Bounds on the Stability Number of a Graph via the Inverse Theta Function

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1 Acta Cyberetica 22 (206) Bouds o the Stability Number of a Graph via the Iverse Theta Fuctio Miklós Ujvári Abstract I the paper we cosider degree, spectral, ad semidefiite bouds o the stability umber of a graph. The bouds are obtaied via reformulatios ad variats of the iverse theta fuctio, a otio recetly itroduced by the author i a previous work. Keywords: stability umber, iverse theta umber Itroductio I this paper we provide several ew descriptios ad variats of the iverse theta fuctio, a otio recetly itroduced by the author (see [0]). We also preset some applicatios i the stable set problem, bouds o the cardiality of a maximum stable set i a graph. We start the paper with describig sadwich theorems o the iverse theta umber ad its predecessor, the theta umber (see [4]). First we fix some otatio. Let N, ad let G = (V (G), E(G)) be a udirected graph, with vertex set V (G) = {,..., }, ad with edge set E(G) {{i, j} : i j}. Let A(G) be the 0- adjacecy matrix of the graph G, that is let { A(G) := (a ij ) {0, } 0, if {i, j} E(G),, where a ij :=, if {i, j} E(G). The complemetary graph G is the graph with adjacecy matrix A(G) := J I A(G), where I is the idetity matrix, ad J deotes the matrix with all elemets equal to oe. The disjoit uio of the graphs G ad G 2 is the graph G + G 2 with adjacecy matrix A(G + G 2 ) := ( A(G ) 0 0 A(G 2 ) ). H-2600 Vác, Szet Jáos utca., Hugary. ujvarim@cs.elte.hu DOI: /actacyb

2 808 Miklós Ujvári Let (δ,..., δ ) be the sum of the row vectors of the adjacecy matrix A(G). The elemets of this vector are the degrees of the vertices of the graph G. We defie similarly the values δ,..., δ i the complemetary graph G istead of G. Let G (resp. µ G ) be the maximum (resp. the arithmetic mea) of the degrees i the graph G. Note that µ G = µ G, µ G+G 2 = µ G + 2 µ G () By Rayleigh s theorem (see [7]) for a symmetric matrix M = M T R the miimum ad maximum eigevalue, λ M resp. Λ M, ca be expressed as λ M = mi u = ut Mu, Λ M = max u = ut Mu. By the Perro-Frobeius theorem (see [6]) for a elemetwise oegative symmetric matrix M = M T R + the maximum is attaied for a oegative uit (eige)vector: we have Λ M = u T Mu for some u R +, u T u =. Furthermore, if M = M T R +, the λ M Λ M. The maximum (resp. miimum) eigevalue of the adjacecy matrix A(G) is deoted by Λ G (resp. λ G ). By Exercise.4 i [5], we have µ G, G Λ G G, µ G ( ). (2) The set of the by real symmetric positive semidefiite matrices will be deoted by S +, that is S + := { M R : M = M T, u T Mu 0 (u R ) }. For example, the Laplacia matrix of the graph G, L(G) := D δ,...,δ A(G) S +. (Here D δ,...,δ deotes the diagoal matrix with diagoal elemets δ,..., δ.) It is well-kow (see [7]), that the followig statemets are equivalet for a symmetric matrix M = (m ij ) R : a) M S+; b) λ M 0; c) M is Gram matrix, that is m ij = vi T v j (i, j =,..., ) for some vectors v,..., v. Furthermore, by Lemma 2. i [9], the set S+ ca be described as ( ) S+ a T i = a j (a i a T j ) m N, a i R m ( i ) a T i a i = ( i ). (3) i,j= The stability umber, α(g), is the maximum cardiality of the (so-called stable) sets S V (G) such that {i, j} S implies {i, j} E(G). The chromatic umber, χ(g), is the miimum umber of stable sets coverig the vertex set V (G). Let us defie a orthoormal represetatio of the graph G (shortly, o.r. of G) as a system of vectors a,..., a R m for some m N, satisfyig a T i a i = (i =,..., ), a T i a j = 0 ({i, j} E(G)).

