On $(P_5,bar{P_5})$-sparse graphs and other families

On ( P 5 , P 5 ) -sparse graphs and other families Jean-Lu F ouquet and Jean-Marie V anherpe email : {Jean-Lu.F ouquet, Jean-Marie.V anherp e}univ-orleans.fr W e extend the notion of P 4 -sparse graphs previously in tro dued b y Ho àng in [11 ℄ b y onsidering F -sparse graphs w ere F denotes a nite set of graphs on p v erties. Th us w e obtain some results on ( P 5 , P 5 ) -sparse graphs already kno wn on ( P 5 , P 5 ) -free graphs. Finally w e ompletely desrib e the struture of ( P 5 , P 5 , bul l )-sparse graphs, it follo ws that those graphs ha v e b ounded lique-width. 1. In tro dution P 4 -free graphs, also alled Co gr aphs , w ere designed to b e ompletely deomp osable b y omplemen tation and motiv ated resear hers for studying graph lasses  haraterized with forbidden ongurations. In addition, a n um- b er of optimization problems on a graph an b e redued to their w eigh ted v ersion on the set of prime graphs also alled the set of rep- resen tativ e graphs (reall that the represen- tativ e graph of graph G is obtained from G b y on trating ev ery maximal prop er mo d- ule of G in to a single v ertex)(see [13℄). Th us sub-lasses of P 5 - free graphs w ere in tensiv ely studied (see e.g. [2,3 ,4℄), in partiular F ou- quet in [7℄ onsider ( P 5 , P 5 ) -free graphs and the sub lass of ( P 5 , P 5 , B ul l ) -free graphs (see Figure 1). Later Giak oumakis and Rusu [10℄ pro vide eien t solutions for some optimiza- tion problems on ( P 5 , P 5 ) -free graphs. Hoàng in tro dued in [11 ℄ the P 4 -sparse graphs (ev ery indued subgraph on 5 v er- ties on tains at most one P 4 ) and sev eral extensions of this notion ha v e arisen in the litterature (see for examples [1,8 ,9,14℄). W e are onerned here with ( P 5 , P 5 )-graphs and ( P 5 , P 5 , B ul l )-graphs where these lasses of graphs are dened in the same w a y (ev ery subgraph on 6 v erties on tains at most one subgraph in the family). In this pap er, w e extend the notion of P 4 - sparse in the follo wing w a y : A graph G is said to b e F -sparse, where F denotes a set of graphs of order p , whenev er an y indued subgraph of G on p + 1 v erties on tains at most one graph of F as indued subgraph. W e study F -sparse when F = { P 5 , P 5 } ) and when F = { P 5 , P 5 , B ul l } ). Those graphs lasses are dened with ongurations whi h are prime with resp et to mo dular deomp o- sition (see Figure 1 ) and whi h prop erly in- terset graphs lasses su h that P L -graphs or some ( q , t ) -graphs lasses. W e obtain some results on ( P 5 , P 5 ) -sparse graphs already kno wn on ( P 5 , P 5 ) -free graphs and w e ompletely desrib e the struture of ( P 5 , P 5 , bul l )-sparse graphs. This sho ws that those graphs ha v e b ounded lique-width. Basis Let G = ( V , E ) b e a graph, the omple- men tary graph of G is denoted G . If x and y are t w o adjaen t v erties of G , x is said adja- en t to y and y is a neighb or of x . A graph on 2 n v erties su h that all of them ha v e exatly one neigh b or is a nK 2 . Let X b e a set of v erties and x b e a v ertex su h that x / ∈ X , the set of neigh b ors of x that b elong to X is said the neighb orho o d