Blow-up structure of graphs excluding a tree or an apex-tree as a minor

BLO W-UP STR UCTURE OF GRAPHS EX CLUDING A TREE OR AN APEX-TREE AS A MINOR QUENTIN CLA US, GWENAËL JORET, AND CLÉMENT RAMBAUD Abstra ct. W e pro ve blo w-up structure theorems for graphs excluding a tree or an apex-tree as a minor. First, we sho w that for ev ery t -vertex tree T with t ⩾ 3 and radius h , and every graph G excluding T as a minor, there exists a graph H with pathwidth at most 2 h − 1 suc h that G is con tained in H ⊠ K t − 2 as a subgraph. This improv es on a recent theorem of Dujmo vić, Hickingbotham, Joret, Micek, Morin, and W o o d (2024), who prov ed the same result but with a larger b ound on the order of the complete graph in the pro duct. Second, we show that for every t -vertex tree T with t ⩾ 2 , radius h and maximum degree d , and every graph G excluding the ap ex-tree T + as a minor, where T + is the tree obtained b y adding a univ ersal vertex to T , there exists a graph H with treewidth at most 4 h − 1 such that G is contained in H ⊠ K 2( t − 1) d . The b ound on the treewidth of H is b est possible up to a factor 2 , and improv es on a 2 h +2 − 4 b ound that follows from a recent result of Dujmović, Hic kingb otham, Ho dor, Joret, La, Micek, Morin, Rambaud, and W o o d (2024). 1. Intr oduction In their Gr aph Minors series of pap ers, Rob ertson and Seymour show ed many results ab out the structure of graphs excluding a fixed graph as a minor. The very first of these results states that ev ery graph G excluding a t -vertex tree T as a minor has pathwidth b ounded by some function of t [ 10 ]. Later, Biensto ck, Rob ertson, Seymour and Thomas [ 1 ] (see also Diestel [ 3 ]) sho w ed an optimal upp er b ound of t − 2 on the path width of G : Theorem 1 (Biensto c k, Robertson, Seymour and Thomas [ 1 ]) . L et T b e a tr e e on t ⩾ 2 vertic es, and let G b e a gr aph excluding T as a minor. Then pw( G ) ⩽ t − 2 . While an upp er b ound of t − 2 on the pathwidth is b est p ossible in general, Dujmo vić, Hic k- ingb otham, Joret, Micek, Morin, and W o o d [ 4 ] recently show ed that if the tree T has radius h (with p ossibly h ≪ t ), then a graph excluding T as a minor is in fact “not far” from a graph of path width O ( h ) : Theorem 2 (Dujmović, Hic kingbotham, Joret, Micek, Morin, W oo d [ 4 ]) . F or every tr e e T with t vertic es, r adius h , and maximum de gr e e d , for every T -minor-fr e e gr aph G , ther e exists a gr aph H such that pw( H ) ⩽ 2 h − 1 and G is c ontaine d in H ⊠ K ( d + h − 2)( t − 1) . 1 Informally , the theorem shows that G is then contained in a graph of pathwidth 2 h − 1 , where ev ery vertex has b een “blo wn up” b y a clique whose size dep ends only on T . The upp er b ound (Q. Claus) Dép ar tement de Ma théma tiques, Université libre de Bruxelles, Belgium (G. Joret) Dép ar tement d’Informa tique, Université libre de Br uxelles, Belgium (C. Ram baud) Université Côte d’Azur, CNRS, Inria, I3S, Sophia Antipolis, France E-mail addresses : quentin.claus@ulb.be, gwenael.joret@ulb.be, clement.rambaud@inria.fr . Q. Claus and G. Joret are supp orted by the Belgian National F und for Scientific Research (FNRS). 1 Here and in the rest of the pap er, by ‘ G is contained in J ’ w e mean that G is isomorphic to a subgraph of the graph J . 1 2 CLAUS, JORET, AND RAMBAUD of 2 h − 1 on the pathwidth of H is b est p ossible, as shown in [ 4 ]. How ev er, determining the b est p ossible b ound on the size of the clique was left op en. Our first result is the following impro v ement on the latter: Theorem 3. F or every tr e e T with t ⩾ 3 vertic es and r adius h , and for every T -minor-fr e e gr aph G , ther e exists a gr aph H such that p w( H ) ⩽ 2 h − 1 and G is c ontaine d in H ⊠ K t − 2 . The b ound of t − 2 is almost optimal: The graph K t − 1 is T -minor-free and yet, if K t − 1 is con tained in H ⊠ K c for some graph H with pw( H ) ⩽ 2 h − 1 , then we m ust hav e c ⩾ ( t − 1) / 2 h . This is b ecause t − 2 = p w( K t − 1 ) ⩽ pw( H ⊠ K c ) ⩽ c (pw( H ) + 1) − 1 ⩽ 2 ch − 1 . Next, w e turn our atten tion to graphs excluding a fixed graph T + as a minor, where T is a tree and T + denotes the graph obtained from T by adding a new v ertex adjacent to all other vertices. Since T + is planar, it follows from the Grid-Minor Theorem of Rob ertson and Seymour [ 11 ] that graphs excluding T + as a minor hav e b ounded treewidth. Later, Leaf and Seymour [ 8 ] ga ve the following b ound on the treewidth: Theorem 4 (Leaf and Seymour [ 8 ]) . L et T b e a tr e e and let G b e