set is a maximal in - ScienceDirect

set X of vertices of G is an independent dominating set if no two vertices of X are tex nut in X is adjacent to at lest omtevertex in X of G arc cliques of the complement G of Q;_and conver concealed with the existence of disjoint independent d A new wameter, the maximum number of disjoin kdependent dominating ied and the class of graphs whose vertex sets p: 4tion into independent ted.

set is a maximal in&

eter, b(G), of a graph G as the maximum number of disjoint independent ominating sets of G. In thjs paper we svjU initiate a stutiy of Phis parameter of giraptls, parto the above ccx~jecture, and we wiH inve::tigtlte the ievcrtiees can Ik partitioned co-npletety into indepenL Such graphs were termed irt&~&~bk in 161. ,m!e number, d(G), of a graph C is the maximum order of a o $aminatiq #sets.In ftil it EAras shown that c(G)., lest clique of G, is a lower lbounrf for the domatic nunslrat Ieither G or c is indominaMe. ari X is an independent dorninat::ng set of G if and on!~ mplement of G, arid hence this paper could be nte$ in terms of cliq WS. which concept :S wrhaps traditikx&ly mare offer, Once our work forms a part of the larger theory of lustratzs the mergi:tg of this theay nwiththat encience and co!curing, ~42prefer to remain with the notion of ndent dominating sets. Several proofs ktowever >re plxsented in / descriytion. rented termin&qgy or notation used in this G), .ivGS)denotes the set of vertice’s which are a4_iacent to

introduction, the paraneter II(G) of a graph G is urn number of Zsjoint ind ndeilt dominating sets i~ G. Our

EJ. Got

ne, S T Hedetniemi i Indepertdant dbminatirrg sets

215

c, provides us with a straightforward equality for &C) Involving the clique (G) of 6 (see [rr, p. xl] ).

)a Filr any

graph

G, b(G

Coroitary 2.1 (b). If G has an isolated vertex, then b(C) =

I.

This simple result thus provides us with our first class of graphs not having two disjoint independent dominating sets. It may be generalised as follows: Theorem 2.2. If G /XV an independent set S such that there are no two disjuint iardependenr sets contained in N(S) whose WI~QLdominates S, hen b(G) = I. hoof. Assume 4(G) 3 2 and let L+ , D2 be diisjoint i Idependent dominating sets.. For ;my independent set S define Ik = Y(S) r-~Dk (k = 1,2). Then I, ,I2 are disjoint independent sets cctntained in N(S). For any s E S, if lk does not dominate {r}, s E D,. !Since III, D, are disjoint, either I, or 1, dominates {s) and I, w Iz domiinates S. Carotbty 2.2. JfGhas an indeperz nit set S of vertices ofdegree 1 such that the subgraph iirduced by N(S) itt G is K, for n 3 3, therr b(G) =;’1. erge has indicated i raphs of degre ar graphs of lar

vate communic

hat he has proved that e now eseablish the result

iarofdegrt)e >p

- 7 > 0, t

Let G kwe sufficient to show that

sof III).

6j vertices and be regular of degree p--- 5. It is hich is regular of degre 3 4, has at least

e following proof, “adjacent”” means adjacent nd (2) wilt denote the subgraph of c inducf,d by vertex set 2:. A has at most five vertices. 5 , then c is the disjoint urticm of this KS and slome of F and the k’, constitute disjoint cliques of i?. maximum clique of c of size! four and W = VIIi?%---- U. arity and since p 2 6, there are at leas! 12 edges in c of which and 3 join U to IV. Hence W) has at least 2 edges. Let X be a ), then 1x12 2. But no vertex of U is adjacent to two vertices clique of c disjoint from U. ? 0 be a maximum clique of C of size 3 and W = V(C) - !f: G is A& with a i-factor removed and has disjoint cliques a triangle X, X Is a clique of c otherwise c has assume M9 has no triangle. IE has at feast 14 edges of which 3 czrein W) and 6 join s zi least 5 edges. There are at most three of these such that for some u E U, {tr, Q,B} is a clique of c. Thus for some , d) and U are disjoint cliques of c. triangle, it clearly has disjoint cliques. This r:omf >

3 and

iw

k=4s

. Let C h;l,vep (3 8) vertices and be regular of degree p .--.7. We that c which is regular of degree 6 has tw3 disjoint cliques. :st clique in c, W = V(c) -- U and X be a 1~.s~st clique ir

and U are disjoint cliques C. edges from each idE

the

S ver:iceS Of U t0 Vertices in W, there are at most 5 edges

[ W’i, tr:J

in

for sO111e U E U, {U, Wi, My) is a trianlgle Of iTe HeJlCe f(.X nd U are discjoint cliques of G. some I\%‘& ‘t “2 ] f&zE(ch=w, {Wl, wz ) G2sc 4. f Uf = 4, IXi = 2. in this case ( ) has p -- 4 vertices and 3p - ] $ (where [ f denotes edges. Ifp = 8.9, or 10, then 3p-- IS > [)(p--4)2 teger ft;nction). Therefore, by Turan’s theorem [ 8, p. 141, gle contrary to hypothr:~~k. For p G%: 11, (IV) has at least 15 edges w’rlich are cliques of W. Since tkre aYe3 edges joining each u E CI to h’, there are at most three edges [ wi, wi J in W such that {u, wi, u;.) is of i?. Thus there are at most I2 triangles with one side in (IV>. 14~2 1 E EWiO), { aq, w2 } is a clique of c_ which

(IV)