of 1 2 P 5 B ull P 5 Figure 1. The forbidden ongurations in a ( P 5 , P 5 , B ul l ) -free graph x in X and is denoted N X ( x ) , if N X ( x ) = ∅ x is said indep endent of X and total for X when N X ( x ) = X , if x is not indep enden t of X nor total for X , x is said p artial for X . If x is indep enden t of X (resp. total for X ), x is said isolated in X ∪ { x } (univ ersal for X ∪ { x } ). Let X and Y b e t w o disjoin t sets of v er- ties, the set S y ∈ Y N X ( y ) is denoted N X ( Y ) and alled the neighb orho o d of Y in X . If there is no edge onneting a v ertex of X to a v ertex of Y , the sets X and Y are inde- p enden t while X is total for Y when there is all p ossible edges onneting v erties of X to v erties of Y . 2. On ( P 5 , P 5 )-sparse graphs. In this setion w e onsider F -sparse graphs when F = { P 5 , P 5 } and w e all those graphs ( P 5 , P 5 ) -sparse. Reall that in a su h graph ev ery indued subgraph on 6 v erties on tains at most one P 5 or P 5 . Theorem 2.1 A prime ( P 5 , P 5 ) -sp arse gr aph is either C 5 -fr e e or isomorphi to a C 5 . Pro of Let G b e a prime F -sparse graph ha ving at least 6 v erties. Observ e rst that a v ertex, sa y x , whi h is partial to a C 5 of G is either adjaen t to exatly t w o non-adjaen t v erties of the C 5 or to three onseutiv e v erties of the C 5 . In all other ases of adjaenies the subgraph in- dued b y the v erties of the C 5 together witrh x on tains t w o P 5 or P 5 , a on tradition. Let abcde b e a C 5 of G , sine G is prime there m ust exist in G a v ertex, sa y x whi h is partial to abcde . Without loss of gener- alit y w e an assume that x is adjaen t to a and c and indep enden t of d and e . Let A b e the set of v erties of G whi h are adjaen t to a and c and indep enden t of d and e . Sine A on tains at least t w o v erties ( { b, x } ⊆ A ) and G is prime there m ust b e a v ertex, sa y y , outside of A whi h distinguishes t w o v erties of A sa y b 1 and b 2 . But no w, the v ertex y annot b e outside of A and satisfy the ab o v e observ ation with b ots C 5 ab 1 cde and ab 2 cde , a on tradition.  W elsh-P o w ell p erfet graphs are p erfetly or- derable and are  haraterized with 17 forbid- den ongurations (see [5 ℄). It is a straigh t- forw ard exerise to see that ( P 5 , P 5 ) -sparse graphs whi h are also C 5 -free are W elsh- P o w ell p erfet . In [12℄, Hoàng, giv es al- gorithms to solv e the Maximum W eighte d Clique problem as w ell as the Minimum W eighte d Coloring problem on p erfetly or- 3 derable graphs within O ( nm ) time omplex- it y . Th us, as w ell as for ( P 5 , P 5 ) -free graphs (see [10℄), there exists algorithms running in O ( nm ) time, for omputing a Maximum W eigte d Clique and a Minimum W eighte d Coloring in a w eigh ted ( P 5 , P 5 ) -sparse graph. Sine the lass of ( P 5 , P 5 ) -sparse graphs is auto-omplemen tary the parameters Max- imum W eighte d Stable Set and Minimum W eighte d Clique Cover an b e omputed within the same time omplexit y . 