3 Bouds o the Stability Number of a Graph via the Iverse Theta Fuctio 809 I the semial paper [4] L. Lovász proved the followig result, ow popularly called sadwich theorem, see [2]: α(g) ϑ(g) χ(g), (4) where ϑ(g) is the Lovász umber of the graph G, defied as { } ϑ(g) := if max i (a i a T i ) : a,..., a o.r. of G. The Lovász umber has several equivalet descriptios, see [4]. For example, by (3) ad stadard semidefiite duality theory (see e.g. [8]), it is the commo optimal value of the Slater-regular primal-dual semidefiite programs ad (T P ) mi λ, (T D) max tr (JY ), x ii = λ (i V (G)), x ij = ({i, j} E(G)), X = (x ij ) S +, λ R tr (Y ) =, y ij = 0 ({i, j} E(G)), Y = (y ij ) S +. (Here tr stads for trace.) Reformulatig the program (T D), Lovász derived the followig dual descriptio of the theta umber (Theorem 5 i [4]): { } ϑ(g) = max (b i b T i ) : b,..., b o.r. of G. (5) Aalogously, the iverse theta umber, ι(g), satisfies the iverse sadwich iequalities, 2 /ϑ(g), (α(g)) 2 + α(g) ι(g) ϑ(g), (6) see [0], ad (9) for a extesio. Here the iverse theta umber, defied as { } ι(g) := if (a i a T i ) : a,..., a o.r. of G, equals the commo attaied optimal value of the primal-dual semidefiite programs (T P ) if tr (Z) +, z ij = ({i, j} E(G)), Z = (z ij ) S+, m ii = (i =,..., ), (T D ) sup tr (JM), m ij = 0 ({i, j} E(G)), M = (m ij ) S+. Moreover, rewritig the feasible solutio M of the program (T D ) as the Gram matrix M = (b T i b j) for some vectors b,..., b R m, we obtai the followig

4 80 Miklós Ujvári aalogue of (5): ι(g) = max b T i b j : b,..., b o.r. of G. (7) i,j= The structure of the paper is as follows: I Sectio 2 we will describe a refiemet of (7) ad also several ew descriptios of the iverse theta fuctio (with well-kow aalogues i the theory of the theta fuctio). Some of these results will be applied i Sectio 3, where we preset two ew lower bouds for the stability umber of a graph, ad examie their additivity properties. Fially, i Sectio 4 we study three variats of the iverse theta fuctio, ad derive further bouds i the stable set problem. 2 New descriptios of ι(g) I this sectio we will describe three reformulatios of the iverse theta umber of a graph G. The results have aalogues i the theory of the theta fuctio, which we will metio i chroological order. Let us deote by A G the followig set of matrices: A G := A = (a ij) R a ii = 0 (i =,..., ), a ij = 0 ({i, j} E(G)), a ij = a ji ({i, j} E(G)). We will describe bouds for the miimum eigevalue λ A with A A G. First, we have for A A G the lower bouds λ A Λ A Λ G max a ij, (8) i,j by Rayleigh s theorem ad the Perro-Frobeius theorem. (Here A R deotes the elemetwise maximum of the matrices A ad A.) O the other had, usig a equivalet form of the reformulatio ϑ(g) = max Λ M m ii = (i =,..., ), m ij = 0 ({i, j} E(G)), M = (m ij ) S + (see for example [2], [0]), L. Lovász proved i Theorem 6 of [4] the upper boud λ A, Λ A ϑ(g) (A A G). (9) Aalogously, as a cosequece of the ext theorem, we have also the upper boud λ A tr (JA) ι(g) (A A G). (0)

5 Bouds o the Stability Number of a Graph via the Iverse Theta Fuctio 8 (Note that by Rayleigh s theorem tr (JA) Λ A, ad by the iverse sadwich theorem ι(g) (ϑ(g) ) so there is o obvious domiace relatio betwee the bouds i (9) ad (0).) Theorem 2.. The program (P ) : sup + has attaied optimal value ι(g). Proof. The variable trasformatios tr (JA), {A A G λ A M A := I + λ A A, A M := M I show that programs (T D ) ad (P ) are equivalet: if A ad M are feasible solutios of (P ) ad (T D ), respectively, the M A ad A M are feasible solutios of the other program such that betwee the correspodig values the iequalities tr (JM A ) + tr (JA), + tr (JA M ) tr (JM) λ A λ AM hold. Hece, the two programs have the same (attaied) optimal value. A differet approach leads to aother descriptio of the iverse theta umber. Karger, Motwai, ad Suda proved the reformulatio ϑ(g) = mi ν ii = (i =,..., ), ij = ν ({i, j} E(G)), N = ( ij ) S+, ν R ad used a variat of this theorem i their graph colourig algorithm. (See [3] for a summary of related results.) By the iverse sadwich theorem we have the lower boud ϑ(g) ι(g) ; () we will show that this latter value ca be obtaied as the optimal value of a semidefiite program, too. Let us cosider the primal-dual semidefiite programs (P 2 ) : sup tr B, (D 2 ) : if γ,, b b ii = 0 (i = 2,..., ), b ij = 0 ({i, j} E(G)), tr ((J I)B) =, B = (b ij ) S +, tr C =, c ij = γ ({i, j} E(G)), C = (c ij ) S +, γ R. The programs have commo attaied optimal value by stadard semidefiite duality theory, see for example [8].

