On the cubicity of AT-free graphs and circular-arc graphs

On the Cubicit y of A T-free graphs and Circular-arc graphs L. Sunil Chandran, Mathew C. F ra ncis ⋆ , and Na veen Siv adasan ⋆⋆ Abstract. A u nit cub e in k dimensions ( k -cu b e) is defined as the t he Cartesian p ro du ct R 1 × R 2 × · · · × R k where R i (for 1 ≤ i ≤ k ) is a closed interv al of the form [ a i , a i + 1] on the real line. A graph G on n no des is said to b e represen table as the intersection of k -cub es (cub e representa tion in k dimensions) if each vertex of G can be mapp ed to a k - cu b e suc h th at t w o vertice s are adjace nt in G if and only if t h eir correspondin g k -cub es ha ve a non- empty in tersection. The cubici ty of G denoted as cub( G ) is the minim um k for which G can b e represented as the intersection of k -cu b es. An in t eresting asp ect about cubicit y is that man y problems know n to b e N P-complete for general graphs hav e p olynomial time deterministic algorithms or hav e goo d approximati on ratios in graph s of lo w cubicity . In most of these algorithms, computing a low dimensional cub e repre- senta tion of th e giv en graph is usually the first step. W e give an O ( bw · n ) algori thm to compute t he cu b e representation of a general graph G in bw + 1 dimensions giv en a band width ordering of the vertices of G , where bw is the b andwidth of G . As a consequence, w e get O ( ∆ ) upp er b oun ds on the cubicity of many well-kno wn graph classes such as A T-free graphs, circular-arc graphs and co-comparabilit y graphs whic h hav e O ( ∆ ) bandwidth. Thus we ha ve: 1. cub( G ) ≤ 3 ∆ − 1, if G is an A T-free graph. 2. cub( G ) ≤ 2 ∆ + 1, if G is a circular-arc graph. 3. cub( G ) ≤ 2 ∆ , if G is a co-comparabilit y graph. Also for these graph classes, there are constant factor approximation algorithms for bandwidth computation that generate orderings of v ertices with O ( ∆ ) width. W e can thus generate the cub e representation of such graphs in O ( ∆ ) dimensions in p olyn omial time. Keywords : Cubicit y , bandwidth, in tersection graphs, A T-free graphs, circular-arc graphs, co-comparabilit y graphs. 1 In tro duction Let F = { S x ⊆ U : x ∈ V } b e a family of subsets of a univ e r se U , where V is an index set. The intersection g raph Ω ( F ) of F ha s V as vertex set, and tw o distinct ⋆ Indian I nstitute of Science, Dept. of Computer S cience and Automation, Bangalore– 560 012, India. email: mathew,sunil@csa.i isc.ernet.in ⋆⋆ Adv anced T echnolog y Centre, TCS, Deccan Park, Madhapur, Hyderabad–500 081, India. email: s.nave en@atc.tcs.c om vertices x and y are adjacent if a nd only if S x ∩ S y 6 = ∅ . Representations of graphs as the in ter section g raphs of v arious geometrica l ob jects is a well studied topic in graph theory . Pro bably the most well studied cla ss of intersection gr aphs ar e the interval gr aphs , where each S x is a clo sed int erv a l on the real line. A res tricted form of int erv al graphs, that allow only interv als of unit length, ar e indiffer enc e gr aphs . A well k nown concept in this area of g raph theory is the cubicity , which was int ro duced by F. S. Rob e rts in 1 969 [11]. This c o ncept gener alizes the conce pt of indifference gr aphs. A unit cube in k dimensions ( k - cub e ) is a Cartesia n pr o duct R 1 × R 2 × · · · × R