3. ( P 5 , P 5 , B ul l )-sparse graphs. In this setion w e will study F -sparse graphs where F = { P 5 , P 5 , B ul l } , namely the ( P 5 , P 5 , B ul l ) -sparse graphs. W e will  hara- terize the prime graphs of this family and giv e some onsequenes. Let's rst reall a main result on ( P 5 , P 5 , B ul l ) -free graphs. Theorem 3.1 ([7℄) A prime gr aph G is ( P 5 , P 5 , bul l ) -fr e e if and only if one of the fol- lowing holds : 1. G is isomorphi to a C 5 2. G or its  omplement is a bip artite P 5 - fr e e gr aph. Sine Theorem 2.1 also holds for ( P 5 , P 5 , B ul l ) -sparse graphs w e onsider heneforth only C 5 -free graphs. Theorem 3.2 L et G b e a prime C 5 -fr e e whih  ontains an indu e d P 5 (r esp. P 5 ). G is ( P 5 , P 5 , B ul l ) -sp arse if and only if G (r esp. G ) is isomorphi to one of the gr aphs depite d in Figur e 2. Pro of of Theorem 3.2. It is easy to see that the graphs depited in Figure 2 are prime ( P 5 , P 5 , B ul l ) -sparse graphs, onsequen tly in the follo wing w e onsider the only if part of the theorem. Assume without loss of generalit y that G on tains a P 5 , namely abcde . Observ e rst that a v ertex partial to this P 5 an only b e adjaen t to c , all other adjaeny ases lead to a on tradition Let's denote C the set of v erties in G whose neigh b orho o d in { a, b, c, d, e } is { c } , in addition w e denote I the set of v erties of G whi h ha v e no neigh- b or in { a, b, c, d, e } while T denotes the set of v erties of G whi h are total for { a, b, c, d, e } , note that V ( G ) = { a, b, c, d, e } ∪ C ∪ T ∪ I . Moreo v er w e supp ose heneforth that C is not empt y , otherwise b y the primalit y assump- tion, G w ould b e the P 5 abcde itself (one of the graphs depited in Figure 2). Claim 1 If I has a neighb or in C then N C ( I ) ∪ N I ( C ) is a nK 2 , the verti es of N I ( C ) ar e isolate d in I and the verti es of N C ( I ) ar e isolate d in C . Mor e over T is total for C and N I ( C ) . Pro of Let's assume that x ∈ C has a neigh- b or i ∈ I , so { a, b, c, x, i } is a P 5 . Then x (resp. i )has no other neigh b or in I (resp. C ). Moreo v er N I ( C ) is isolated in I b eause if i has a neigh b or i ′ in I , then { a, b, c, x, i, i ′ } is a P 6 , a on tradition; and N C ( I ) is isolated in C b eause if x has a neigh b or x ′ in C then { a, b, c, x, x ′ , i indues a P 5 and a bull , a on tradition. Let t ∈ T , assume that t isn't a neigh b or of x or i . Let rst t isn't a neigh b or of i then { i, x, c, t, a, e } indues 2 P 5 or 2 bul l . Otherwise if t isn't a neigh b or of x then { i, x, c, t, b, d } indues 2 P 5 , a on tradi- tion. Let x ′ ∈ C − N C ( I ) , reall that x isn't adjaen t to x ′ ; if x ′ is not adjaen t to t , the graph G [ { a, t, c, x ′ , x, d } ] on tains t w o indued bulls, a on tradition, then the v erties of T are all adjaen t to the v erties of C \ N C ( I ) .  4 y optional v ertex A bundle of P 5 s x 0 t 0 y x x optional K 2 s x optional v ertex Figure 2. The 2 t yp es of prime ( P 5 , P 5 , B ul l ) -sparse graphs whi h are C 5 -free and on tain a P 5 . Sine G is a prime graph, when I has a neigh b or in C it follo ws that the sets T and I \ N I ( C ) are empt y or { a, b, c, d, e } ∪ C ∪ N I ( C ) w ould b e a non trivial mo dule of G . Similarly C \ N C ( I ) on tains at most one v ertex and th us G is a bund le of P 5 's , one of the graphs depited in