6 82 Miklós Ujvári Theorem 2.2. The programs (P 2 ) ad (D 2 ) have (commo attaied) optimal value /( ι(g)). Proof. Similarly as i the proof of Theorem 2., the variable trasformatios M B := tr B B, B M := tr (JM) M show the equivalece of programs (P 2 ) ad /( (T D )), where the latter program ca be obtaied from (T D ) formally exchagig its value fuctio tr (JM) for /( tr (JM)) ad addig the extra costrait tr (JM) >. It is left to the reader to prove that the program if Λ R, tr R =, r ij = ({i, j} E(G)) R = (r ij ) = (r ji ) R is equivalet with both (D 2 ) ad /( (T P )). Now, we tur to the third descriptio of the iverse theta umber. We will use the followig lemma, a slight modificatio of (7). Lemma 2.. For ay graph G, ι(g) = sup i,j= with e deotig the vector (, 0,..., 0) T. ˆbT i ˆbj ˆb,..., ˆb o.r. of G (ˆb i ) = e T ˆb i > 0 for i =,..., Proof. Let (b i ) be a orthoormal represetatio of G such that ι(g) = b T i b j i,j= (that is a optimal solutio i (7)). For 0 < ε <, let us defie a orthoormal represetatio (ˆb i (ε)) of G the followig way: ( ) ε2 O (ˆb i (ε)) :=, εb,..., εb where O R is a orthogoal matrix satisfyig e T O > 0. Note that the e Tˆb i (ε) > 0 holds for all i. O the other had, it ca easily be verified that ˆbT i (ε)ˆb j (ε) ι(g) (ε ). i,j=,

7 Bouds o the Stability Number of a Graph via the Iverse Theta Fuctio 83 Hece, we have proved ι(g) sup i,j= which is the otrivial part of the lemma. ˆbT i ˆbj ˆb,..., ˆb o.r. of G e T ˆb i > 0 for i =,..., Applyig the variable trasformatio described i (3) to the program i Lemma 2., as a immediate cosequece we obtai a aalogue of Theorem 2.2 i [9]., Theorem 2.3. The optimal value of the program { d ij + (P 3 ) : sup (dii + ) (d jj + ), dij = ({i, j} E(G)), D = (d ij ) S+ equals ι(g). i,j= We will apply Theorem 2.3 i the ext sectio for obtaiig lower bouds i the stable set problem. 3 Lower bouds o α(g) I this sectio we will describe two lower bouds o the stability umber of a graph G, ad examie their additivity properties. Note that the Z := L(G), Z 2 := Λ G I A(G) feasible solutios i (T P ) give the iequalities ι(g) (µ G + ), (Λ G + ). (2) By Exercises.20 ad.4 i [5], we have χ(g) Λ G + µ G ( ) +, µ G Λ G. O the other had, easy calculatio verifies µg ( ) + (µ G + ). Hece, we have besides (2) also χ(g) (µ G + ) (Λ G + ). (3) O the dual side istead of ι(g), χ(g) we ca approximate ι(g)/, α(g). Note that D := L(G), D 2 := Λ G I A(G) are feasible solutios of the program (P 3 ) i Theorem 2.3. This fact implies the versio of the followig theorem, where α(g) is exchaged for ι(g)/. (For aalogous results with ϑ(g), see [9].)