SUCh &h&t

c‘as@5. f UI = 3,fXI = 2. This is similar to Cases 3 and 4, t.e omit the details. Case 6. f Ut = 4,1x1 = 3. As in Case 4, W) as p -- 4 vertices and 312- fi8 edges. If p = 8, W = Kd and if p = 9, (W>= 5 with an edge remove:& Each of these contradict fXi = 3. Ifp = k 0, W has 6 vertices and 12 edges. Since W) contains no K4 q one may show that W it equal to K, with a l-factor removed, which has 8 triangles. There are :,I most 4 triangks CM+, M>,k+ ) in W for which there exist ti E U and {zd,uljt We,wk ) is a clique of CZ.Hence for some triangle { wl, I-+ w3 ) iii Of), -:wl, W2.W3l and 61are disjoint cliques of %. Thus p :Z 11 and (I+‘)has at least 15 edges. As in Ca,se 4, we deduce that 2 f of OV) does not form a triangie with any vertex; of U. sume edge ( s a clique of c disjoint from U, or for some w3 E W, Either {We, b’, * \4’?, w3 ) is a clique of c disjoint from U. Case; ! to 6 exhaust the possibilities of (1). This completes the proof. Yet another class of regular graphs satisfying the the cltas:;of star poi ygons Qcf.

erge conjecture is

’

.

Ed

we define

this secti~oinwith a conjecture of CM own, which is weaker ofijecture but still seems difficula to settle.

h G & indomitable if its vertex set may be partitkxxd irlto indendent dominating sets. Iln this section we list some: classes of indcrmins. St is a simple mattes to see that the followirrg classes are inphs alnd their complements.

sses of indominable graphs, ( f ) FWxJbes,Qn [ 8, v*23 J .

(g) k-trees [ I, I?.ZOO] (Ii) F& which is equal to K,, with a l-fac.or removed. In the thesry of cc&xing, a graph (3 is critical if x(G) = n, yet for each h G’ uf G, x(G’) < IL One is therefore led to define if d(G) =: k but fur every edge x of G, &G-x) < k. date virtually no work has been ddne on &critical grap s, but they are of same inlrerest here because sf thi following result. l

opasition 3 .2. lj’ G is a d+Wcizl _graph,then 6 is indominable.

oof.Let

D,, D, be a partition of V(G) into dominating sets. If contains an edge x, then d(G-.x) = d( ) contrary to hypothesis. Hence, .,.I

each L?, is an independent set, as asserted. In [6] 9 it was shown that for all graphs G, d(G) 6 6(C) + was termed (&ma&~~ll~~)ftdl if d(G) attained this kund. tion Z&3.If G is pegular

1

and a graph

and furl/, then @ is itrdominable.

oaf. Suppox that G is regular of degree k and full, i.e., there exists a partitiqn D,, .. . . Dk+t of V(G) int9 dominating sets. Consider any 24E Di, i = 1 r . . . . k + I. Since each Di dominates {u), each Di must contain an element of {u) u N(u). But i {TV} u .N(u)~ = k + 1. Hence each Di contains exactly one element of (tl) U N(tc). But u E Di, therefore N(tcl n Di = $8and we deduce that Di is an independent set for each i = 1 q . . . . k f 1. Hence G is indominable.

t success in locating non-indoriainable graghs and the class. If G satisfies any of tile following contains an isolated vertex, G # I&.

condition i is a non-indsminable ISlargest partition into dominating am! m&lest parrtitisn into independent

(b) which are not indamaph for wSrich x(G, ) = as {l,4,7), {2,5,8), sets { 1 92,3}, {4,S, 61,

u) GM-a graph G2 with 12 vertices which has G2 has Ia est dominating parti-

tion (t,2,3), C4,5,6L I7,8,9), {lQ,lk,12) and smallesltin partition ( 1 ,4,7,lW, CL~,l~~I, 0,6,9,12). h[6] it was shown that although e inequality c(G) & d(G) does not always RoP r an arbitrary graph C, it does hold for indominable graphs; in fact: is dndcminabk then c(G) < d(G). he next result shows that the joi indominable property.

operation on graphs

sition 3.6. FW ant* m two gruphsG, and G2, G, and G2 are hdorrtirlabbe if and mly if G, + G2 is indmniraable. On the other hand one can easily construct examples which show that if 6, and G2 are indominable then nei her G, LJG2 (the disjoint union of G, and G,), nor G, - G2 (graphs 6, and (;, joi ed by a bridge), nor G1 *G2 (graphs G, and G2 joined by a cut vertex) n d be indominable. A graph G can also maintain iits sndominability if I Mces are identified which belong ~:othe same subset of an indominable ]rartition of G. reverse is also true. opositicsn 3.7. IfG is int&mhrrble md

is a$!independent set of new vtvtices, em% laf which is @ined to at least me element in every set of an independqn t dumintz&qg yattitim of G, the esu8tinggraph is indomincble. We end’the paper with several open questio s for future r tuitively, one feels that the edge-connectivity, Ik(G), of a gra be related to d(G).

e

G is in

esearchCouncil

C., Canad.a,far its h to thank the referees for their constmctive criticisms.

rt, The number of l&eL:d Mimensionai

trees, J. Combin. Theory 6

s N-xth-Hotiand, Amsttgpdam, Ik 973). , A gen rd class of invulnerable graphs, Melwotrks 2 (1932) UEI,Uniquefy colorable p&m-ugraphs, 3. Combin. Theory 6 ! 1969)

edetnicmi, Endepetience gaphs, Pro+:.5th SE Confkesrce on Corn , Boca Raton, Florida ( 1974 :. mi, Towards P theory of domjpatiow irr graphs, Networks, nterpufation systems, Pzoc. 3rd SE. Conference on ry and Computing, Boca Raton, Florida, Huffman, Levwv and

Theory (Addison-We&y, e&ding, Mass., 1959). e?n kmi and 8. Robinson, Wquely colorable graphs, J. Combin. Theory 6