Figure 2. F rom no w on, w e assume that I has no neigh b or in C , moreo v er w e ma y assume that a v ertex of T has a non-neigh b or in C other- wise the set T w ould b e empt y ( G is a prime graph) and one again, G w ould b e a bundle of P 5 's. Claim 2 Ther e is a unique non-e dge c 0 t 0 suh that c 0 ∈ C and t 0 ∈ T , c 0 is adja ent to al l other verti es of C , t 0 is adja ent to al l other verti es of T and has no neighb or in I . Pro of Observ e rst that a v ertex of T annot ha v e t w o non-neigh b ors in T , other- wise a su h v ertex sa y t together with t w o non-neigh b ors in C , sa y c 1 and c 2 and the v erties a , c and d w ould indue t w o bulls, a on tradition. Similarly , a v ertex of C , sa y x annot ha v e t w o non-neigh b ors t 1 and t 2 in T or t w o bulls w ould b e indued with the v erties x , c , d , a , t 1 and t 2 , a on tradi- tion. If there is t w o non-edges c 1 t 1 and c 2 t 2 su h that c 1 , c 2 ∈ C and t 1 , t 2 ∈ T , those v erties together with a and e w ould indue t w o P 5 's or t w o bulls or t w o P 5 's or t w o C 5 's aording to the onnetions b et w een c 1 and c 2 and b et w een t 1 and t 2 , a on tradition. If t 0 w ould ha v e a non-neigh b or in T , sa y t , the v erties c 0 , c , t 0 , t , a and e w ould indue t w o P 5 , a on tradition. A neigh b or i of t 0 in I together with c 0 and the v erties b , c and d w ould indue t w o bulls in G , a on tradition. If c 0 is indep enden t of another mem b er of C sa y x , the graph indued b y the v erties x 0 , x , t 0 , c and a indues a bull, as w ell as G [ { x 0 , x, t 0 , c, e } ] , a on tradition.  No v ertex of I ∪ T \ { t 0 } an distinguish the v erties of { a, b, c, d, e } ∪ C ∪ { t 0 } , onse- quen tly I ∪ T \ { t 0 } = ∅ . Moreo v er, C \ { c 0 } on tains at most one v ertex, it follo ws that G has either 7 or 8 v erties aording to the fat that C \ { c 0 } is empt y or not and is isomorphi to a graph depited in Figure 2.  Theorem 3.3 L et G b e a prime ( P 5 , P 5 , C 5 ) - fr e e gr aph whih  ontains an indu e d bul l. G is ( P 5 , P 5 , B ul l ) -sp arse if and only if G or 5 optional v erties i 0 c 0 5 1 2 3 4 a c d 0 c 0 5 1 2 3 4 c d i 0 4 3 2 1 5 c 0 d 0 d c a 4 3 2 1 5 Graph G 4 Graph G 3 Graph G 2 Graph G 1 a , c and d a , c and { c 0 , i 0 } c , d and { c 0 , d 0 } optional v erties optional v erties Figure 3. The 4 t yp es of prime ( P 5 , P 5 , B ul l ) -sparse graphs whi h are ( P 5 , P 5 , C 5 )-free and on tain a B ul l . G is isomorphi to one of the gr aphs depite d in Figur e 3. Pro of It is easy to  he k that all graphs in Figure 3 are ( P 5 , P 5 , B ul l ) -sparse. Let's onsider an indued bull in G whose v erties are n um b ered 1 , 2 , 3 , 4 , 5 in su h a w a y that { 1 , 2 , 3 , 4 } indues a P 4 whose end- p oin ts are 1 and 4 and 5 is preisely adjaen t to 2 and 3 and not to 1 nor 4 . W e onsider the 6 follo wing subsets of V \ { 1 , 2 , 3 , 4 , 5 } . Let T b e the set of v erties whi h are ad- jaen t to all the mem b ers of { 1 , 2 , 3 , 4 , 5 } and I b e the set of v erties ha ving no neigh b or among { 1 , 2 , 3 , 4 , 5 } . Let A b e the set of v er- ties b eing adjaen t to 2 and 5 and indep en- den t of 1 , 3 and 