k where R i (for 1 ≤ i ≤ k ) is a clo sed in terv al of the form [ a i , a i + 1 ] on the r e a l line. Two k - cub e s, ( x 1 , x 2 , . . . , x k ) and ( y 1 , y 2 , . . . , y k ) are said to hav e a non- empty intersection if and only if the interv als x i and y i hav e a non-empty intersection for 1 ≤ i ≤ k . F or a gra ph G , its cu bicity is the minimum dimension k , such that G is repres entable as the intersection g r aph o f k -cub es. W e denote the cubicity of a graph G by cub( G ). The graphs of cubicity at most 1 a re exactly the cla ss of indifference g raphs. If we require tha t ea ch vertex co rresp ond to a k -dimensio na l axis-par allel box R 1 × R 2 × · · · × R k where R i (for 1 ≤ i ≤ k ) is a clo sed in terv al of the form [ a i , b i ] on the real line, then the minimum dimension required to represe n t G is called its boxicity denoted a s b ox( G ). Clea r ly box( G ) ≤ c ub( G ) for any g raph G beca use cubicity is a stricter notion than b oxicit y . It has b een s hown that deciding whether the cubicity of a given gra ph is at least 3 is NP- hard [15 ]. In man y alg orithmic pro blems rela ted to graphs, the av ailability of certain conv enie nt representations turn out to b e ex tremely useful. Pr obably , the mos t well-kno wn and imp or tant e x amples are the tree decomp ositions and path de- comp ositions. Ma ny NP-hard pro blems are known to b e p olynomial time so lv able given a tree(path) dec omp osition of the input gra ph that has bounded width. Similarly , the representation of g raphs as intersections of “disks” o r “ spheres” lies at the core o f s o lving problems related to freq uency assignments in r a dio net- works, c omputing molecular confor mations etc. F or the maximum indep endent set problem which is hard to approximate within a factor of n (1 / 2) − ǫ for general graphs, a PT AS is known for disk g r aphs g iven the disk representation [4, 1] and an FPT AS is known fo r unit disk g raphs [14 ]. In a similar wa y , the av ailability of cub e or box representation in low dimensio n make s ome well known NP har d problems like the max-clique problem, p olynomia l time solv able since ther e are only O ((2 n ) k ) max imal cliques if the b oxicit y or cubicity is at most k . Thoug h the complexity of finding the ma ximum indep endent set is ha rd to a ppr oximate within a factor n (1 / 2) − ǫ for general graphs, it is a pproximable to a log n fa c to r for b oxicit y 2 graphs (the problem is NP-hard even fo r b oxicit y 2 gra phs) given a b ox o r c ub e r epresentation [2, 3]. It is easy to see that the problem of repr e senting g raphs using k - cub e s can b e equiv alently for mu lated as the following geometr ic embedding problem. Given an undirected un weigh ted graph G = ( V , E ) and a threshold t , find an em b edding f : V → R k of the v ertices of G in to a k - dimensional spa c e (for the minimum 2 po ssible k ) such that for any tw o vertices u and v o f G , || f ( u ) − f ( v ) || ∞ ≤ t if and only if u and v a re adjacent . The no rm || || ∞ is the L ∞ norm. Clearly , a k -cub e representation of G yields the required embedding of G in the k -dimensional space. The minim um dimension requir ed to embed G as above under the L 2 norm is calle d the sphericity of G . Refer [9] for a pplications wher e such an embedding under