a T + -minor-fr e e gr aph. Then t w( G ) ⩽ 3( | V ( T + ) |− 1) 2 . The upp er b ound on the treewidth w as recen tly impro ved to | V ( T + ) | − 2 by Liu and Y oo [ 9 ], whic h is b est p ossible: Theorem 5 (Liu and Y oo [ 9 ]) . L et T b e a tr e e, and let G b e a T + -minor-fr e e gr aph. Then t w( G ) ⩽ | V ( T + ) | − 2 . Indep enden tly , we found that a slight mo dification of the pro of of Theorem 4 gives a differen t pro of of Theorem 5 that is shorter than the one given in [ 9 ]. This pro of is given in the app endix. Observ e that Theorem 5 is similar to Theorem 1 except that treewidth is b eing b ounded instead of path width. It is natural to wonder whether a ‘blow-up’ analogue of Theorem 5 exists, which w ould b e similar to Theorem 2 . In a lo ose sense, this is known to b e the case, as follo ws from the follo wing v ariant of the Grid Minor Theorem: Theorem 6 (Dujmović, Hic kingb otham, Ho dor, Joret, La, Micek, Morin, Ramb aud, W o o d [ 5 ]) . F or every planar gr aph X , ther e exists a p ositive inte ger c such that for every X -minor- fr e e gr aph G , ther e exists a gr aph H of tr e ewidth at most 2 td( X ) − 4 such that G is c ontaine d in H ⊠ K c . In the ab ov e theorem, td( X ) denotes the treedepth of X . T aking X = T + for some tree T of radius h in Theorem 6 , and using the fact that T has treedepth at most h + 1 , and thus T + has treedepth at most h + 2 , we obtain the follo wing corollary: Corollary 7 (Dujmović et al. [ 5 ], implicit) . F or every tr e e T of r adius h , ther e exists a p ositive inte ger c such that for every T + -minor-fr e e gr aph G , ther e exists a gr aph H of tr e ewidth at most 2 h +2 − 4 such that G is c ontaine d in H ⊠ K c . Our second contribution is to show that the graph H in Corollary 7 can b e chosen so that H has treewidth 4 h − 1 , whic h is b est p ossible up to a factor 2 , and answ ers p ositiv ely a sp ecial case of Question 1 in [ 5 ]. W e also obtain an explicit b ound on the clique size c in the blow-up, namely c ⩽ 2( t − 1) d . BLOW-UP STRUCTURE OF GRAPHS EXCLUDING A TREE OR AN APEX-TREE AS A MINOR 3 Theorem 8. F or every tr e e T with t ⩾ 2 vertic es, r adius h , and maximum de gr e e d , for every T + -minor-fr e e gr aph G , ther e exists a gr aph H of tr e ewidth at most 4 h − 1 such that G is c ontaine d in H ⊠ K (2 t − 1) d . A straigh tforw ard mo dification of Prop osition 3 in [ 4 ] shows that H m ust ha v e treewidth at least 2 h in the ab o v e theorem, thus the upp er b ound of 4 h − 1 is within a factor 2 of optimal. W e susp ect that the upp er b ound of (2 t − 1) d on the clique size in the blow-up in Theorem 8 could b e further impro v ed to O ( t ) , as in Theorem 3 , how ev er the pro of of the latter result do es not seem to b e easily adaptable to this setting. The pap er is organized as follo ws. In Section 2 , we giv e the necessary definitions and notation. In Section 3 , we prov e Theorem 3 . Next, in Section 4 , w e prov e Theorem 5 and Theorem 8 . Finally , in Section 5 , we conclude with some op en problems. 2. Preliminaries W e consider simple, finite, undirected graphs. Given a graph G , w e denote by V ( G ) its vertex set, and by E ( G ) its edge set. F or c ∈ N , w e denote b y K c the complete graph on c v ertices. F or h, d ∈ N , w e denote b y T h,d the complete d -ary tree of radius h . F or a tree T , we denote b y T + the ap ex-tree obtained b y adding a universal vertex to T . A r o ote d tr e e is a tree where a v ertex is sp ecified to b e the ro ot. The height of a ro oted tree is the maxim um distance b etw een the ro ot and a v ertex of the tree. A r o ote d for est is a disjoint union of ro oted trees. A vertic al p ath of F is a path contained in some ro oted tree T of F ha ving the ro ot of T as one of its endp oints. F or a graph G , and a set S ⊆ V ( G ) , w e define the neighb orho o d of S in G as the set of vertices in V ( G ) \ S that are adjacen t to at least one v ertex in S , and we denote it b y N G ( S ) . If S con tains only one vertex v , we sometimes write N G ( v ) instead of N G ( { v } ) . W e will denote b y ∂ G ( S ) the set of v ertices in S that are adjacen t to at least one v ertex not in S . F or b oth notations N G ( S ) and ∂ G ( S ) , w e drop the subscript when the graph is clear from the context. F or a graph G , and set S ⊆ V ( G ) , we denote b y G − S the subgraph of G obtained by removing all the vertices of G contained in S . If S con tains only a vertex v , we simply write G − v instead of G − { v } . A graph H is a minor