4 , while B denotes the set of v erties whi h are adjaen t to 3 and to 5 and indep enden t of 1 , 2 , and 4 . Let C b e set of v erties whi h are adjaen t to 1 , 2 and 3 and indep enden t of 4 and 5 . D denotes the set of v erties b eing adjaen t to 2 , 3 and 4 and indep enden t of 1 and 5 . It is easy to see that a v ertex x whi h is par- tial with resp et to { 1 , 2 , 3 , 4 , 5 } m ust b elong to A ∪ B ∪ C ∪ D , in other ases of adjaeny the subgraph indued b y { 1 , 2 , 3 , 4 , 5 , x } w ould not b e ( P 5 , P 5 , B ul l ) -sparse. Conse- quen tly V ( G ) = { 1 , 2 , 3 , 4 , 5 } ∪ T ∪ I ∪ A ∪ B ∪ C ∪ D . Claim 1 C is total for A and T , C is inde- p endent of B . Pro of Let c ∈ C . When c has a non- neigh b or in A , sa y a , the set { a, 5 , 3 , c, 1 } in- dues a P 5 , a on tradition sine G is assumed to b e P 5 -free. The v ertex c annot ha v e a neigh b or in B , or this neigh b or together with c , 1 , 2 , and 5 w ould indue a P 5 , a on tra- dition. When c has a non-neigh b or t in T , 43 ct 1 is a P 5 , a on tradition.  Let f b e an edge preserving mapping su h that f (1) = 4 , f (4) = 1 , f (2) = 3 , f (3) = 2 and f (5) = 5 , w e ha v e f ( A ) = B , f ( B ) = A , f ( C ) = D , f ( D ) = C while f ( T ) = T and 6 f ( I ) = I . It follo ws that w e an deriv e from Claims 1 to Claim 5 b elo w man y analogous results b y onsidering the mapping f and/or the omplemen tary graph of G . F or example the assertion C is total for A b eomes D is total for B when onsidering the mapping f , while C is total for T b eomes B is indep en- dent of I when applied in G and A is inde- p endent of I when onsidering the mapping f in G . Let's no w examine the onnetions b et w een v erties of C and D and b et w een v erties of C and I . Claim 2 If A 6 = ∅ then ther e is no e dge  on- ne ting a vertex of C to a vertex of D , nor a vertex of D to a vertex of I . Pro of Let a b e a v ertex of A . Assume that cd is an edge ( c ∈ C and d ∈ D ), the v erties c , d , 3 , 5 , a indue a P 5 , a on tradition. Supp ose that d ∈ D has a neigh b or i in I , then id 35 a is a P 5 , a on tradition  Claim 3 If a vertex of C has a neighb or in I then the verti es of N I ( C ) ar e isolate d in I , the verti es of N C ( I ) ar e isolate d in C , T is total for A ∪ N I ( C ) , in addition ther e is a unique e dge c 0 i 0  onne ting a vertex of C to a vertex of I and i 0 is isolate d in I .. Pro of Let c ∈ C and i ∈ I b e adjaen t v erties. If i has a neigh b or in I , sa y i ′ , i ′ ic 34 is a P 5 when i ′ is indep enden t of c while { i, i ′ , c, 2 , 3 , 4 } indues 2 bulls when i is ad- jaen t to c , a on tradition. If c has a neigh b or in C , sa y c ′ , the v erties i , c , c ′ , 2 , 3 , 4 indues t w o bulls, a on tradi- tion. Let at ( a ∈ A , t ∈ T ) b e a non edge of G , then ict 5 a is a P 5 of G , a on tradition. Moreo v er, observ e that a v ertex of C an- not ha v e t w o neigh b ors i and i ′ in I , other- wise the v erties c , i , i ′ , 2 , 3 , 4 w ould in- due t w o bulls, a on tradition. On the same manner, a v ertex in I annot ha v e t w o neigh- b ors in C , sa y c and c ′ or one again t w o bulls are indued in G [ { i, c, c ′ , 2 , 3 , 4 } ] , on- tradition. Consequen tly , aording to Claim 3 , t w o edges onneting v erties of C to v er- ties of I w ould indue a 2 K 2 and th us this 2 K 2 together with the v ertex 2 w ould indue a P 5 in G , a on tradition. Let c 0 i 0 b e the unique edge onneting a v ertex of C to a v ertex of I , if i 0 has a neigh- b or, sa y i ′ in I , i ′ i 0 c 0 34 w ould b e P 5 of G , a on tradition.  