L ∞ norm is ar gued to b e more appropriate than em bedding under L 2 norm. The connectio n b etw een cubicity a nd spher icity of gr aphs were studied in [6, 8]. As far as we know, the only known upp er bo und fo r the cubicity o f genera l graphs (existential or constructive) is by Rob erts [11], who show ed that cub( G ) ≤ 2 n/ 3 for any graph G on n vertices. The c ub e representation of sp e c ial class of graphs like h y pe r cub es and co mplete multipartite graphs were investigated in [11, 8, 10]. Line ar Or dering and Bandwid th. Giv en an undirected graph G = ( V , E ) on n vertices, a line ar or dering of G is a bijection f : V → { 1 , . . . , n } . The width of the linea r ordering f is defined as ma x ( u,v ) ∈ E | f ( u ) − f ( v ) | . The b andwidth mini- mization pr oblem is to compute f with minim um poss ible width. The b andwidth of G denoted a s bw( G ) is the minimum p ossible width achieved by any linear ordering of G . A b andwidth or dering of G is a linear or dering of G with width bw( G ). Our a lg orithm to compute the cube representation o f a graph G tak es as input a linear ordering of G . The smaller the width of this ordering, the lesser the num b er of dimensions o f the cub e representation of G co mputed by our al- gorithm. It is NP-har d to approximate the bandwidth o f G within a r atio better than k for every k ∈ N [13]. F eige [5] gives a O (log 3 ( n ) √ log n lo g log n ) appr oxi- mation algo rithm to compute the bandwidth (and a lso the cor resp onding linear ordering) of genera l g r aphs. F or bandwidth computation, several algorithms with go o d heuristics are known that per form very well in prac tice [12 ]. 1.1 Our results W e summarize b elow the r e sults o f this pap er . 1. F or any gra ph G , cub ( G ) ≤ bw( G ) + 1 2. F or an A T- free graph G with max imum de g ree ∆ , cub( G ) ≤ 3 ∆ − 1 3. F or a circular-a rc graph G with maximum degree ∆ , c ub( G ) ≤ 2 ∆ + 1 4. F or a co-compar ability gr aph G with maximum degree ∆ , cub( G ) ≤ 2 ∆ 1.2 Definitions and N o tations All the graphs that we consider will b e simple, finite and undirected. F or a gra ph G , we denote the v ertex set of G by V ( G ) and the edge set of G by E ( G ). F or a vertex u ∈ V , let d ( u ) denote its degree (the n umber of outer neighbors o f u ). The maximum degree of G is denoted by ∆ ( G ) or simply ∆ when the g r aph under consideratio n is clear . F or a v ertex u ∈ V ( G ), we denote the s e t of neighbours of u by N G ( u ). By definition, N G ( u ) = { v ∈ V ( G ) | ( u, v ) ∈ E ( G ) } . Again, 3 for ease of nota tion, we use N ( u ) instead o f N G ( u ) when there is no scop e for ambiguit y . Let G ′ be a g raph such that V ( G ′ ) = V ( G ). Then G ′ is a su p er gr aph of G if E ( G ) ⊆ E ( G ′ ). W e define the interse ction of tw o graphs as follows. If G 1 and G 2 are tw o gra phs such that V ( G 1 ) = V ( G 2 ), then the intersection of G 1 and G 2 denoted as G = G 1 ∩ G 2 is the gra ph with V ( G ) = V ( G 1 ) = V ( G 2 ) and E ( G ) = E ( G 1 ) ∩ E ( G 2 ). An indifference gra ph is an interv al g raph which has an in terv al r epresenta- tion that maps the vertices to unit length interv als on the real line such that tw o vertices are adjacent in the graph if and only if the interv als mapp ed to them ov er lap. Definition 1 (Unit in terv