on characterizations of the middle graphs

With author's compliment

ON CHARACTERIZATIONS OF THE MIDDLE GRAPHS BY

JrN AKIYAMA,TAKASHI HAMADA AND lZUMI YOSHIMuRA

Printed from

TRU Mathemalics, Vol.

ll,

1975

ON CHARACTERIZATIONS OF THE MIDDLE GRAPHS BY

JnI AKIYAMA, Tersru HAMADA

Abstrsct

The middle graph

.cND lzurvg

YOSHIMURA

M(A

of a graph G has been introduced by T. Hamada and 1. Yoshimura in [4]. S. B. Chikkodimath and E. Sampathkumar also studied it independ.ently, and they called it the semitotal graph ft(G) of a graph G in [3]. We define a graph M(G) as an intersection graph g(F) on the point set ,/(G) of any graph G. I-et X(G) be the line set (each line taken as a set of two points) of G and F :

Y'(G) U .X(G), where Y'(@ indicates the family of all one poiot subsets of the set Z(G;. Let M(O : O(n. M(G) is called the middle graph of G. In this paper.we study the characterizations of middle graphs of any graph (Theorem 1) by using Krausz's Theorem [5, Theorem 8.4(2)], [, Theorem l], characterization of the middle graph of a tree (Theorem 2) by using Chartrand's Theorem [5, Theorem 8.5], and characterization of the middle eraph of a complete graph (Theorem 4) by using characterization of the linc graph of a complete graph (fheorem 3).

1.

Introducfion In this paper, we shall consider a graph as finite, undirected, with All definitions trot presented here can be found in [5].

single lines and no loops.

1.

DrrrNrtox l*ta1, uz, . . . . .rupixurxz, . . . . .,xqdenotethepointsandlines ,of a Qr, C) gaph G respectively. We define the middle graph of G, denoted M(G), as an in:tersection graph O(F) on Z(G) where F Y'(Q U X(q, Y'(G) {{ui luz} ,

{url}, X(C)

=

lxy xz,. . . . .,xc}.

:

=

,

2.

Let G be any (p, C) gaph arid Y(G), X(G) bc the point set of G and the line set of G respectively. We add to Gp points vr andp lines {ur, ur} (i: 7,2, . . ., p), 'where yr are different from any point of G and from each other. Tlen we obtain a (2p, p * .c) graph which consists of the point set y(G+) , , ., up, yL, . . ., vpl and the line

DrrnrmoN

-

lur,

setX(G+): {xt,...,xq, {zr,vr}, ...,{up, ve}}.LctusdenotethisgraphbyG+and ,call this the endline grap,ft of a graph G.

From the Definitions above, following Propositions

1

and 2 are e4qfly verified. i

Pnoposrnox l. Let G+,II+ be the cndline graphs of grapls G, H respectively. Then G+ .and H+ are isomorphic,i,e., G+ H+, if and only if G and H are isomorplric, i,e,, G = H.

=

■Received June l,1975

36

J. AKIYAMA, T. HAMADA, AND I. YOSHIMURA

PnoposrrioN 2. A gr@h G is an endline graph if and only if each point of G is incidenr tvith one and only one endline.

we state here three theorems which shall be used to prove our theorems, Tnroneu A. A graph G is a line graph if and only if its lines can be partitioned into complete subgraphs in such a way that no point lies in more than two of these subgraphs 15, Theorem B.4(2)1,11, Theorem ll.

B.

TsronEv

Let G be any graph and G+ be the endline graph of G (see Definition 2). M(G), where L(G+) is the line graph of G+ 14, Theorem 4).

Then we have, L(G+)

=

Tsronrrq C. A graph is the line graph of a tree if and only if it in which eaclt cutpoint is on exactly two blocks15, Theorem 8.5),

2.

is a connected block graph

Characterizations of The Middle Graphs

DrrrnrroN 3. A graph G is called a middle graph if it is isomorphic to the middle $aph M(H) ol some graph IL Our main theorem is as follows: Tireonpt'r I

1.

A graph G is a middle graph if and only

. Its lines can be partitioned into complete subgraphs

if

in such a way that no

point lies in more.

than two of these subgraphs, 2. Each of such subgraphs contains one and only one point which is not contained in any one of other subgraphs.

Pnoop. Let

G be the middle graph of a graph I1, i.e., G

= M(H).BV Theorem B, G is isomorphic to the line graph ol the endline graph 11+ of a graph H, G = L(H*). Hence, G satisfies condition 1, by Theorem A. Furthermore G satisfies condition 2, since each point ol If+ is incident with unique endline. Next, let us prove sufficiency. Assume that G satisfies both conditions 1 and 2. Given a decomposition of the lines of a graph G into complete subgraphs Sr, Sz, . ,, S, satisfying condition 1, the constructions of a graph 11 whose line graph is G, L(H) = 6, is given as the intersection graph as follows:

Let,S:

. . . ., S"} , where Sl (i:7,2, . , .,n),are subgraphs ol the decomposition and let U : .!,Ut, where Ur : {{v}ly e r/(&) and v € v(Si, i*i}. Then the points of If correspond to the members & of family S together with the member {u} of U. Thus ,S U U is the point set of Il and two points are adjacent whenever they have a nonempty intersection, that is, I/ is the intersection eraph O(S U (]), H: A(S U U){Sr, Sz,

The condition 2 implies that every Ur is a singleton. Therefore there is one and only one. endline at each point of If. Hence Il is an endline graph by Proposition 2, i.e., there exists,

ON CHARACrERIZAnONS OF THE MIDDLE GRAPHS a graph F+ such that the proof.

