Journal of Couter and Matheatal Senes, Vol.66,354-36, June 015 An Internatonal Researh Journal, www.oath-ournal.org ISSN 0976-577 Prnt ISSN 319-8133 Onlne On the Blok-ut Transforaton Grahs B. Basaanagoud and Chetana S. Gal Deartent of Matheats, Karnatak Unersty, Dharwad, INDIA. eal:.asaanagoud@gal.o, hetanagal19@gal.o. Reeed on: June 4, 015 ABSTRACT In ths note, we ntrodue four lok-ut transforaton grahs. We nestgate soe as roertes of lok-ut transforaton grahs suh as order, sze, onnetedness and daeter. 010 Matheats Suet Classfaton: 05C40. Keywords: Blok-ut transforaton grahs; lok grah; tree; dstane; daeter. 1. INTRODUCTION Throughout the aer we only onsder sle grah, wthout solated ertes. The dstane etween two ertes and, denoted y, s the length of the shortest ath etween the ertes and n. The shortest ath s often alled geodes. The daeter of a onneted grah s the length of any longest geodes. A utont of a onneted grah s the one whose reoal nreases the nuer of oonents. A nonsearale grah s onneted, nontral and has no utonts. A lok of a grah s a axal nonsearale sugrah. A lok s alled endlok of a grah f t ontans exatly one utont of. The lok grah of a grah s the grah whose ertex set s set of loks of, n whh two ertes are adaent wheneer the orresondng loks ontan a oon utont of. Ths onet was frst studed y Harary n 3 and further ths was studed y Kull n 6,7,8. For a onneted grah wth loks { } and utonts { }, the lok-utont grah of, denoted y,s defned as the grah hang ertex set, wth two ertes adaent f one orresonds to a lok and the other to a utont and s n. Ths onet was ntrodued n 4. June, 015 Journal of Couter and Matheatal Senes www.oath-ournal.org
355 B. Basaanagoud, et al., J. Co. & Math. S. Vol.66, 354-36 015 Let,, s a lok of and s a utont of denote the ertex set, the edge set, the lok set and the utont set of a grah resetely. The ertes and loks of a grah are alled ts eers. If s a lok of a grah wth ertex set,,..., ;, then we say that the ertex and lok are ndent wth eah other, 1. If two dstnt loks are ndent wth a utont, then they are adaent loks. Ths dea was ntrodued y Kull n 9. We follow 5,11 for unexlaned ternology and notaton.. BLOCK-CUT TRANSFORMATION GRAPHS Insred y the defnton of total transforaton grahs 1 and lok-transforaton grahs, we ntrodue the grah alued funtons, naely, lok-ut transforaton grahs and we defne as follows: Defnton: Let e a grah wth lok set and utont set,, e two eleents of. We say that the assoatty of and s true f they are adaent or ndent n, otherwse false. Let e a -erutaton of the set,. We say that and orresond to the frst ter of f oth and are n and and orresond to the seond ter of f one of and s n and the other n. The lok-ut transforaton grah of s defned on the ertex set. Two ertes and of are oned y an edge f and only f ther assoatty n s onsstent wth the orresondng ter of. Sne there are four dstnt -erutaton of, we otan four lok-ut transforaton grah of naely,,,,. In other words, let e a grah, and, e two arales takng alues or. The lok-ut transforaton grah s the grah hang as the ertex set, and for,, and are adaent n f and only f one of the followng holds:,., are adaent n G f ; and are not adaent n G f.,., are ndent n G f ; and are not ndent n G f. It s surrse to see that s tree. The tree s ntrodued y V. R. Kull and N. S. Annger 10. The tree, usually denoted y, of, whose ertes an e ut n one-toone orresondene wth the set of loks and utonts of n suh a way that two ertes of G are adaent or the orresondng eers of are ndent. The as roertes and haraterzaton of tree and total tree grahs are studed n 1,10. The ertex of orresondng to lok utont of and s refered as lok-ertex utont-ertex. In Fg. 1 lght rles reresents lok-ertes and dark rles exet reresents utont-ertes. June, 015 Journal of Couter and Matheatal Senes www.oath-ournal.org
B. Basaanagoud, et al., J. Co. & Math. S. Vol.66, 354-36 015 356 June, 015 Journal of Couter and Matheatal Senes www.oath-ournal.org Fgure 1 Theore.1 Let e a grah wth ertex set,,...,, lok set,,...,, and utont set =,,...,. Then order of s and sze of = + sze of ] [ = n + sze of = n + sze of, ] [ = n n + where s the nuer of loks to whh ertex olongs and s the nuer of loks to whh utont s ndent. Proof. Note that has lok grah and lok-utont grah as lne dsont ndued sugrahs. Therefore sze of = sze of B + sze of. One an easly erfes that lok grah B has edges. In, and are adaent f and only f s ndent wth and s the nuer of loks to whh ndent. The total nuer of edges etween lok-ertes and utont-