al representation). Given an indiffer enc e gr aph I ( V , E ) , the unit interval r epr esentation is a mapping f : V → R such that for any two vertic es u, v , | f ( u ) − f ( v ) | ≤ 1 if and only if ( u, v ) ∈ E . Note that this is equiv alent to mapping each vertex of I to the unit interv al [ f ( u ) , f ( u ) + 1] so that t wo vertices ar e adjacent in I if and o nly if the unit int erv als mapp ed to them ov erlap. Now, co nsider the ma pping g : V → R given by g ( u ) = xf ( u ) where x ∈ R . It ca n b e easily seen that for any tw o vertices u, v , | g ( u ) − g ( v ) | ≤ x if and only if ( u, v ) is an edge in I . g th us corresp onds to an int erv al r epresentation of I using interv als of length x . W e call such a ma pping g a u n it interval r epr esentation of I with interval length x . Definition 2 (Indiff e rence graph representation). The indiffer enc e gr aphs I 1 , . . . , I k c onstitut e an indiffer enc e gr aph r epr esentation of a gr aph G if G = I 1 ∩ · · · ∩ I k . Theorem 1 (Ro b erts[11 ] ). A gr aph G has cub( G ) ≤ k if and only if it has an indiffer en c e gr aph r epr esentation with k indiffer enc e gr aphs. 2 Cubicit y and bandwidth 2.1 The construction W e show that given a linear order ing of the vertices of G with width b , we can construct an indifference g raph representation of G using b + 1 indifference graphs. Theorem 2. If G is any gr aph with b andwidth b , then cub( G ) ≤ b + 1 . Pr o of. Le t n denote | V ( G ) | and let A = u 1 , u 2 , . . . , u n be a linear order ing of the vertices of G with width b . i.e., if ( u i , u j ) ∈ E ( G ), then | i − j | ≤ b . W e constr uct b + 1 indifference graphs I 0 , I 1 , . . . , I b , such that G = I 0 ∩ I 1 ∩ · · · ∩ I b . Let f i denote the unit interv al representation of I i . Construction o f I 0 : Since I 0 has to b e a sup ergra ph of G , we have to ma ke sure that every edge in E ( G ) has to b e pr esent in E ( I 0 ). b b eing the bandwidth of the linear o rdering 4 A of vertices taken, a vertex u j is not adjacent in G to any vertex u k when | j − k | > b . Now, we define f 0 in such a wa y that E ( I 0 ) = { ( u j , u k ) | | j − k | < b } ∪ { ( u j , u k ) | | j − k | = b and ( u j , u k ) ∈ E ( G ) } . The definition of f 0 can b e explained as the following pro cedure. W e first assign the interv al [ j, j + b ] to vertex u j , for a ll j . This makes sur e that u j is no t a djacent to any vertex u k , if k > j + b . Now, each vertex is adja c ent in I 0 to exa ctly the b vertices preceding and following it in A . Now, for each vertex u j where j > b , we shift f 0 ( u j ), the unit interv al for u j , slightly to the right (b y ǫ ) if u j is not adjacent to u j − b in G so that f 0 ( u j ) bec omes disjoint from f 0 ( u j − b ). Along with f 0 ( u j ), all the int erv als that start after f 0 ( u j ) a r e also shifted r ight by ǫ . This pro cedure is done for v ertices u b +1 , . . . , u n in that order. O ur choice of a sma ll v alue for ǫ e ns ures that I 0 is still a s upe r graph of G . f 0 is a unit interv al representation for I 0 with interv a l le ng th b defined as follows. Let ǫ = 1 /n 2 . f 0 ( u j ) = j, for j ≤ b f 0 ( u j ) = f ( u j − b ) + b, for j > b and ( u j − b , u j ) ∈ E ( G ) f 0 ( u j ) = f ( u j − b ) + b + ǫ, for j > b and ( u j − b , u j ) 6∈ E ( G ) Construction o f I i , for 1 ≤ i ≤ b : W e split the sequence of vertices A in to blocks B i 0 , . . . , B i p − 1 of vertices of size b s tarting fr om the v ertex i w he r e the last block B i p − 1 may hav e less than b vertices. F ormally , B i t = { u i + bt , . . . , u i + b ( t +1) − 1 } , for 1 ≤ t < p − 1, and B i p − 1 = { u i + b ( p − 1) , . . . , u n } . Let s i t denote the vertex u i + bt , or the first v ertex (in the o rdering A ) in blo ck B i t . W e now define f i , the unit interv al representa- tion for I i with in terv al length 2, as follows: f i ( u j ) = 2 , if j < i Let u b e a vertex in V ( G ) − { u j | j < i } and let u ∈ B i t . f i ( u ) = t , if u = s i t = t + 2, if ( u, s i t ) ∈ E ( G ) = t + 3, if ( u, s i t ) 6∈ E ( G ) Claim. I 0 is a n indifference sup er graph of G . Pr o of. Fir st we observe that for any vertex u j , j ≤ f 0 ( u j ) ≤ j + 1 /n . This is b ecause f 0 ( u j ) ≤ f 0 ( u j − b ) + b + ǫ where ǫ = 1 / n 2 . Now, consider an edge ( u j , u k ) of G where j < k . Since the width o f the input linear orde r ing A is b , we ha ve k − j ≤ b . Now we consider the following tw o cases. If k − j ≤ b − 1 then f ( u k ) − f ( u j ) ≤ k + 1 /n − j ≤ b − 1 + 1 /n < b . Since each interv al in I 0 has length b , it follows that ( u j , u k ) ∈ E ( I 0 ). If k − j = b then from the definition of f 0 , it follows that f 0 ( u k ) = f 0 ( u k − b ) + b = f 0 ( u j ) + b . Thus f 0 ( u k ) − f 0 ( u j ) ≤ b implying that ( u j , u k ) ∈ E ( I 0 ). 5 Claim. I i for 1 ≤ i ≤ b is a n indifference s uper graph of G . Pr o of. Co nsider the indifference gra ph I i . Let ( u j , u k ) b e any edge in E ( G ). W e assume witho ut loss of g e nerality that j < k . If j < i , then k < i + b and therefore, u k ∈ B i 0 . In this c a se, f i ( u j ) = 2 and 0 ≤ f i ( u k ) ≤ 3 and so we have | f i ( u j ) − f i ( u k ) | ≤ 2. Now, consider the case when j ≥ i . Let u j ∈ B i t . Since | j − k | ≤ b , we have u k ∈ B i t ∪ B i t +1 . F rom the definition of f i , it is clea r that | f i ( u j ) − f i ( u k ) | ≤ 2 if u j 6 = s i t . Now, if u j = s i t , then either u k ∈ B i t , in w hich case f i ( u k ) = t + 2, o r u k = s i t +1 , in whic h case f i ( u k ) = t + 1. But in bo th ca ses, | f i ( u j ) − f i ( u k ) | ≤ 2 . Therefore, we have f i ( u j ) ∩ f i ( u k ) 6 = ∅ which implies that ( u j , u k ) ∈ E ( I i ). It r emains to show that G = I 0 ∩ · · · ∩ I b . T o do this, it suffices to show that for any ( u j , u k ) / ∈ E ( G ), there exis ts an I i , 0 ≤ i ≤ b such that ( u j , u k ) / ∈ E ( I i ). Let j < k . Ca se k − j ≥ b . In this case, w e claim that ( u j , u k ) / ∈ E ( I 0 ). This is bec ause of the following. If k − j = b then clearly f 0 ( u k ) − f 0 ( u j ) = b + ǫ and th us ( u j , u k ) / ∈ E ( I 0 ). Now, if k − j > b then f 0 ( u k ) ≥ f 0 ( u k − b ) + b > f 0 ( u j ) + b observing that f 0 ( u 1 ) < f 0 ( u 2 ) < · · · < f 0 ( u n ). Thus ( u j , u k ) / ∈ E ( I 0 ). Now the remaining case is k − j < b . Co nsider the graph I l where l = j mo d b . Let t be such that u j ∈ B l t . Therefore, bt + l ≤ j < b ( t + 1 ) + l . This implies that bt + l = bt + j mo d b = j and so we hav e s l t = u j . Therefore, u k ∈ B l t since k < j + b = b ( t + 1) + l . No w, from the definition of f l , we hav e f l ( u j ) = t and f l ( u k ) = t + 3. Th us, | f l ( u j ) − f l ( u k ) | > 2 a nd hence ( u j , u k ) / ∈ E ( I l ) as required. Thu s I 0 , . . . , I b is a v alid indifference gra ph repre sentation of G using b + 1 indifference graphs which establishes tha t cub( G ) ≤ b + 1. 