Claim 4 If C has a neighb or in D then N D ( C ) is universal in D , ther e is a unique e dge  onne ting a vertex of C to a vertex of D , the verti es of I do not distinguish c 0 fr om d 0 and D \ { d 0 } is indep endent of I . Pro of Assume that a v ertex c ∈ C has t w o neigh b ors in D , namely d and d ′ . In this ase the graph indued b y the v erties c , d , d ′ , 1 , 3 , 5 on tains t w o bulls, a on tradition. Symmetrially , a mem b er of D annot ha v e t w o neigh b ors in C . Moreo v er, if d is indep enden t of some other v ertex of D , namely d ′ , the set { c, 3 , d, 1 , 5 , d ′ } indues t w o bulls, a on tradition, th us N D ( C ) is univ ersal in D , similarly N C ( D ) is univ ersal in C . If cd and c ′ d ′ ( c, c ′ ∈ C , d, d ′ ∈ D ) are t w o distint edges, G [ { c, c ′ , d, d ′ , 4 } ] is a P 5 , a on tradition whi h pro v es the uniqueness of an edge onneting C to D . Assume without loss of generalit y that i ∈ I is adjaen t to c 0 and not to d 0 . the sub- graph indued b y 1 , i , c 0 , d 0 , 3 and 5 on tains t w o bulls, a on tradition. 7 Finally , supp ose that d ∈ D , distint from d 0 is adjaen t to i ∈ I , then G on tains a P 5 ( idd 0 35 if i is adjaen t to d 0 and dic 0 25 if i is not adjaen t to d 0 ), a on tradition.  Claim 5 A t le ast one of the sets A , B , C , D is empty. Pro of Let a ∈ A , b ∈ B , c ∈ C , d ∈ D . W e kno w b y Claim 1 that a is onneted to c and not to d and that b is onneted to d and not to c , Claim2 asserts that c is not adjaen t to d while a and b are onneted. Consequen tly 1 cabd is a P 5 , a on tradition.  A ording to Claim 5 w e will no w disuss on the n um b er of empt y sets among A , B , C and D and pro v e that G or G is isomorphi to one of the graphs depited in Figure 3. Case 1 : The sets A , B , C and D are all empt y . Reall that G is prime, th us the sets T and I are also empt y , for otherwise { 1 , 2 , 3 , 4 , 5 } w ould b e a non-trivial mo dule. Consequen tly G is a bull, a graph isomorphi to G 1 in Fig- ure 3 when a , c and d are missing. Case 2 : Three of the sets A , B , C and D are empt y . Assume without loss of generalit y that C 6 = ∅ . W e kno w b y Claim 1 that C is total for T If C has no neigh b or in I , no v ertex of T ∪ I an distinguish the mem b ers of { 1 , 2 , 3 , 4 , 5 } ∪ C and b y the primalit y of G the sets T and I are empt y while C is redued to a single v ertex. In this ase G is isomorphi to G 1 where a and d are missing. If C has a neigh b or in I w e kno w b y Claim 3 that there is a unique edge, namely c 0 i 0 onneting C to I . W e onsider the follo wing deomp osition of C and I : C = { c 0 } ∪ ( C \ { c 0 } ) , I = { i 0 } ∪ ( I \ { i 0 } ) . By onstrution C \ { c 0 } is indep enden t of I and I \{ i 0 } is indep enden t of C while i 0 has no neigh b or in I \ { i 0 } and c 