Tnronrrra 1

. It

2.

2.

Il = F+. Thuswehave G = L(H) = L(F+) M(n,this completes =

A graph G is the middle graph of a tree tf and only

if

is a connected block graph in which each cutpoint is on exactly two blocks,

Each block which is a complete subgraph contains one and only one non-cutpoint,

Pnoor.

Suppose that G

= M(T), Tis

some tree. By ThcoremB,M(T)= l(f+), 1+ to the line graph of a tree r+, G satisfies the condition 1 according to Theorem C. Since each non-cutpoint x of G corresponds to one and only one endline at a point of 7+, each block of G containo one and only one non-cutpoint. Next, Iet us prove sufficiency. Suppose that G satisfies both conditions I and 2. From condition I and rheorem c, there exists a tree rsuch that G r,(r). Now we show that = this tree ris an endline graph. That is, there exists a tree I, such that T = T,+.By Theorem C, the tree lwith I(7) = G is given as follows: f : O(^S U U), ,S - {Br, Bz, . . ., Bt l& is a block of G}, is also a tree. As G is isomorphic

U

: (: (Jt, (Jt: f=l

.

{[v] lv

e t4Br) and v g

V(81), (i

+h].

However from condition 2, each Ur consists of one and only one singleton. By these facts

and Proposition2, T is the endline graph of L(T'+) - M(T), and this completes the proof.

g(S)= I'.

Thus, we have

G:L(T):

3. Characterizations of lte Mirldle Graphs of Comptete Graphs Characterization of the line graphs of complete graphs was independently settled by chaog [2] and Hoffman [6], as the following Theorem D; Tnronru

if

D.

unlessp:8,ag:raphGisthelinegraphofacompletegraphKeif andonly

1. G has (!) wints,

2.

i. 4,

G is regular of degree 2(p 2), Every two nonadjacen t points are mutually adjacent ro exactly four points, Every two adiacent points are mutually adjacent to exactly p-2 points 15, Theorem g,6l-

-

The above Theorem gives the pointwise characterizatio a of L(Kp),we give another characterization of L(.Kr) from the constructive viewpoint which excludes an exceptional case as in Theorem D. THeonrN,r

3.

A graph G is the line graph of Ke tf and only

l,The lines of Gcanbepartitioned into p complete order p- I, in such a way thot 2. Any two distinct such complete subgrophs mon,

ff

subgrophs

K[L1, K]jLl,have

fsiti:1,2,.

one

. .,p)of

andonly one point incom-

38

ェ

AKIYAMA,T.HAMADA,AND I.YOSmMuRA

3.助 ″′ ο レ′ グc燃

"′

χ α ε ″ r"び 詔 ω″″″山8r… 撃 1“ ″K」21. “

PR90F. Suppose that c=zKKp),and denOte the point set Of xЪ

。 Each … .,′ 〕 (メ

by/α ttp)=={1,2,

star対 ち _l atthe pOintF ofる indu∝ s a cOmoldesubgraph弓 と10f二 に,),

=1,2,...,P)o

Shce nO Hne of G Hcs h bo■

4Ll and』 学 ゃ the Hnes of c can bも partidoned htO′ complcte subgraphs OfOrder p‐ -1.Thisleads tO cOndition l.MOreoverpthe Factthat each f K2ヽ

un。

(F,プ )。 contdned hjusttwO dsdnct stars K夕 )_l and K場 _l Ladsto cOnddOn 2.Shce anytwO dゞ hd stars対 _b対 _l have one and Onぃ nc Hne(1ノ )h CO― on, ち ら

cach pOht Of c hes hjusttw0 0Fsuch cOmptte subgraphs x馨 condition 3. sumciency.

l and]攣 1.This tads tO

suppOsc that C satisnes conditions l,2,and 3.

∞ ぬぱ 田 ぱ attLⅧ

GКna“ m"鋼

h∝

1,2,...,′ )Satisfying cOnditiOn l,the gra intersection graph,that is,

冒TFluttFrttf二

″=ρ ∪び),where κ={石μュ￨′ =1,2,… .,〃 },υ i、1)andッ ∈ 7● %),(ノ ≠ノ)}. ={磁 =1,2,… .,″ 磁 ={{ツ “}￨ッ ∈ け However by cOndition 3;we see that each sct r/t has nO elements.That is,″ =o(κ ),κ = {ろ 井11ノ =1,2,...,′ 〕 ;″ Oonsists of′ pOints`μ l(′ =1,2,...,p)and any two ￨′

pdmS 41,製 1,a“

attace■ by cond“ On

ThuS 27 is a cOmpletc graph馬

}

2

,such that z(⊃

completes Itお easy tO venfy that TheOrem D andTheOrem 3 are=G.This equivdent untss′

the pr00■

=8.