of a graph G if H can b e obtained from a subgraph of G by contracting edges. A graph G is H -minor-fr e e if H is not a minor of G . An H -mo del in a graph G , also called a mo del of H in G , consists of pairwise-disjoint v ertex sets { W v ⊆ V ( G ) | v ∈ V ( H ) } that each induce a connected subgraph in G , and such that for every edge v w of H , then there exists x ∈ W v and y ∈ W w suc h that x and y are adjacen t in G . Note that H is a minor of G if and only if G contains an H -mo del. A tr e e-de c omp osition of a graph G is a collection D := { B x ⊆ V ( G ) | x ∈ V ( T ) } of v ertex subsets of G called b ags , where T is a tree, and suc h that • for each vertex v ∈ V ( G ) , the set { x ∈ V ( T ) | v ∈ B x } induces a nonempty subtree of T , and • for each edge v w ∈ E ( G ) , there exists x ∈ V ( T ) such that b oth v and w are contained in the bag B x . W e will sa y that T is the tree asso ciate d to this decomp osition, and for every v ertex x ∈ V ( T ) , that B x is the bag asso ciate d to x . The width of the tree-decomp osition is max {| B x | | x ∈ 4 CLAUS, JORET, AND RAMBAUD V ( T ) } − 1 . The tr e ewidth of a graph G , denoted by tw( G ) , is the minimal width of a tree- decomp osition of G . A p ath-de c omp osition of a graph G , the width of a path-decomp osition and the p athwidth of a graph G , denoted by pw( G ) , are defined the same w a y except that the tree T is required to b e a path. It will b e con v enient to denote path-decomp ositions simply as a sequence B 1 , B 2 , . . . , B m of bags. In this case, we will call B 1 the first bag of the decomp osition. Let G b e a graph. A p artition of V ( G ) is a collection P of nonempt y subsets of V ( G ) such that every vertex of G is in exactly one set of the collection. Each elemen t of P is called a p art . The width of P is the maximum size of a part of P . The quotient G/ P is the graph with v ertex set P and where distinct parts P , P ′ ∈ P are adjacent in G/ P if and only if there exists a v ertex of P which is adjacent to a vertex of P ′ in G . The str ong pr o duct of tw o graphs G and H , denoted b y G ⊠ H , is the graph with v ertex set V ( G ⊠ H ) := V ( G ) × V ( H ) and such that t wo distinct vertices ( v , w ) and ( v ′ , w ′ ) in V ( G ⊠ H ) are adjacen t if and only if the following tw o conditions are verified: (i) v = v ′ , or v and v ′ are adjacen t in G (ii) w = w ′ , or w and w ′ are adjacen t in H . W e will use the follo wing easy observ ation throughout the rest of the pap er (often implicitly): Prop osition 9. Given two gr aphs G, H and a p ositive inte ger c , the gr aph G is c ontaine d in H ⊠ K c if and only if ther e exists a p artition P of V ( G ) of width at most c such that G/ P is isomorphic to a sub gr aph of H . 3. Ex cluding a tree In this section w e prov e Theorem 3 , which we restate here for conv enience. Theorem 3. F or every tr e e T with t ⩾ 3 vertic es and r adius h , and for every T -minor-fr e e gr aph G , ther e exists a gr aph H such that p w( H ) ⩽ 2 h − 1 and G is c ontaine d in H ⊠ K t − 2 . Let us recall that the statemen t of this result is the same as that of Theorem 2 pro ved in [ 4 ], except the clique size in the blow-up is decreased from ( d + h − 2)( t − 1) to t − 2 . In [ 4 ], the authors deriv e Theorem 2 as a consequence of the following lemma (Lemma 8 in [ 4 ]). Lemma 10 (Dujmo vić et al. [ 4 ]) . F or every h, d ∈ N with h + d ⩾ 3 , for every T h,d -minor- fr e e gr aph G , for every vertex r in V ( G ) , the gr aph G has a p artition P of width at most ( d + h − 2)(pw( G ) + 1) such that { r } ∈ P , and G/ P has a p ath-de c omp osition of width at most 2 h − 1 such that the first b ag c ontains { r } . T o see that Lemma 10 implies Theorem 2 , observ e that if G excludes a t -v ertex tree T of radius h and maximum degree d as a minor then in particular G excludes T h,d as a minor, and furthermore p w( G ) ⩽ t − 2 by Theorem 1 . The heart of the pro of of Lemma 10 in [ 4 ] is the following technical lemma, which is not stated explicitly in [ 4 ] but follows from their pro ofs. F or completeness, we note that this lemma can also b e derived as a sp ecial case of Lemma 19 (with R = V ( G ) ) introduced later in this pap er. Lemma 11 (Dujmo vić et al. [ 4 ], implicit) . L et G b e a gr aph, let x, y ∈ N , and let r ∈ V ( G ) . L et R := N G ( r ) . Supp ose that ther e exists a set X ⊆ V ( G ) \{ r } of size at most x satisfying the fol lowing c onditions: BLOW-UP STRUCTURE OF GRAPHS EXCLUDING A TREE OR AN APEX-TREE AS A MINOR 5 (i) F or e ach vertex v ∈ X , ther e exists a p ath in G fr om r to v internal ly disjoint fr om X . (ii) F or e ach c onne cte d c omp onent C of G − X − r c ontaining a vertex of R , ther e exists a p artition P C of V ( C ) of width at most x such that p w( C / P C ) ⩽ y . (iii) If G ′ is a minor of G with | V ( G ′ ) | < | V ( G ) | and ther e exists a G ′ -mo del in G such that r is c ontaine d in the br anch set of a vertex r ′ of G ′ , then ther e exist a p artition P G ′ of V ( G ′ ) of width at most x such that { r ′ } ∈ P G ′ , and a p ath-de c omp osition D ′ of G ′ / P G ′ of width at most y + 2 such that { r ′ } is in the first b ag of D ′ . Then ther e exist a p artition P of V ( G ) of width at most x such that { r } ∈ P , and a p ath- de c omp osition D of G/ P of width at most y + 2 such that { r } is in the first b ag of D . In order to prov e Theorem 3 , we will reuse the ideas from [ 4 ], and in particular Lemma 11 , in com bination with the follo wing lemma, whic h is a sligh t mo dification of a recent result in [ 6 ]. T o state this lemma, w e need the follo wing terminology: Giv en a ro oted forest F , a graph G , and a set R of v ertices of G , we sa y that an F -model in G is we akly R - r o ote d if the branch set of eac h of the ro ots of the trees of F con tains at least one vertex of R . T o keep notations light, it will b e conv enien t to also allow R to contain v ertices not in G in this definition, and thus w e do so. (This allows us for instance to write “weakly R -ro oted” instead of “weakly ( R \ X ) -ro oted” in the follo wing lemma.) Lemma 12. L et G b e a gr aph, let R ⊆ V ( G ) , and let F b e a r o ote d for est. If ther e is no we akly R -r o ote d mo del of F in G , then ther e exists a c onne cte d c omp onent T of F and a set X ⊆ V ( G ) with | X | < | V ( F ) | such that G − X do es not c ontain any we akly R -r o ote d mo del of T . Before proving Lemma 12 , let us we show that it implies the following lemma, which directly implies Theorem 3 , b y applying it with an arbitrary vertex r on every connected comp onen t of G , and c ho osing the ro ot of T in such a wa y that the height of T is the radius of T . Lemma 13. F or every r o ote d tr e e T with t ⩾ 3 vertic es and height h ⩾ 1 , for every c onne cte d gr aph G , for every r ∈ V ( G ) such that G do es not c ontain any we akly { r } -r o ote d mo del of T , ther e is a p artition P of V ( G ) of width at most t − 2 such that { r } ∈ P , and G/ P has a p ath-de c omp osition of width at most 2 h − 1 such that the first b ag c ontains { r } . Pr o of of L emma 13 , assuming L emma 12 . W e pro v e the lemma by induction on ( h, | V ( G ) | ) , in lexicographic order. The base case h = 1 is done similarly as in the pro of of Theorem 2 in [ 4 ], w e describ e it here for completeness. F or every i ∈ N , let V i := { v ∈ V ( G ) | dist G ( r , v ) = i } . If | V i | ⩾ t − 1 for some i ∈ N , then the set V 0 ∪ V 1 ∪ . . . ∪ V i − 1 (whic h is connected in G ) together with t − 1 vertices of V i tak en as singletons give a weakly { r } -ro oted T -model in G , a contradiction. Th us, | V i | ⩽ t − 2 , for all i ∈ N . Let m b e the largest in teger such that V m  = ∅ . Because G is connected, { V i | i ∈ N , i ⩽ m } is a partition of V ( G ) of width at most t − 2 . Moreo ver, V 0 = { r } , and it is easy to verify that  { V i , V i +1 } | 0 ⩽ i < m  is a path decomp osition of G/ P of width at most 1 = 2 h − 1 , where { r } = V 0 is in the first bag. This concludes the base case. No w for the inductiv e case, assume h ⩾ 2 . First, w e sho w that the h yp otheses of Lemma 11 are fulfilled for x = t − 2 and y = 2 h − 3 . Let F b e the ro oted forest obtained from T b y deleting its ro ot, where for each connected component of F , the ro ot of that tree is the only v ertex that was a child of the ro ot of T . Let R := N G ( r ) . Observ e that G − r cannot con tain a weakly R -ro oted mo del of F , b ecause otherwise G would contain a weakly { r } -ro oted mo del of T . Applying Lemma 12 to G − r , we obtain that there exists a connected comp onen t T ′ of F and a set X ⊆ V ( G − r ) of size at most t − 2 suc h that G − ( X ∪ { r } ) has no weakly R -ro oted T ′ -mo del. Without loss of generality , we may c ho ose X to b e inclusion-wise minimal 6 CLAUS, JORET, AND RAMBAUD with this prop erty . Notice also that by definition of F , the height h ′ of T ′ is strictly less than h . Let us p oin t out that the set X could b e empty , which is fine for our purp oses. In case X is not empt