0 has no neigh b or in C \ { c 0 } (Claim 3). Moreo v er i 0 is ompletely adjaen t to T (Claim 3). Consequen tly T ∪ ( I \ { i 0 } ) = ∅ or the set { 1 , 2 , 3 , 4 , 5 , c 0 , i 0 } ∪ ( C \ { c 0 } ) w ould b e a non trivial mo dule of G , a on tradition. In ad- dition C \ { c 0 } is either a singleton, sa y { c } or empt y and G is isomorphi to G 2 without the v ertex a and where c is p ossibly missing if C \ { c 0 } = ∅ (see Figure 3 ). Case 3 : Among A , B , C and D ex- atly t w o sets are empt y . Due to symmetries w e only onsider three dif- feren t situations. Let rst supp ose that B = C = ∅ . W e kno w (Claim 1) that A and D are inde- p enden t, D is total for T and A is indep en- den t of I . Moreo v er D is indep enden t of I (Claim 2) and th us A is total for T . Beause of the primalit y of G the set T ∪ I is empt y and A as w ell as D is a singleton. Conse- quen tly G is isomorphi to G 2 without the v ertex c (Figure 3). Assume in a seond stage that B = D = ∅ . W e kno w b y Claim 1 that C is total for T and A is indep enden t of I . If there is an edge b et w een C and I , it is unique (Claim 3), let's denote this edge c 0 i 0 . In this ase A is totally adjaen t to T (Claim 3 ), the set C \ { c 0 } is indep enden t of I and b y onstrution c 0 is indep enden t of I \ { i 0 } , i 0 is indep enden t of I \ { i 0 } and c 0 has no neigh b or in C \ { c 0 } (Claim 3 again). It follo ws that the prime graph G is isomorphi to G 2 (Figure 3) where c an miss if C \ { c 0 } is empt y . If there is no onnetion b et w een C and I , some v ertex of A an ha v e a non neigh b or in T , w e are then in a similar situation than ab o v e in the omplemen tary graph of G . 8 When C is indep enden t of I and A is total for T the graph is isomorphi to graph G 1 in Figure 3 . Finally let's study the ase A = B = ∅ . W e kno w that C and D are totally adjaen t to T (Claim 1). If C and D are not onneted it is easy to see that C and D are not adjaen t to I . As a matter of fat, supp ose on the on trary that c 0 i 0 is an edge ( c 0 ∈ C and i 0 ∈ I ) and that d is some v ertex in D . If d and i 0 are not onneted, i 0 c 0 2 d 4 is a P 5 and i 0 c 0 3 d 4 is a P 5 if d and i 0 are adjaen t, a on tradition in b oth ases. Consequen tly , G b eing prime is isomorphi to the graph G 1 in Figure 3 where d misses. When C has a neigh b or in D , w e onsider the unique edge onneting C to D , namely c 0 d 0 ( c 0 ∈ C, d 0 ∈ D ). By Claim 4 , c 0 is univ ersal in C and d 0 is univ ersal in D . W e kno w (Claim 4) that only c 0 and d 0 an ha v e a neigh b or in I . If it is not the ase G is isomorphi to G 3 in Figure 3 without c or d if C \ { c 0 } or D \ { d 0 } is empt y . If, on the on trary , c 0 and d 0 ha v e a neigh b or, sa y i o in I , C = { c 0 } (or { 1 , c, c 0 , i 0 , d 0 , 4 } where c is a v ertex of C dis- tint from c 0 indues t w o bulls, a on tradi- tion) and similarly D = { d 0 } . Consequen tly G is isomorphi to the graph G 4 of Figure 3. Case 4 : Among A , B , C and D ex- atly one set is empt y . F or on v eniene w e will supp ose that B = ∅ . By Claim 1, A is ompletely adjaen t to C and indep enden t of D . There is no edge onneting a v ertex of C to