On the Block-cut Transformation Graphs

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1 Journal of Couter and Matheatal Senes, Vol.66,354-36, June 015 An Internatonal Researh Journal, ISSN Prnt ISSN Onlne On the Blok-ut Transforaton Grahs B. Basaanagoud and Chetana S. Gal Deartent of Matheats, Karnatak Unersty, Dharwad, INDIA. 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

2 355 B. Basaanagoud, et al., J. Co. & Math. S. Vol.66, 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

3 B. Basaanagoud, et al., J. Co. & Math. S. Vol.66, June, 015 Journal of Couter and Matheatal Senes 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-

4 357 B. Basaanagoud, et al., J. Co. & Math. S. Vol.66, 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

5 B. Basaanagoud, et al., J. Co. & Math. S. Vol.66, 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

6 359 B. Basaanagoud, et al., J. Co. & Math. S. Vol.66, 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

7 4. DIAMETERS OF B. Basaanagoud, et al., J. Co. & Math. S. Vol.66, 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

8 361 B. Basaanagoud, et al., J. Co. & Math. S. Vol.66, 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 /01 dated : June, 015 Journal of Couter and Matheatal Senes

9 B. Basaanagoud, et al., J. Co. & Math. S. Vol.66, 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, B. Basaanagoud, Jashr B. Veeragoudar, On the lok-transforaton grahs, grah euatons and daeters, Internatonal Journal of Adanes n Sene and Tehnology,, F. Harary, A haraterzaton of lok grah, Canad. Math. Bull., 6, F. Harary, G. Prns, The lok-utont-tree of a grah, Pul. Math. Dereen 13, F. Harary, Grah Theory, Addson-Wesley, Readng, Mass V.R. Kull, Soe relatons etween lok grahs and nterhange grahs, J. Karnatak Unersty S., 16, V.R.Kull, Interhange grahs and lok grahs, J. Karnatak Unersty S., 16, V.R.Kull, On lok-utertex trees, nterhange grahs and lok grahs, J. Karnatak Unersty S., 18, V.R. Kull, The setotal lok grah and the total-lok grah of a grah, Indan J. Pure Al. Math., 7, V. R. Kull, N. S. Annger, The tree and total tree grah of a grah, Vnana Ganga,, V.R. Kull, College Grah Theory, Vshwa Internatonal Pulatons, Gularga, Inda B. Wu, J. Meng, Bas roertes of total transforaton grahs, J. Math. Study, 34, June, 015 Journal of Couter and Matheatal Senes