2.2 The algorithm Our a lgorithm to compute the cub e repr esentation of G in b + 1 dimensions given a linear order ing of the v ertices of G with width b constructs the indifference sup e rgraphs of G , namely , I 0 , . . . , I b using the constructive pro cedure used in the pro of of Theorem 2. It is easy to verify that this algorithm runs in O ( b · n ) time where b is the width of the input linear a rrang ement and n is the num ber of vertices in G . 3 Applying our results Theorem 2 can b e used to derive upp er b ounds for the cubicity of s everal sp ecial classes of g raphs suc h as circular arc graphs, co-compar ability gr a phs and A T- free g raphs. Corollary 1. If G is a cir cular-ar c gr aph, cub( G ) ≤ 2 ∆ + 1 , wher e ∆ is the maximum de gr e e of G . 6 Pr o of. Le t an a rc on a circle cor r esp onding to a vertex u be denoted by [ h ( u ) , t ( u )] where h ( u )(called the he ad of the arc) is the starting p oint o f the a rc when the circle is trav er sed in the clo c kwise order a nd t ( u ) (called the tail of the arc) is the e nding p oint of the arc when trav ersed in the clockwise order . W e assume without loss of generality that the end-p oints of all the arcs are distinct and that no a rc cov er s the whole circle. If a ny of these ca ses o c c ur, the end-p o ints of the arcs ca n be s hifted s lightly so that o ur assumption ho lds tr ue. Cho ose a vertex v 1 ∈ V ( G ). Start fr o m h ( v 1 ) and trav erse the cir c le in the clo ckwise order . W e order the vertices of the gra ph (other than v 1 ) as v 2 , . . . , v n in the or der in which the heads of their corres po nding ar cs a re encountered during this traversal. Now, we define a n or de r ing f : V ( G ) → { 1 , . . . , n } of the vertices of G a s follows: f ( v j ) = 2 j , if 1 ≤ j ≤ ⌊ n/ 2 ⌋ . f ( v j ) = 2 ( n − j ) + 1, if ⌊ n/ 2 ⌋ < j ≤ n . W e no w prov e that the w idth of this order ing is at most 2 ∆ . W e claim tha t if h ( v j ) and h ( v k ) ar e tw o cons e cutive heads enco unt ered during a clo ckwise trav ersal of the circle, | f ( v j ) − f ( v k ) | ≤ 2 . T o see this, we will consider the different cases that can o cc ur : Case : When 1 ≤ j < j + 1 = k ≤ ⌊ n/ 2 ⌋ . Here, f ( v j ) = 2 j and f ( v k ) = 2 ( j + 1). Therefore, | f ( v j ) − f ( v k ) | = 2 . Case : When ⌊ n/ 2 ⌋ < j < j + 1 = k ≤ n . In this ca se, f ( v j ) = 2( n − j ) + 1 and f ( v k ) = 2 ( n − ( j + 1)) + 1 , which means that | f ( v j ) − f ( v k ) | = 2 . Case : When j = ⌊ n / 2 ⌋ < j + 1 = k , Subca se : If n is even. f ( v j ) = 2 j = n a nd f ( v k ) = 2( n − ( j + 1)) + 1 = 2 n − 2 j − 1 = n − 1. Subca se : If n is o dd, f ( v j ) = 2 j = n − 1 a nd f ( v k ) = 2 n − 2 j − 1 = n . In bo th these cas es, | f ( v j ) − f ( v k ) | = 1 . Case : When j = n and k = 1. W e then hav e f ( v j ) = 1 and f ( v k ) = 2 . Therefor e, | f ( v j ) − f ( v k ) | = 1 . Now, consider a ny edge ( v j , v k ) ∈ E ( G ). Assume without loss of gener ality that h ( v j ) occur s first when we tr averse the circle in c lo ckwise dir ection starting from h ( v 1 ). Now, if we trav erse