y , b y minimality of X , for eac h v ertex v ∈ X the graph G − r − ( X \ { v } ) contains a weakly R -ro oted mo del of T ′ . This mo del has to use v , so in particular it contains a path from v to a vertex of R , and thus G − ( X \ { v } ) contains a path from v to r . This path has to b e in ternally disjoin t from X , so the first condition of Lemma 11 is verified. Next, supp ose that C is a connected comp onent of G − ( X ∪ r ) containing a v ertex v ∈ R . By definition of X , C do es not con tain any weakly { v } -ro oted mo del of T ′ . Thus, since h ′ < h , w e may apply induction, and by the induction hypothesis, the second condition of Lemma 11 is v erified for the set X as well. Finally , suppose that G ′ is a minor of G such that | V ( G ′ ) | < | V ( G ) | , and such that there exists a mo del of G ′ in G and a vertex r ′ in G ′ whose branc h set in the mo del contains r . If there is a weakly { r ′ } -ro oted mo del of T in G ′ , then there is a weakly { r } -rooted mo del of T in G , a contradiction . So there is no weakly { r ′ } -ro oted mo del of T in G ′ , and b ecause G ′ has less vertices than G , we ma y apply the induction hypothesis, and w e deduce that the third condition of Lemma 11 is also verified. Therefore, w e may apply Lemma 11 to G and X , which yields the desired result. □ No w w e pro v e Lemma 12 . W e will need the following lemma due to Diestel [ 3 ]. W e remark that it is not stated explicitly in [ 3 ] but it follows from the main pro of in that pap er. Lemma 14 (Diestel [ 3 ], implicit) . L et G b e a c onne cte d gr aph, let v ∈ V ( G ) , and let T b e a r o ote d tr e e on t > 1 vertic es. If pw( G ) ⩾ t − 1 , then ther e exists Y ⊆ V ( G ) such that r ∈ Y , G [ Y ] c ontains a we akly { r } -r o ote d T -mo del, and G [ Y ] has a p ath-de c omp osition of width at most t − 1 whose last b ag c ontains ∂ G ( Y ) . The follo wing lemma, whose pro of dep ends on Lemma 14 , is a sligh t mo dification of a result in [ 6 ] (stated at the b eginning of the pro of of Theorem 2 in that pap er). The only difference is that, in [ 6 ], the mo dels are not ro oted. W e include a pro of of the lemma for completeness; w e emphasize that it is the same pro of as in [ 6 ], except for minor c hanges to make sure that the mo dels we build are ro oted in the appropriate w a y . Lemma 15. L et G b e a gr aph, let R ⊆ V ( G ) , let c b e a p ositive inte ger, let t 1 , . . . , t c b e p ositive inte gers with t 1 ⩽ . . . ⩽ t c , let T 1 , . . . , T c b e r o ote d tr e es such that | V ( T i ) | = t i for every i ∈ [ c ] , let x 1 , . . . , x c b e nonne gative inte gers at le ast one of which is nonzer o, and let I := { i ∈ N | 1 ⩽ i ⩽ c and x i ⩾ 1 } . Then, at le ast one of the two fol lowing c onditions ar e satisfie d (i) G c ontains p airwise vertex-disjoint sub gr aphs { M i,j | 1 ⩽ i ⩽ c, 1 ⩽ j ⩽ x i } such that for e ach i ∈ [ c ] and e ach j ∈ [ x i ] , M i,j c ontains a we akly R -r o ote d T i -mo del. (ii) Ther e exists X ⊆ V ( G ) with | X | ⩽ ( P i ∈ I x i t i ) − t max( I ) , and i ∈ I such that G − X do es not c ontain any we akly R -r o ote d T i -mo del. Pr o of. W e call the tuple ( G, R, c, T 1 , . . . , T c , x 1 , . . . , x c ) an instanc e . Let ( G, R, c, T 1 , . . . , T c , x 1 , . . . , x c ) b e an instance, and let m := min( I ) . The proof is b y induction on ( P c i =1 x i , | V ( G ) | ) , in lexicographic order. In the base case, P c i =1 x i = 1 , implying that x m = 1 and x i = 0 for every i ∈ { 1 , . . . , c } \ { m } . Either there is a weakly R -ro oted mo del BLOW-UP STRUCTURE OF GRAPHS EXCLUDING A TREE OR AN APEX-TREE AS A MINOR 7 of T m in G , and the first statemen t holds with M m, 1 = G , or there is no R -ro oted mo del of T m is G , and then the second statement holds with X = ∅ . F or the inductive case, assume that P c i =1 x i ⩾ 2 , and that the statemen t holds for strictly smaller v alues of the sum, or equal v alues of the sum and strictly smaller v alues of the num b er of vertices. If, for ev ery integer i ∈ I , there is no w eakly R -ro oted mo del of T i in G , then the sec- ond statement holds with X = ∅ . Th us, we may assume that there exists i ∈ I such that G con- tains an R -ro oted mo del of T i . If there exists a connected comp onent C of G suc h that V ( C ) ∩ R = ∅ , then we apply induction on the instance ( G − V ( C ) , R \ V ( C ) , c, T 1 , . . . , T c , x 1 , . . . , x c ) . If the first statement holds for this instance, then it holds also for ( G, R , c, T 1 , . . . , T c , x 1 , . . . , x c ) . If the seconds statement holds for ( G − V ( C ) , R \ V ( C ) , c, T 1 , . . . , T c , x 1 , . . . x c ) , then it also holds for ( G, R, c, T 1 , . . . , T c , x 1 , . . . x c ) with the same set X . Thus we ma y assume that every connected comp onen t of G con tains at least a vertex of R . If pw( G ) < t m − 1 , let Y := V ( G ) . Notice that in this case ∂ G ( Y ) = ∅ . If p w( G ) ⩾ t m − 1 , then there exists a connected comp onen t C of G such that pw( C ) ⩾ t m − 1 . Then we apply Lemma 14 on C , T m and an arbitrary vertex r ∈ R ∩ V ( C ) . In this case, let Y be the resulting subset of v ertices. Notice that, in b oth cases in the definition of Y , the follo wing tw o prop erties hold: • G [ Y ] contains a weakly R -rooted mo del of T i for some i ∈ I , and • G [ Y ] has a path-decomp osition ( B 1 , B 2 , . . . , B q ) of width at most t m − 1 suc h that ∂ G ( Y ) ⊆ B q . (In case Y = V ( G ) , the first prop ert y holds b ecause it holds for G .) Let ℓ b e the smallest in teger j b et ween 1 and q such that there exists i ∈ I such that G ℓ := G [ S ℓ j =1 B j ] contains a w eakly R -rooted mo del of T i . Let i ′ ∈ I b e such that G ℓ con tains a weakly R -ro oted mo del of T i ′ . W e claim that there is no edge b etw een any vertex of G ℓ − B ℓ and any vertex of G − V ( G ℓ ) . Supp ose for a con tradiction that there exists an edge uv ∈ E ( G ) with u ∈ V ( G ℓ ) \ B ℓ and v ∈ V ( G ) \ V ( G ℓ ) . First, notice that u ∈ S ℓ − 1 j =1 B j . If v ∈ Y , then v app ears only in bags of ( B 1 , . . . , B q ) with indices strictly bigger than ℓ . How ev er, u app ears in at least one bag of index strictly smaller than ℓ . Then, it follows from the definition of a path-decomp osition that u ∈ B ℓ , a con tradiction. Th us, v ∈ Y , and then u ∈ ∂ G ( Y ) and so u ∈ B q . Because u app ears in B q as w ell as in a bag of index strictly smaller than ℓ , w e should hav e u ∈ B ℓ , a con tradiction. Let G ′ := G − V ( G ℓ ) . Let x ′ i := x i for ev ery i ∈ { 1 , . . . , c }\{ i ′ } , and let x ′ i ′ := x i ′ − 1 . Let I ′ := { i ∈ I | x ′ i ⩾ 1 } . Observe that I ′ is not empty , since P i ∈ I x i ⩾ 2 . Apply induction to the instance ( G ′ , R \ V ( G l ) , c, T 1 , . . . , T c , x ′ 1 , . . . , x ′ c ) . If there exists a collection of vertex-disjoin t subgraphs { M ′ i,j | i ∈ I ′ , 1 ⩽ j ⩽ x ′ i } such that M ′ i,j con tains a weakly R -ro oted mo del of T i for every i ∈ I ′ and j ∈ { 1 , . . . , x ′ i } , then setting M i,j := M ′ i,j for every i, j such that 1 ⩽ i ⩽ c , i  = i ′ and 1 ⩽ j ⩽ x i , or i = i ′ and 1 ⩽ j ⩽ x i − 1 and setting M i ′ ,x i = G ℓ satisfies the first statemen t of the lemma. Otherwise, there is a set X ′ of size at most ( P i ∈ I ′ x ′ i t i ) − t max( I ′ ) and an integer z ∈ I ′ suc h that G ′ − X ′ do es not contain any weakly R -ro oted mo del of T z . Let X := X ′ ∪ B ℓ . W e hav e | X | = | X ′ | + | B ℓ | ⩽ X i ∈ I ′ x ′ i t i ! − t max( I ′ ) ! + t m = X i ∈ I x i t i ! − ( t max( I ′ ) + t i ′ − t m ) . 8 CLAUS, JORET, AND RAMBAUD If max( I ′ ) = max( I ) , then t max( I ′ ) + t i ′ − t m = t max( I ) + ( t i ′ − t m ) ⩾ t max( I ) b ecause m = min( I ) , and th us i ′ ⩾ m and so t i ′ ⩾ t m . If max( I ′ )  = max( I ) , then max( I ) = i ′ (and x i ′ = 1 ), and then t max( I ′ ) + t i ′ − t m = t max( I ) + ( t max( I ′ ) − t m ) ⩾ t max( I ) , b ecause max( I ′ ) ∈ I , thus max( I ′ ) ⩾ m , and thus t max( I ′ ) ⩾ t m . Th us, in b oth cases, w e hav e t max( I ′ ) + t i ′ − t m ⩾ t max( I ) , and hence | X | ⩽  P i ∈ I x i t i  − t max( I ) . T o fulfill the second statement, it suffices to show that G − X do es not con tain any weakly R -ro oted mo del of T z . Because B ℓ ⊆ X , and since we ha ve already shown that there is no edge in G b et ween an y vertex of G ℓ − B ℓ and any vertex of G − V ( G ℓ ) , it suffices to show that there is no w eakly R -ro oted mo del of T z in G ℓ − B ℓ and no weakly R -ro oted mo del of T z in G − V ( G ℓ ) − X . The first prop ert y holds b ecause of the definition (in particular, the minimalit y) of ℓ . The second prop ert y holds b ecause there is no weakly R -ro oted mo del of T z in G ′ − X ′ . This concludes the pro of. □ Lemma 12 follo ws from Lemma 15 b y letting T 1 , . . . , T c b e the connected comp onen ts of the forest F , and letting x 1 = · · · = x c = 1 . This concludes the pro of of Theorem 3 . 