357 B. Basaanagoud, et al., J. Co. & Math. S. Vol.66, 354-36 015 ertes n = sze of =. Hene sze of = + has lok garh as an ndued sugrah. In, and are adaent f and only f s not ndent wth. If s the nuer of loks to whh ndent, then s the nuer of loks to whh s not ndent. Therefore total nuer of edges etween lok-ertes and utont-ertes n s [ n ]. Hene sze of = + [ n ]. Note that has oleent of lok grah and lok-utont grah as lne dsont ndued sugrahs. Therefore sze of = sze of + sze of. Hene sze of n = + has oleent of lok garh as an ndued sugrah. The total nuer of edges etween lok-ertes and utont-ertes n s [ n ]. Hene sze of n = + [ n ]. Reark.1 For any grah, s tral f and only f s a lok. Reark. Let e a onneted grah wth loks. Then f and only f has exatly one utont. Reark.3 For any grah, has solated ertex f and only f has a lok adaent to all other loks of. Reark.4 If, then B B, where and are onneted grahs wth exatly one utont. 3. CONNECTEDNESS OF Theore 3.1 For any gen grah, s onneted f and only f s onneted. Proof. If s onneted, then s onneted. Conersely, f s onneted, then s onneted. For eery utont, there exst a lok, ndent wth. Then and are adaent n. Thus s onneted. June, 015 Journal of Couter and Matheatal Senes www.oath-ournal.org
B. Basaanagoud, et al., J. Co. & Math. S. Vol.66, 354-36 015 358 Theore 3. For any grah, s onneted f and only f has atleast two utonts, where and are onneted grahs wth exatly one utont. Proof. If has atleast two utonts, then we hae the followng three ases: Case 1. Let If and are adaent n, then and are adaent n. Suose and are not adaent. Then onsder followng two suases: Suase 1.1 If there exst a utont lok not ndent adaent wth and, then and are onneted n through. Suase 1. Otherwse and are onneted y a ath,,,, of length four n see Fg.. Fgure Case. Let If and are not ndent wth a oon lok n, then and are onneted n through. Suose and are not ndent wth loks and resetely. Then onsder followng two suases: Suase.1 If and are adaent, then and are onneted y ath of length 3. Suase. Otherwse and are onneted y a ath,,,, of length 4 n see Fg. 3. Fgure 3 Case 3. Let If and are not ndent n, then and are adaent n. Otherwse we hae followng two suases: Suase 3.1 If s not endlok, then there exst a lok, whh s not adaent wth n. Thus and are onneted n through. June, 015 Journal of Couter and Matheatal Senes www.oath-ournal.org
359 B. Basaanagoud, et al., J. Co. & Math. S. Vol.66, 354-36 015 Suase 3. If s endlok, then there exst a lok, whh s adaent to and ndent wth. Consder another lok, whh s adaent to ut not ndent wth n G. Hene and are onneted n y a ath,,, of length three see Fg 3. Fro aoe three ases, eery ar of onts n are onneted. Thus s onneted. Conersely, onsder s onneted. Assue has exatly one utont. Then y Reark., s dsonneted, a ontradton. Thus has atleast two utonts. Assue. Then y Reark.4, B B s dsonneted, a ontradton. Thus, where and are onneted grahs wth exatly one utont. Theore 3.3 For any gen grah, s onneted. Proof. Suose s not onneted. then s not onneted. Oously, s onneted. For eery utont, there exst a lok, ndent wth. By defnton of, and are adaent. Thus s onneted. Suose s onneted. Then we hae the followng two ases. Case 1. If no lok of G s adaent to all other loks of G, then s onneted. For eery utont, there exst a lok a lok, ndent wth. By defnton of, and are adaent. Thus s onneted. Case. If has a lok adaent to all other loks of, then s ndent wth all utonts of. Exet all other loks are onneted n and s adaent to all of. Thus we otan two oonents and for eah utont, there exst a lok a lok, ndent wth. By defnton of, and are adaent. Hene the two oonents are onneted. Thus s onneted. Theore 3.4 For any gen grah, s onneted f and only f no lok of s adaent to all other loks of. Proof. Suose s not onneted. Then s not onneted. Oously, s onneted. For eery utont, there exst a lok, whh s not ndent wth. By defnton of, and are adaent n. Thus s onneted. Suose s onneted and no lok of G s adaent to all other loks of G. Then s onneted. For eah utont, there exst a lok a lok not ndent wth. By defnton of, and are adaent. Thus s onneted. Conersely, suose s onneted. Assue has a lok, adaent to all other loks of. Then y Reark.3, s solated ertex of. Thus s dsonneted, whh s ontradton. Hene no lok of G s adaent to all other loks of G. June, 015 Journal of Couter and Matheatal Senes www.oath-ournal.org
