annamalai Univrsity,
More informationEquidistribution and Weyl s criterion
Euidistribution and Wyl s critrion by Brad HanniganDaly W introduc th ida of a sunc of numbrs bing uidistributd (mod ), and w stat and prov a thorm of Hrmann Wyl which charactrizs such suncs. W also discuss
More informationOuterplanar graphs and Delaunay triangulations
CCCG 011, Toronto ON, August 10 1, 011 Outrplanar graphs and Dlaunay triangulations Ashraful Alam Igor Rivin Ilana Strinu Abstract Ovr 0 yars ago, Dillncourt [1] showd that all outrplanar graphs can b
More informationComputing and Communications  Network Coding
89 90 98 00 Computing and Communications  Ntwork Coding Dr. Zhiyong Chn Institut of Wirlss Communications Tchnology Shanghai Jiao Tong Univrsity China Lctur 5 Nov. 05 0 Classical Information Thory Sourc
More information1 InputOutput Stability
InutOutut Stability Inutoutut stability analysis allows us to analyz th stability of a givn syst without knowing th intrnal stat x of th syst. Bfor going forward, w hav to introduc so inutoutut athatical
More informationObjective Mathematics
x. Lt 'P' b a point on th curv y and tangnt x drawn at P to th curv has gratst slop in magnitud, thn point 'P' is,, (0, 0),. Th quation of common tangnt to th curvs y = 6 x x and xy = x + is : x y = 8
More informationSECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.
SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain
More informationHardyLittlewood Conjecture and Exceptional real Zero. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R.
HardyLittlwood Conjctur and Excptional ral Zro JinHua Fi ChangLing Company of Elctronic Tchnology Baoji Shannxi P.R.China Email: fijinhuayoujian@msn.com Abstract. In this papr, w assum that HardyLittlwood
More informationSearch sequence databases 3 10/25/2016
Sarch squnc databass 3 10/25/2016 Etrm valu distribution Ø Suppos X is a random variabl with probability dnsity function p(, w sampl a larg numbr S of indpndnt valus of X from this distribution for an
More informationFSA. CmSc 365 Theory of Computation. Finite State Automata and Regular Expressions (Chapter 2, Section 2.3) ALPHABET operations: U, concatenation, *
CmSc 365 Thory of Computtion Finit Stt Automt nd Rgulr Exprssions (Chptr 2, Sction 2.3) ALPHABET oprtions: U, conctntion, * otin otin Strings Form Rgulr xprssions dscri Closd undr U, conctntion nd * (if
More informationSupplementary Materials
6 Supplmntary Matrials APPENDIX A PHYSICAL INTERPRETATION OF FUELRATESPEED FUNCTION A truck running on a road with grad/slop θ positiv if moving up and ngativ if moving down facs thr rsistancs: arodynamic
More informationVishnu V. Narayan. January
A 17 12 approimation algorithm for 2rtconnctd spanning subgraphs on graphs with minimum dgr at last arxi:1612.047902 [cs.ds] 17 Jan 2017 Vishnu V. Naraan Januar 17 2017 W obtain a polnomialtim 17 approimation
More informationSome Inequalities for General Sum Connectivity Index
MATCH Counications in Mathatical and in Coputr Chistry MATCH Coun. Math. Coput. Ch. 79 (2018) 477489 ISSN 03406253 So Inqualitis for Gnral Su Connctivity Indx I. Ž. Milovanović, E. I. Milovanović, M.