8 84 Miklós Ujvári Theorem 3.. For ay graph G, a) b) α (G) := + {i,j} E(G) α (G) := + 2/ (δi + ) (δ j + ) α(g); µ G Λ G + α(g). Proof. By Exercise.4 i [5] we have µ G Λ G. Usig this relatio it is immediate that Λ G + α (G) µ G +. (4) We will show that the iequalities µ G + α (G) hold also, from which the theorem follows, as δ i + (5) α(g) (6) δ i + by the Caro-Wei theorem (see [], or for aother proof [9]). First, usig the obvious iequality we obtai 2 δi + δ j + δ i + + δ j +, (7) α (G) + = δ i +. O the other had, we will verify the relatio α (G) δ i + ( δ i) µ G +. (8) Usig the arithmetic mea-harmoic mea iequality, it is easy to show that α (G) + 4 δ i + + δ j + {i,j} E(G) + (µ G )2 / {i,j} E(G) (δ i + + δ j + ).

9 Bouds o the Stability Number of a Graph via the Iverse Theta Fuctio 85 Hece, to prove (8), it is eough to verify that µ G (µ G + ) {i,j} E(G) holds. This iequality ca be rewritte as ( δ i ) (δ i + ) (δ i + + δ j + ) (δ i + )( δ i ), ad thus is a cosequece of the Cauchy-Schwarz iequality. The proof of (8) is complete, as well. The followig theorem describes additivity properties of the bouds α, α. (For additivity properties of ϑ(g), see Sectios 8, 9 i [2].) Theorem 3.2. With the lower bouds l = α, α we have a) l(g + G 2 ) l(g ) + l(g 2 ), b) l(g + G 2 ) max{l(g ), l(g 2 )}, for ay graphs G, G 2. Proof. Case : l = α. a) Rewritig the statemet, we have to verify i V (G ), j V (G 2) 2 (δi + )(δ j + ) α (G ) 2 + α (G 2 ), that is (without loss of geerality assumig G = G 2 = G) ( δi + ) 2 α (G). I other words, we have to prove the iequality δ i + + {i,j} E(G) which follows immediately applyig (7). b) is obvious, as 2 (δi + )(δ j + ), hold. α (G + G 2 ) α (G ) + α (G 2 ) max{α (G ), α (G 2 )}

10 86 Miklós Ujvári Case 2: l = α. By Rayleigh s theorem the formulas Λ G+G 2 max{λ G, Λ G2 }, Λ G+G 2 max{λ G, Λ G2 } hold. The statemets a) ad b), respectively, are straightforward cosequeces of these iequalities, after applyig (): For example, a) ca be reduced this way to the iequality 2 µ G + µ G2 + 2 max{λ G, Λ G2 }, which holds true, as µ G Λ G for ay graph G, by Exercise.4 i [5]. Additivity properties of a lower boud o the stability umber ca be applied for stregtheig the boud if the give graph or its complemeter is ot coected. I fact, if G = G + G 2 (or G = H + H 2 ) with some graphs G, G 2 (H, H 2 ), the α(g) is equal to α(g ) + α(g 2 ) (max{α(h ), α(h 2 )}). Hece, both l(g) ad the, by additivity stroger, boud l(g ) + l(g 2 ) (max{l(h ), l(h 2 )}) are lower bouds o α(g). It is left to the reader to adapt this boud-stregtheig method to upper bouds u(g) o the chromatic umber χ(g). Summarizig, the so-called weak sadwich theorems (see [9]) ivolve the bouds l(g) α(g) χ(g) u(g) l(g) = α (G), α (G), u(g) := (µ G + ), (Λ G + ) i iverse theta umber theory. I the ext sectio we tur to the iverse sadwich theorem ad its stregtheed versio. 4 Upper bouds o α(g) I this sectio we itroduce three variats of the iverse theta umber. costitute bouds for the stability umbers of G ad G. They