4. Ex cluding an apex-f orest Let us define some notions introduced in [ 7 ] (see also [ 2 ]) that we will use in this section: Let G and H be graphs and let S, R ⊆ V ( G ) . An H -mo del in G is said to b e S -r o ote d if the branch set of eac h v ertex of H con tains at least one v ertex of S . If H is a ro oted forest, an H -mo del in G is said to be ( S , R ) -r o ote d if it is S -ro oted and w eakly R -ro oted. T o k eep notations light when considering subgraphs, it will b e con venien t to also allow S and R to con tain vertices not in G in this definition, and thus we do so. A p ath-de c omp osition of ( G , S ) consists of an induced subgraph H of G such that S ⊆ V ( H ) and a path-decomp osition D of H such that for every connected comp onen t C of G − V ( H ) , there exists a bag of D containing all of N G ( V ( C )) . The p athwidth of ( G , S ) , denoted by pw( G, S ) , is the minimum width of a path-decomp osition of ( G, S ) . The authors of [ 7 ] pro ved the following theorem. Theorem 16 (Ho dor et al. [ 7 ]) . F or every for est F with at le ast one vertex, for every gr aph G and for every S ⊆ V ( G ) , if G has no S -r o ote d mo del of F , then pw( G, S ) ⩽ 2 | V ( F ) | − 2 . Using a slight mo dification of their pro of, one can obtain the following ro oted v ariant of their theorem, whic h we will use in this section: Theorem 17. F or every r o ote d tr e e T with at le ast one vertex, for every c onne cte d gr aph G and for every S, R ⊆ V ( G ) such that R is nonempty, if G has no ( S, R ) -r o ote d mo del of T , then p w( G, S ) ⩽ 2 | V ( T ) | − 2 . Next, taking the p oint of view of graph blow-ups, w e introduce a generalization of the definition of the pathwidth of ( G, S ) which will b e helpful for our purp oses. Let G b e a graph, let S ⊆ V ( G ) , and let k b e a p ositive integer. A k - p artition-p ath-de c omp osition of ( G , S ) consists of an induced subgraph H of G such that S ⊆ V ( H ) , a partition P H of V ( H ) of width at most k , and a path-decomp osition D H of H / P H suc h that for every connected comp onen t C of G − V ( H ) , there exists a bag B in D H suc h that N G ( V ( C )) is contained in the union of the parts of P H con tained in B . W e will sa y that the induced subgraph H , the partition BLOW-UP STRUCTURE OF GRAPHS EXCLUDING A TREE OR AN APEX-TREE AS A MINOR 9 P H and the path-decomp osition D H are asso ciate d to this decomp osition. W e define the k - p artition-p athwidth of ( G , S ) as the minim um width of a path-decomp osition asso ciated to a k -partition-path-decomp osition of ( G, S ) , and w e denote it by ppw( k , G, S ) . Observe that for ev ery k ′ ⩽ k , we hav e pp w( k ′ , G, S ) ⩾ ppw( k , G, S ) . W e will use this fact later. No w we mov e on to the pro of of Theorem 8 . Doing so, w e will pro v e the follo wing analogue of Theorem 3 , in the setting of S -ro oted mo dels. The arguments are strongly inspired b y the pro of of Theorem 2 in [ 4 ] and our pro of of Theorem 3 : Theorem 18. L et G b e a gr aph, let S ⊆ V ( G ) , and let T b e a t -vertex tr e e with t ⩾ 2 , maximum de gr e e d , and r adius h . If G do es not c ontain any S -r o ote d mo del of T , then pp w((2 t − 1) d, G, S ) ⩽ 2 h − 1 . First, we prov e an analogue of Lemma 11 (whic h in fact implies Lemma 11 in the case S = V ( G ) ). The pro of is an adaptation of the pro of of Lemma 8 in [ 4 ]. Lemma 19. L et G b e a c onne cte d gr aph, let S ⊆ V ( G ) , let x, y b e inte gers with x ⩾ 1 and y ⩾ 0 , let r ∈ V ( G ) , and let R := N G ( r ) . Supp ose that ther e exists a set X ⊆ V ( G ) \{ r } of size at most x such that: (i) F or every v ∈ X , ther e exists a p ath in G fr om r to v internal ly disjoint fr om X . (ii) F or every c onne cte d c omp onent C of G − X − r c ontaining a vertex of R , we have pp w( x, C, S ∩ V ( C )) ⩽ y . (iii) F or every minor G ′ of G with | V ( G ′ ) | < | V ( G ) | and such that ther e exists a mo del M of G ′ in G and a vertex r ′ of G ′ whose br anch set in M c ontains r , letting S ′ b e the set of vertic es of G ′ whose br anch sets in M c ontain at le ast one vertex of S , ther e exists an x -p artition-p ath-de c omp osition of ( G ′ , S ′ ) of width at most y + 2 such that the asso ciate d p artition c ontains { r ′ } and the first b ag of the asso ciate d p ath-de c omp osition c ontains { r ′ } . Then ther e exists an x -p artition-p ath-de c omp osition of ( G, S ) of width at most y + 2 such that the asso ciate d p artition c ontains { r } and the first b ag of the asso ciate d p ath-de c omp osition c ontains { r } . Pr o of. First, we note that if R is empty then G consists of only the vertex r , and it is easy to define an x -partition-path-decomp