More informationChapter Finding Small Vertex Covers. Extending the Limits of Tractability. Coping With NPCompleteness. Vertex Cover
Coping With NPCompltnss Chaptr 0 Extning th Limits o Tractability Q. Suppos I n to solv an NPcomplt problm. What shoul I o? A. Thory says you'r unlikly to in polytim algorithm. Must sacriic on o thr
More informationABEL TYPE THEOREMS FOR THE WAVELET TRANSFORM THROUGH THE QUASIASYMPTOTIC BOUNDEDNESS
Novi Sad J. Math. Vol. 45, No. 1, 2015, 201206 ABEL TYPE THEOREMS FOR THE WAVELET TRANSFORM THROUGH THE QUASIASYMPTOTIC BOUNDEDNESS Mirjana Vuković 1 and Ivana Zubac 2 Ddicatd to Acadmician Bogoljub Stanković
More informationSlide 1. Slide 2. Slide 3 DIGITAL SIGNAL PROCESSING CLASSIFICATION OF SIGNALS
Slid DIGITAL SIGAL PROCESSIG UIT I DISCRETE TIME SIGALS AD SYSTEM Slid Rviw of discrttim signals & systms Signal: A signal is dfind as any physical quantity that varis with tim, spac or any othr indpndnt
More informationChapter 10. The singular integral Introducing S(n) and J(n)
Chaptr Th singular intgral Our aim in this chaptr is to rplac th functions S (n) and J (n) by mor convnint xprssions; ths will b calld th singular sris S(n) and th singular intgral J(n). This will b don
More informationApproximation and Inapproximation for The Influence Maximization Problem in Social Networks under Deterministic Linear Threshold Model
20 3st Intrnational Confrnc on Distributd Computing Systms Workshops Approximation and Inapproximation for Th Influnc Maximization Problm in Social Ntworks undr Dtrministic Linar Thrshold Modl Zaixin Lu,
More information64. A Conic Section from Five Elements.
. onic Sction from Fiv Elmnts. To raw a conic sction of which fiv lmnts  points an tangnts  ar known. W consir th thr cass:. Fiv points ar known.. Four points an a tangnt lin ar known.. Thr points an
More informationThe graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the
Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More informationCramérRao Inequality: Let f(x; θ) be a probability density function with continuous parameter
WHEN THE CRAMÉRRAO INEQUALITY PROVIDES NO INFORMATION STEVEN J. MILLER Abstract. W invstigat a onparamtr family of probability dnsitis (rlatd to th Parto distribution, which dscribs many natural phnomna)
More informationApplication of Vague Soft Sets in students evaluation
Availabl onlin at www.plagiarsarchlibrary.com Advancs in Applid Scinc Rsarch, 0, (6):4843 ISSN: 0976860 CODEN (USA): AASRFC Application of Vagu Soft Sts in studnts valuation B. Chtia*and P. K. Das Dpartmnt
More informationOn the irreducibility of some polynomials in two variables
ACTA ARITHMETICA LXXXII.3 (1997) On th irrducibility of som polynomials in two variabls by B. Brindza and Á. Pintér (Dbrcn) To th mmory of Paul Erdős Lt f(x) and g(y ) b polynomials with intgral cofficints
More informationFurther Results on Pair Sum Graphs
Applid Mathmatis, 0,, 6775 http://dx.doi.org/0.46/am.0.04 Publishd Oli Marh 0 (http://www.sirp.org/joural/am) Furthr Rsults o Pair Sum Graphs Raja Poraj, Jyaraj Vijaya Xavir Parthipa, Rukhmoi Kala Dpartmt
More informationAbstract Interpretation: concrete and abstract semantics
Abstract Intrprtation: concrt and abstract smantics Concrt smantics W considr a vry tiny languag that manags arithmtic oprations on intgrs valus. Th (concrt) smantics of th languags cab b dfind by th funzcion
More informationFigure 1: Closed surface, surface with boundary, or not a surface?
QUESTION 1 (10 marks) Two o th topological spacs shown in Figur 1 ar closd suracs, two ar suracs with boundary, and two ar not suracs. Dtrmin which is which. You ar not rquird to justiy your answr, but,
More informationCSE303  Introduction to the Theory of Computing Sample Solutions for Exercises on Finite Automata
CSE303  Introduction to th Thory of Computing Smpl Solutions for Exrciss on Finit Automt Exrcis 2.1.1 A dtrministic finit utomton M ccpts th mpty string (i.., L(M)) if nd only if its initil stt is finl
More informationThus, because if either [G : H] or [H : K] is infinite, then [G : K] is infinite, then [G : K] = [G : H][H : K] for all infinite cases.