11 Bouds o the Stability Number of a Graph via the Iverse Theta Fuctio 87 First, let us derive a boud from the origial versio ι(g). Let S V (G) be a stable set with cardiality #S = α(g), ad let ε > 0. Let us defie the matrix Z = Z(ε) R the followig way: let Z := (z ij ), where z ij := ε( #S) + 0, if i, j S, /ε + ( #S ), if i = j S, 0 + ( ), if i, j S, i j, ( ) + 0 otherwise. It ca easily be verified usig Schur complemets (see [6]) that Z S+. (This statemet holds eve without addig the secod terms i the defiitio of the elemets z ij.) For ε = / #S the value of Z i (T P ) satisfies tr (Z) + = As this value is at least ι(g), so we obtaied Propositio 4.. For ay graph G, we have ι(g) ( α(g) + α(g)) 2. ( α(g) + α(g)) 2, (9) i other words α(g) 4 ( ( ι(g)) )2 (20) holds. We remark that the upper boud i (20) is betwee the values + ι(g), + ι(g) as it ca easily be verified. Propositio 4. allows a stregtheig: a ι, ι + exchage i (20), where we add the z ij costraits i (T P ). Let us deote by ι + (G) the commo attaied optimal value of the Slater regular primal-dual semidefiite programs z + (P + ) : if + tr Z + ij = ({i, j} E(G)),, z + ij ({i, j} E(G)), Z + = (z + ij ) S +, m + (D + ) : sup tr (JM + ii = (i =,..., ), ), m + ij 0 ({i, j} E(G)), M + = (m + ij ) S +. (Stadard semidefiite duality theory ca be foud for example i [8].)

12 88 Miklós Ujvári Theorem 4.. For ay graph G, ( holds. α(g) ( ι (G)) )2 + (2) For a aalogue i theta fuctio theory, see the results of Szegedy ad Meurdesoif cocerig the variat ϑ + (G) := sup tr (JY tr Y + =, + ) y + ij 0 ({i, j} E(G)), Y + = (y + ij ), S + the relatios ϑ(g) ϑ + (G) χ(g), e.g. i [3]. Now, we tur to the lower variats of ι(g). Let us cosider the primal-dual semidefiite programs { z (P ) : if + tr Z, ij ({i, j} E(G)), Z = (z ij ) S +, m (D ) : sup tr (JM ii = (i =,..., ), ), m ij = 0 ({i, j} E(G)), M = (m ij ) S + R +. The programs have commo attaied optimal value by stadard semidefiite duality theory (see for example [8]), we will deote this value by ι (G). Obviously, ι (G) ϑ (G), where ϑ (G) is a sharpeig of the theta umber, due to McEliece, Rodemich, Rumsey, ad Schrijver (α(g) ϑ (G) ϑ(g), see for example [3]), defied as ϑ (G) := sup tr (JY tr Y =, ) y ij = 0 ({i, j} E(G)), Y = (y ij ) S + R + Besides the metioed relatios. ϑ (G) ι (G)/, α(g), (22) we have also ( + ) 4(ι 2 (G) ) + ι (G)/, α(g) (23) as the followig theorem shows. (For aalogous results with ι(g), see [0].) Theorem 4.2. For ay graph G, we have i other words holds. ι (G) α(g) 2 + α(g), α(g) 2 ( + ) 4(ι (G) ) +

13 Bouds o the Stability Number of a Graph via the Iverse Theta Fuctio 89 Proof. Let S be a stable set i G with cardiality #S = α(g). Let us defie the matrix M := (m ij ) R the followig way: let m ij := if i, j S or i = j, ad let m ij := 0 otherwise. The, the matrix M is a feasible solutio of the program (D ) with correspodig value Hece, the statemet follows. The boud i Theorem 4.2 implies (#S) 2 + (#S) ι (G). α(g) ι (G), (24) ad also, by ι (G) ι(g), the relatios α(g) ( + ) 4(ι(G) ) + ι(g) (25) 2 from [0]. It is a ope problem whether ay of these bouds ca be less tha ϑ(g) or eve ϑ (G) for some graphs. We metio a related result, see also Theorems 3 ad 6 i [4] ad Propositio 2. i [0], where the bouds i (26) appear as lower ad upper bouds for ϑ(g) ad ι(g), respectively. Propositio 4.2. For ay graph G, the iequalities + Λ ( G + µ ) G, Λ G + (µ G + ) (26) λ G λ G hold. Proof. By (2), it suffices to prove (26) after substitutig µ G with Λ 2 G /( ). The, the first iequality follows by 2 Λ G 2 + ( ΛG 2 ) 2 + 4( ) Λ G λ G (ote that λ G ad Λ G ), the secod iequality is immediate. This fiishes the proof. Fially, we metio aother variat of the iverse theta umber, which leads to a iterestig weak sadwich theorem. Let us defie ι (G) as the commo attaied optimal value of the primal-dual semidefiite programs m (P ) : if tr (JM ii = (i =,..., ), ), m ij = 0 ({i, j} E(G)), M = (m ij ) S +, { z (D ) : sup tr Z ij = ({i, j} E(G)),, Z = (z ij ) S +.