4. DIAMETERS OF B. Basaanagoud, et al., J. Co. & Math. S. Vol.66, 354-36 015 360 In ths seton, we onsder the daeters of. Theore 4.1 If s a onneted grah wth atleast one utont, then 1. Proof. Let e a onneted grah. We onsder the followng two ases: Case 1. Assue has exatly one utont. Then s a olete grah. Hene 1. Case. Assue s a grah wth atleast two utonts. If eah utont of s adaent wth atleast one utont of G, then 1. Otherwse, 1. Theore 4. For any grah wth atleast two utonts. 4,, 1 3, where and are onneted grahs wth exatly one utont and s a onneted grah wth exatly two utonts. Proof. Consder the followng three ases: Case 1. Let Suose and are adaent n. Then, 1. Suose and are not adaent n. Then there exst a utont lok not ndent adaent wth and, then,. In, f and, then there does not exst a utont not ndent wth and and oously,, 4 see Fg.. Case. Let Then fro ase of Theore 3., we hae, f and are not ndent wth a lok. F, 3 f and are not ndent wth adaent loks and resetely. In, there does not exst a lok, whh s not ndent wth and. Oously,, 4 see Fg. 3. Case 3. Let Suose and are not ndent n. Then,. If and are ndent n, then fro Suases 3.1 and 3. of Theore 3., we hae, f s not endlok of., 3 f s an endlok of. Theore 4.3 For any grah wth loks. 1, = 3, June, 015 Journal of Couter and Matheatal Senes www.oath-ournal.org
361 B. Basaanagoud, et al., J. Co. & Math. S. Vol.66, 354-36 015 Proof. If, then and hene 1. Now onsder the followng three ases: Case 1. Let Suose and are not adaent n. Then, 1. If and are adaent n, then there exst a utont lok ndent not adaent wth and, suh that,. Case. Let If and are ndent wth a oon lok B n, then,. Otherwse,,,, 3, where ndent wth s not adaent wth ndent wth. Case 3. Let Suose and are ndent n. Then, 1. If and are not ndent n, then there exst a lok, whh s not adaent wth ut ndent wth, suh that,. Theore 4.4 If no lok of s ndent wth all utonts of, then 3. Proof. If, then and hene 1. Now onsder the followng three ases: Case 1. Let If and are not adaent n, then, 1. Suose and are adaent n and no lok of G s ndent wth all utonts of G. Then there exst a utont lok not ndent not adaent wth and suh that,. Case. Let If there exst a lok, whh s not ndent wth and, then,. Otherwse, 3. Case 3. Let Suose s not ndent wth. Then, 1. Suose s ndent wth and no lok of G s ndent wth all utonts of G. Then there exst a lok not ndent wth and not adaent wth, suh that, 5. ACKNOWLEDGEMENT Ths researh s suorted y UGC-MRP, New Delh, Inda: F. No. 41-784/01 dated : 17-07-01. June, 015 Journal of Couter and Matheatal Senes www.oath-ournal.org
B. Basaanagoud, et al., J. Co. & Math. S. Vol.66, 354-36 015 36 Ths researh s suorted y UGC-UPE Non-NET-Fellowsh, K. U. Dharwad No. KU/Sh/UGC-UPE/014-15/895, dated: 4 No 014. REFERENCES 1. B. Basaanagoud, Charaterzaton of nally nonouterlanar tree grahs, J. Karnatak. Un. S. 4, 9-33 1998.. B. Basaanagoud, Jashr B. Veeragoudar, On the lok-transforaton grahs, grah euatons and daeters, Internatonal Journal of Adanes n Sene and Tehnology,, 6-74 011. 3. F. Harary, A haraterzaton of lok grah, Canad. Math. Bull., 6, 1-6 1963. 4. F. Harary, G. Prns, The lok-utont-tree of a grah, Pul. Math. Dereen 13, 103-107 1966. 5. F. Harary, Grah Theory, Addson-Wesley, Readng, Mass 1969. 6. V.R. Kull, Soe relatons etween lok grahs and nterhange grahs, J. Karnatak Unersty S., 16, 59-6 1971. 7. V.R.Kull, Interhange grahs and lok grahs, J. Karnatak Unersty S., 16, 63-68 1971. 8. V.R.Kull, On lok-utertex trees, nterhange grahs and lok grahs, J. Karnatak Unersty S., 18, 315-30 1973. 9. V.R. Kull, The setotal lok grah and the total-lok grah of a grah, Indan J. Pure Al. Math., 7, 65-630 1976. 10. V. R. Kull, N. S. Annger, The tree and total tree grah of a grah, Vnana Ganga,, 10-3 1989. 11. V.R. Kull, College Grah Theory, Vshwa Internatonal Pulatons, Gularga, Inda 01. 1. B. Wu, J. Meng, Bas roertes of total transforaton grahs, J. Math. Study, 34, 109-116 001. June, 015 Journal of Couter and Matheatal Senes www.oath-ournal.org