a v ertex of D (Claim 2). Moreo v er C and D are ompletely adjaen t to T and A is indep enden t of I (Claim 1 ). In addition, there is no onnetion b et w een D and I (Claim 2) and similarly A is total for T . If there is no edge b et w een C and I , the sets T and I m ust b e empt y (or { 1 , 2 , 3 , 4 , 5 } ∪ A ∪ C ∪ D w ould b e a non trivial mo dule of G ) and A , C , D are singletons. In this ase G is isomorphi to G 2 in Figure 3. When there is a unique edge c 0 i 0 b et w een C and I ( c 0 ∈ C, i 0 ∈ I ), one again I \ { i 0 } is ompletely indep enden t of C ∪ { i 0 } while { c 0 , i 0 } has no onnetions with C \ { c 0 } (Claim 3). Consequen tly , G is isomorphi to G 2 where c misses if C = { c 0 } .  It follo ws from Theorem 3.1, 2.1 , 3.1 and 3.3 that a prime ( P 5 , P 5 , B ul l ) -sparse graph or its omplemen t is either a C 5 or a P 5 -free bi- partite graph or a bundle of P 5 's (see Fig- ure 2) or is a graph on less than 10 v erties. This leads to a linear time reognition algo- rithm for ( P 5 , P 5 , B ul l ) -sparse graphs, more- o v er those graphs ha v e b ounded lique-width (see [6℄). REFERENCES 1. L. Bab el and S. Olariu. A new  harater- isation of P 4 -onneted graphs. L e tur e notes in Computer Sien e , 1197,W G 96:1730, 1996. 2. G. Basó and ZS. T uza. Dominating liques in P 5 -free graphs. Perio di a Math- emati a Hungari a , 21:303308, 1990. 3. A. Brandstädt and D. Krats h. On the struture of ( P 5 , g em ) -free graphs. Disr ete Applie d Mathematis , 145(Issue 2):155166, Jan uary 2005. 4. A. Brandstädt and R. Mosa. On the struture and stabilit y n um b er of P 5 and o- hair-free graphs. Disr ete Applie d Mathemaitis , 2003. 5. V. Ch v átal, C.T. Hoàng, N.V.R. Ma- hadev, and D. de W erra. F our lasses of p erfetly orderable graphs. Journal of 9 Gr aph The ory , 11:481495, 1987. 6. B. Courelle, J.A. Mak o wsky , and U. Rotis. Linear time solv able opti- mization problems on graphs of b ounded lique width. L e tur e Notes in Computer Sien e , 1517:116, 1998. 7. J.L. F ouquet. A deomp osition for a lass of ( P 5 , P 5 ) -free graphs. Disr ete Mathe- matis , 121:7583, 1993. 8. J.L. F ouquet and J.M. V anherp e. On bi- partite graphs with w eak densit y of some subgraphs. Disr ete Mathematis , 307(Is- sues 11-12):15161524, Ma y 2007. 9. V. Giak oumakis, F. Roussel, and H. Th uillier. On P 4 -tidy graphs. Dis- r ete Mathematis and The or eti al Computer Sien e , 1:1741, 1997. 10. V. Giak oumakis and I. Rusu. W eigh ted parameters in ( P 5 , P 5 ) -free graphs. Dis- r ete Applie d Mathematis , 80:255261, 1997. 11. C.T. Hoàng. Perfe t Gr aphs . PhD the- sis, S ho ol of Computer Siene, MGill Univ ersit y , Mon treal, 1985. 12. C.T. Hoàng. Eien t algorithms for min- im um w eigh ted olouring of some lasses of p erfet graphs. Disr ete Applie d Math- ematis , 55:133143, 1994. 13. R.H. Möhring and F.J. Raderma her. Substitution deomp osition for disrete strutures and onnetions with om bi- natorial optimization. A nnals of Disr ete Mathematis , 19:257356, 1984. 14. F. Roussel, I. Rusu, and H. Th uillier. On graphs with limited n um b er of P 4 - partners. International Journal of F un- dations of Computer Sien e , 10:103121, 1999.