Homwork 5 M 373K Solutions Mark Lindbrg and Travis Schdlr 1. Prov that th ring Z/mZ (for m 0) is a fild if and only if m is prim. ( ) Proof by Contrapositiv: Hr, thr ar thr cass for m not prim. m 0: Whn
More informationThat is, we start with a general matrix: And end with a simpler matrix:
DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss
More informationMapping properties of the elliptic maximal function
Rv. Mat. Ibroamricana 19 (2003), 221 234 Mapping proprtis of th lliptic maximal function M. Burak Erdoğan Abstract W prov that th lliptic maximal function maps th Sobolv spac W 4,η (R 2 )intol 4 (R 2 )
More informationFunction Spaces. a x 3. (Letting x = 1 =)) a(0) + b + c (1) = 0. Row reducing the matrix. b 1. e 4 3. e 9. >: (x = 1 =)) a(0) + b + c (1) = 0
unction Spacs Prrquisit: Sction 4.7, Coordinatization n this sction, w apply th tchniqus of Chaptr 4 to vctor spacs whos lmnts ar functions. Th vctor spacs P n and P ar familiar xampls of such spacs. Othr
More informationCS 361 Meeting 12 10/3/18
CS 36 Mting 2 /3/8 Announcmnts. Homwork 4 is du Friday. If Friday is Mountain Day, homwork should b turnd in at my offic or th dpartmnt offic bfor 4. 2. Homwork 5 will b availabl ovr th wknd. 3. Our midtrm
More informationTotal Prime Graph. Abstract: We introduce a new type of labeling known as Total Prime Labeling. Graphs which admit a Total Prime labeling are
Itratoal Joural Of Computatoal Egrg Rsarch (crol.com) Vol. Issu. 5 Total Prm Graph M.Rav (a) Ramasubramaa 1, R.Kala 1 Dpt.of Mathmatcs, Sr Shakth Isttut of Egrg & Tchology, Combator 641 06. Dpt. of Mathmatcs,
More informationLINEAR DELAY DIFFERENTIAL EQUATION WITH A POSITIVE AND A NEGATIVE TERM
Elctronic Journal of Diffrntial Equations, Vol. 2003(2003), No. 92, pp. 1 6. ISSN: 10726691. URL: http://jd.math.swt.du or http://jd.math.unt.du ftp jd.math.swt.du (login: ftp) LINEAR DELAY DIFFERENTIAL
More informationElements of Statistical Thermodynamics
24 Elmnts of Statistical Thrmodynamics Statistical thrmodynamics is a branch of knowldg that has its own postulats and tchniqus. W do not attmpt to giv hr vn an introduction to th fild. In this chaptr,
More informationEXPONENTIAL ENTROPY ON INTUITIONISTIC FUZZY SETS
K Y B E R N E T I K A V O L U M E 4 9 0 3, N U M B E R, P A G E S 4 7 EXPONENTIAL ENTROPY ON INTUITIONISTIC FUZZY SETS Rajkumar Vrma and Bhu Dv Sharma In th prsnt papr, basd on th concpt of fuzzy ntropy,
More informationWEIGHTED SZEGED INDEX OF GRAPHS
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 23034874, ISSN (o) 23034955 www.imvib.org /JOURNALS / BULLETIN Vo. 8(2018), 1119 DOI: 10.7251/BIMVI1801011P Formr BULLETIN OF THE
More informationBSc Engineering Sciences A. Y. 2017/18 Written exam of the course Mathematical Analysis 2 August 30, x n, ) n 2
BSc Enginring Scincs A. Y. 27/8 Writtn xam of th cours Mathmatical Analysis 2 August, 28. Givn th powr sris + n + n 2 x n, n n dtrmin its radius of convrgnc r, and study th convrgnc for x ±r. By th root
More informationECE602 Exam 1 April 5, You must show ALL of your work for full credit.
ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closdbook. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b
More information4.5 Minimum Spanning Tree. Chapter 4. Greedy Algorithms. Minimum Spanning Tree. Motivating application
1 Chaptr. Minimum panning Tr lids by Kvin Wayn. Copyright 200 ParsonAddison Wsly. All rights rsrvd. *Adjustd by Gang Tan for C33: Algorithms at Boston Collg, Fall 0 Motivating application Minimum panning
More informationEXST Regression Techniques Page 1
EXST704  Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy
More informationWeak Unit Disk and Interval Representation of Graphs
Wak Unit Disk and Intrval Rprsntation o Graphs M. J. Alam, S. G. Kobourov, S. Pupyrv, and J. Toniskottr Dpartmnt o Computr Scinc, Univrsity o Arizona, Tucson, USA Abstract. W study a variant o intrsction
More informationRELATIONS BETWEEN MEDIAN GRAPHS, SEMIMEDIAN GRAPHS AND PARTIAL CUBES
Univrsity of Ljubljana Institut of Mathmatics, Physics an Mchanics Dpartmnt of Mathmatics Jaranska 19, 1000 Ljubljana, Slovnia Prprint sris, Vol. 36 (1998), 612 RELATIONS BETWEEN MEDIAN GRAPHS, SEMIMEDIAN
More informationMinimum Spanning Trees
Minimum Spnning Trs Minimum Spnning Trs Problm A town hs st of houss nd st of rods A rod conncts nd only houss A rod conncting houss u nd v hs rpir cost w(u, v) Gol: Rpir nough (nd no mor) rods such tht:
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
More informationFirst derivative analysis
Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points
More informationWhat is a hereditary algebra?
What is a hrditary algbra? (On Ext 2 and th vanishing of Ext 2 ) Claus Michal Ringl At th Münstr workshop 2011, thr short lcturs wr arrangd in th styl of th rgular column in th Notics of th AMS: What is?
More informationNote If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark.
. (a) Eithr y = or ( 0, ) (b) Whn =, y = ( 0 + ) = 0 = 0 ( + ) = 0 ( )( ) = 0 Eithr = (for possibly abov) or = A 3. Not If th candidat blivs that = 0 solvs to = 0 or givs an tra solution of = 0, thn withhold
More informationIndependent Domination in Line Graphs
Itratoal Joural of Sctfc & Egrg Rsarch Volum 3 Issu 6 Ju1 1 ISSN 95518 Iddt Domato L Grahs M H Muddbhal ad D Basavarajaa Abstract  For ay grah G th l grah L G H s th trscto grah Thus th vrtcs of LG
More informationUNTYPED LAMBDA CALCULUS (II)
1 UNTYPED LAMBDA CALCULUS (II) RECALL: CALLBYVALUE O.S. Basic rul Sarch ruls: (\x.) v [v/x] 1 1 1 1 v v CALLBYVALUE EVALUATION EXAMPLE (\x. x x) (\y. y) x x [\y. y / x] = (\y. y) (\y. y) y [\y. y /
More informationText: WMM, Chapter 5. Sections , ,
Lcturs 6  Continuous Probabilit Distributions Tt: WMM, Chaptr 5. Sctions 6.6.4, 6.66.8, 7.7. In th prvious sction, w introduc som of th common probabilit distribution functions (PDFs) for discrt sampl
More informationThe second condition says that a node α of the tree has exactly n children if the arity of its label is n.
CS 6110 S14 Hanout 2 Proof of Conflunc 27 January 2014 In this supplmntary lctur w prov that th λcalculus is conflunt. This is rsult is u to lonzo Church (1903 1995) an J. arkly Rossr (1907 1989) an is
More informationGEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES. Eduard N. Klenov* RostovonDon, Russia
GEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES Eduard N. Klnov* RostovonDon, Russia Th articl considrs phnomnal gomtry figurs bing th carrirs of valu spctra for th pairs of th rmaining additiv
More informationObserver Bias and Reliability By Xunchi Pu
Obsrvr Bias and Rliability By Xunchi Pu Introduction Clarly all masurmnts or obsrvations nd to b mad as accuratly as possibl and invstigators nd to pay carful attntion to chcking th rliability of thir
More information