By using TheOrem 3,we obtain the fol10wing Theorem 4 which gives characterization of the middlё graph iイ 区 ,)Of a complete graphる 。

THEOREM 4. ン grapヵ cぉ 所′″J″二 ″ カグ αω ″ ,た た g″″ ヵKp r“ ″。″夕 J `g″ ′ .C aοぉぉ rsぽ′′ ο ″おガ ■g″ ―′″″c)′ ο ″rs″ ″ ヵ&gree 2o_f), `′ `力 2.ル′rirω グc caz b′ ′″′ ′ ′ ′ 0″ ″ル ゎ′ω切ル″sttgrapな ら 0(′ =f,2,:..′ ″∫ ε ヵ 勁 ″

α″の ″αち

i,筋 :‰

″ ′ α ″ ″ ο ″ ο ″ ′ ο ″滋 ″ ケ ο ′ ろ lh‰〃競鶴罵fl力 “° 00″

73a

α ″ ″0″ クο ″ ″r″ ノ ″α響″ りο

`′

PROor・

′

SuppOSe c笙

″

(6).By Theorem B,c=z(4)。

_′ 滋G.

Denote endpoints ofる

′ byム る has pキ c) Ines and each Ofp endlhesお inddenttop tt Hines,andcachOfOtherc)Httes k incident by′

(ノ

=1,2,...,′ ),and

a point"hici is attacent to the point′

to 2o-1)lines.By these facts,∞ ndition (′

l f0110ws.

Since the lines in star`l♭ Of』 け at the pOint Findu∝ =1,2,。 ..,′ ),COndition 2 fouows.

_..

a cOmplete sibgraph K♂

。 fC,

There clsts One and Only Onelneinκ Which is incident tO bOth POintS F andノ ょ .Hence,

ON CHARACrEPATIONS OF THE MIDDLE CRAPHS

39

conddon 3 foHows.MOreover,cach′ of石 iS attaCent tO One and Only one endpoint F, 吉 and each endline{ち sumciellcy.

′ つiS

incident toP-1 lines,thus condition 4 follows.

Suppose that C satisfles conditiOns l,2,3,and 4.Bytheoran A,the graph

l with ια7)笙 C is glven as an intersection graph as follows:

″=0(SUυ ),S={Kダ =1,2,… .,pl,び ={研 ￨′ =1,2,… 。 ,′ 〕 υt=〔 〕￨ッ ∈アαりわandツ こア ダ),′ ≠ノ }. Silnce each point r夕 Of O(s∪ び )iS incidenttooneand onlyoneenlhe“ ダ,“ ),″ お “ the endline graph of ρ (o.0(S)is a cOmplete graph κ ,,sO that I=K方 。 ThereFore,G tt zcr)=二 rヵ )=ν に ThiS cOmpletes the proo■ By the Fact that r飾 )笙 ι “。し+1),we Obtain■ e fol10Wing cOrollary 4-1. ￨′

{ツ

b)・

COROLLARY 4-1. Иgr4,■ Gお ″ ′ α′蓼 `ゎ

α ″ ル″ gr″ 力 α ″″ο″ク ノ `ノ “ bgr"hs Kp(`)(′ ρ ″ル″ ′+I "ヵ r c cα ″ba′ ″riFiO″ θ ル c02″ ″躍 `2グ =′ ,2,・ .., `rirω 力α″cン ″″ ′+2)Й s″ ε 2.И ″ ′ ″ο ω たた sttgF″ お κp“ ),る 0(′ ≠Jil力 αたο ″θ″ロ グο ″夕ο″ ″″ο ″′0″ r, "働 ″ `ω 3.И ″ ″ ο Й ′ r a/Cお cο ″″レ iz`χαcrry′ ″。s″ ω″ セ″sぉ 彎 ヵ s KpC)α ″ グκ,(ノ `α `力 `″ (′ ≠ノ “ 『 ′.動

)

)。

珂   朗   朝   切   到 ｑ

REFERENCES

L' w'

Beineke, characterization of dcrived glaphs, J. combinatorial rheory Ser. 89 (1970), 129-135 L'-c' cttgls: The uniqueness and nonu.iqueness ofthe triangular association scheme. Sci.

Record 3 (1959), 604-6lj

s. B. chikkodimath and E. sampathkumar, semitotal Graphs-Il, ..Graph rheory Research

Report", KarDatak University No. 2 (1973), 5_9 T. Hamada and I. yoshimura, Traversability and connectivity of the Middle Graphs of _ 9ppt, Discrete Mathematics, t4 (1976)

a

T.Y11rry, Graph Theory, (Addison-Wesloy, Reading, Mass., 1969) A' J. Hofman, Oo the uniqueness of the tri-angular association scheme. Ann. Math. Statist. 31(1960), 492497

DER T

鶴 躙 亀黒 lECL

KAC謝1驚討野攀藩LN。 IZUMI YOSIIIMuRA TO螂 'WIVERSrrY, WAKANAME,HIRATSUKA,JAPAN.