the arc co rresp onding to v j from h ( v j ) to t ( v j ), we will enco unter at mos t ∆ − 1 hea ds h ( u 1 ) , h ( u 2 ) , . . . , h ( u ∆ − 1 ) b efore we r each h ( v k ) since v j can b e co nnected to a t most ∆ vertices in G . W e alr eady know that | f ( v j ) − f ( u 1 ) | ≤ 2 and | f ( u i ) − f ( u i +1 ) | ≤ 2, for 1 ≤ i ≤ ∆ − 2. Also , | f ( u ∆ − 1 − f ( v k ) | ≤ 2 . It follo ws that | f ( v j ) − f ( v k ) | ≤ 2 ∆ . Thus f is a n order ing of the vertices of G with width at mo s t 2 ∆ . It follows fro m theore m 2 that cub( G ) ≤ 2 ∆ + 1. Corollary 2. If G is a c o-c omp ar ability gr aph, then cub( G ) ≤ 2 ∆ , wher e ∆ is the maximu m de gr e e of G . Pr o of. Le t V denote V ( G ) and let | V | = n . Since G is a comparability g raph, there exists a par tial order ≺ in G on the no de set V s uch that ( u, v ) ∈ E ( G ) if and only if u ≺ v o r v ≺ u . This partial order gives a direction to the e dg es in E ( G ). W e can run a top ologica l sort o n this partia l order to pro duce a linea r 7 ordering of the vertices, say , f : V → { 1 , . . . , n } . The top o logical sort ens ures that if u ≺ v , then f ( u ) < f ( v ). Now, let ( u, v ) ∈ E ( G ) and le t w be a vertex such that f ( u ) < f ( w ) < f ( v ). W e will show that w is adjacent to either u or v in G . Suppo se not. Then ( u, w ) , ( w , v ) ∈ E ( G ) and there fo re u ≺ w a nd w ≺ v . Now, by tr ansitivity of ≺ , this implies that u ≺ v , whic h mea ns that ( u, v ) ∈ E ( G ) – a contradiction. Ther efore, any vertex w such that f ( u ) < f ( w ) < f ( v ) in the or der ing f is a dja c ent to either u or v . Since the maximum degree of G is ∆ , ther e can b e at mo st 2 ∆ − 2 vertices b etw e en with f ( · ) v alue b e t ween f ( u ) and f ( v ). Thus, the width o f the order ing given by f is at most 2 ∆ − 1 and by Theorem 2, we have our bound o n cubicity . A c aterpil lar is a tree s uch that a path (calle d the spine ) is obtained by removing all its leav es. In the pro of of Theorem 3.16 of [7], Kloks et al. show that ev ery connected A T-free graph G has a spa nning caterpillar subgraph T , such that adjacent no des in G are at a distance at most four in T . Moreover, for any edg e ( u, v ) ∈ E ( G ) suc h that u a nd v are at distance exa ctly four in T , bo th u a nd v are leaves of T . Le t p 1 , . . . , p k be the no des alo ng the spine of G . Corollary 3. If G is an A T-fr e e gr aph, cub ( G ) ≤ 3 ∆ − 1 , wher e ∆ is the maximum de gr e e of G . Pr o of. Le t L i denote the set o f leav e s of T adjacent to p i . Clearly , | L i | ≤ ∆ and L i ∩ L j = ∅ for i 6 = j . F o r a n y s et S o f vertices, let h S i deno te an arbitrary ordering of the vertices in set S . Let < u > denote ordering with just one vertex u in it. If α = u 1 , . . . , u s and β = v 1 , . . . , v t are tw o orderings of vertices in G , then let α ⋄ β denote the order ing u 1 , . . . , u s , v 1 , . . . , v t . Let A = < L 1 > ⋄ < p 1 > ⋄ < L 2 > ⋄ < p 2 > ⋄ · · · ⋄ < L k > ⋄ < p k > b e a linear ordering of the vertices of G . O ne can use the prop erty of T stated b efore the theo rem to easily show that A is a linea r ordering of the vertices of G with width at most 3 ∆ − 2 . The corolla r y will then follow from Theo rem 2 . References 1. P . A fshani and T. Chan. 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