14 820 Miklós Ujvári (See, for example, [8] for stadard semidefiite duality theory.) Theorem 4.3. For ay graph G, the iequalities a) ι (G) α(g), b) ι (G) χ(g) hold. Proof. a) Let us itroduce the otatio if i, j S, i = j, M S := (m ij ) R, where m ij := #S if i, j S, i j, 0 otherwise, for S V (G). Let S,..., S k be a stable set partitio of V (G) such that the cardiality of the idex set {i : #S i 2} is maximal. The, S := k {S i : #S i = } is a stable set i G. Furthermore, the matrix k M Si is feasible i (P ) with correspodig value #S α(g), which completes the proof of statemet a). b) Let S,..., S l be disjoit stable sets i G coverig the vertex set V (G), where l := χ(g). The, there exist o-edges e pq E(G) betwee S p ad S q for each p < q l. Let us defie a symmetric matrix M R by writig i it: o diagoal positios, /(l ) o the positios correspodig to e pq, ad 0 otherwise. By Gerschgori s disc theorem (see [7]) the matrix M is positive semidefiite, a feasible solutio of the program (P ) with correspodig value χ(g). This fiishes the proof of statemet b), too. Summarizig, i this sectio we obtaied the (α(g)) 2 + α(g) ι (G) ι (G) ι(g) ι + (G) ( ι + (G) α(g) + α(g) iverse sadwich theorem as a aalogue of Lovász s sadwich theorem. I the same cotext we metio also the well-kow ) 2 χ(g) ν(g) + α(g) 2 (27)

15 Bouds o the Stability Number of a Graph via the Iverse Theta Fuctio 82 sadwich theorem, where ν(g) deotes the matchig umber of G, that is the largest umber of pairwise disjoit edges i E(G), see Sectio 7 i [5]. This fact, together with the formulas α(g) χ(g), χ(g) + χ(g) + (28) (see Exercise 9.5 i [5]), makes upper bouds for α(g) particularly useful i derivig other (upper ad lower) bouds for α(g), χ(g), for example ad via the sadwich theorem. α(g) 2ϑ(G), + ϑ(g) χ(g) + ϑ(g), + ϑ(g) 2 (29) (30) 5 Coclusio I the paper we studied the iverse theta fuctio: results aalogous to sadwich theorems ad their stregtheed versios from the theory of Lovász s theta umber were derived, based o ew descriptios of the iverse theta umber. Whether the ew bouds o the stability umber ca be tighter tha already kow oes remaied a partly udecided questio. Ackowledgemets I thak oe of the aoymous referees of the paper for callig my attetio to several relevat refereces. Refereces [] Alo, N. ad Specer, J. The Probabilistic Method. 4th editio, The probabilistic les after Chapter 6, Wiley, page 00, Hoboke NJ, 206. [2] Kuth, D. The sadwich theorem. Electroic Joural of Combiatorics, : 48, 994. [3] Lauret, M. ad Redl, F. Semidefiite programmig ad iteger programmig. I: Aardal, K. et al., editors., Hadbook o Discrete Optimizatio, Elsevier B.V., Amsterdam, pages , [4] Lovász, L. O the Shao capacity of a graph. IEEE Trasactios o Iformatio Theory, IT-25(): 7, 979.

16 822 Miklós Ujvári [5] Lovász, L. Combiatorial Problems ad Exercises. Corrected reprit of the 993 secod editio, AMS Chelsea Publishig, Providece, RI, [6] Serre, D. Matrices, Theory ad Applicatios. Traslated from the 200 Frech origial, Graduate Texts i Mathematics vol. 26, Spriger-Verlag, New York, [7] Strag, G. Liear Algebra ad its Applicatios. Academic Press, New York, 980. [8] Ujvári, M. A ote o the graph-bisectio problem. Pure Mathematics ad Applicatios 2():9 30, [9] Ujvári, M. New descriptios of the Lovász umber, ad the weak sadwich theorem. Acta Cyberetica, 20(4):499 53, 202. [0] Ujvári, M. Applicatios of the iverse theta umber i stable set problems. Acta Cyberetica, 2(3):48 494, 204. Received 26th October 204