osition of G with the desired prop erties. Th us, we ma y assume that R is not empty . Next, we deal with the case where X is empty . In this case, every connected comp onen t of G − X − r = G − r con tains a vertex of R since G is connected and R = N G ( r ) . Prop erty (ii) implies then that ppw( x, G − r, S \ { r } ) ⩽ y . T aking an x -partition-path-decomp osition of G − r of width at most y , and adding { r } to the asso ciated partition and to all the bags of the asso ciated path-decomp osition, we obtain an x -partition-path-decomposition of ( G, S ) of width at most y + 1 such that the asso ciated partition has { r } as one of its parts, and the first bag of the asso ciated path-decomp osition contains { r } . The result follows. Thus, for the remainder of the pro of, w e ma y assume that X is nonempty . Let G 1 , . . . , G p b e the connected comp onen ts of G − X − r that contain a vertex of R . By Prop ert y (ii) , for each i ∈ [ p ] , there exists A i ⊆ V ( G i ) , P i a partition of A i of width at most x , and a path-decomp osition D i of G [ A i ] / P i suc h that: (i) S ∩ V ( G i ) ⊆ A i ; (ii) D i has width at most y ; and (iii) for every connected comp onen t C of G i − A i , there exists a bag of D i that contains all the parts of P i in tersecting N G i ( V ( C )) . 10 CLAUS, JORET, AND RAMBAUD Let Z b e the union of the v ertex sets of all connected comp onen ts of G − X − r having no v ertices in R . Consider a v ertex v ∈ X , and let P v b e a path from r to v internally disjoint from X , which exists by Prop erty (i) . Because the neighbor of r in P v b elongs to R , and thus is in G i for some i ∈ [ p ] , and since v is the only v ertex in P v whic h is also in X , P v − { v , r } is fully con tained in G i , and so P v a v oids Z . Thus, G [ S v ∈ X V ( P v )] is connected and av oids Z . Let G ′ b e the graph obtained from G by contracting S v ∈ X V ( P v ) in one vert ex r ′ and deleting all the other vertices of G not in Z , so that V ( G ′ ) = { r ′ } ∪ Z . Observe that X ∪ { r } ⊆ S v ∈ X V ( P v ) . Because X is nonempty , | X ∪ { r }| ⩾ 2 , th us | V ( G ′ ) | < | V ( G ) | , and hence by Prop ert y (iii) , there exists A ′ ⊆ V ( G ′ ) , a partition P ′ of width at most x of A ′ ha ving { r ′ } as one of its parts, and a path-decomp osition D ′ of G [ A ′ ] / P ′ suc h that: (i) ( S ∩ Z ) ∪ { r ′ } ⊆ A ′ ; (ii) D ′ has width at most y + 2 , and its first bag contains { r ′ } ; and (iii) for every connected comp onen t C of G ′ − A ′ , there exists a bag of D ′ that con tains all the parts of P ′ in tersecting N G ′ ( V ( C )) . Let A := { r } ∪ X ∪ ( ∪ 1 ⩽ i ⩽ p A i ) ∪ ( A ′ \ { r ′ } ) and P :=  { r } , X  ∪ ( ∪ 1 ⩽ i ⩽ p P i ) ∪ ( P ′ \  { r ′ }  ) . Observ e that S ⊆ A and that P is a partition of A of width at most x . Let D b e the sequence of subsets of v ertices of G [ A ] / P obtained from the concatenation of {{ r } , X } , D 1 , . . . , D p and D ′ b y adding { r } and X to ev ery bag coming from the D ′ i ’s and replacing { r ′ } by X in ev ery bag of D ′ con taining { r ′ } . W e argue that D is a path- decomp osition of G [ A ] / P . It is easy to chec k that every part of P app ears in consecutive bags. Indeed, the D i ’s and D ′ are path-decompositions, X w as added to all the bags coming from the D i ’s, the only part p ossibly in common betw een the bags coming from the D i ’s and the bags coming from D ′ is X , and X is also in the first bag coming from D ′ . Then, we sho w that ever y part of P is in at least one bag of D . This is true by construction for { r } and X . Ev ery part of P which is in P ′ i for some i ∈ [ p ] is in at least one bag of D i , and thus in at least one bag of D . Every part of P which is in P ′ app ears in at least one bag of D ′ and, since { r ′ } ∈ P , app ears then also in at least one bag of D . Th us, every part of P app ears indeed in at least one bag of D . Next, we show that for ev ery tw o distinct parts P , P ′ ∈ P that are adjacen t in G [ A ] / P , there is a bag of D containing b oth P and P ′ . By construction, one of the following holds: (i) P P ′ ∈ E ( G [ A i ] / P i ) for some i ∈ [ p ] , and then there is a bag W of D i with P , P ′ ∈ W , and b y construction W ∪ {{ r } , X } is a bag in D . (ii) P P ′ ∈ E ( G [ A ′ ] / P ′ ) , and so there is a bag W of D ′ with P , P ′ ∈ W , and b y construction W or ( W \ { r ′ } ) ∪ { X } is a bag in D , and contains b oth P and P ′ . (iii) P = X or P ′ = X , and assume without loss of generalit y that P = X . If P ′ ∈ P i for some i ∈ [ p ] , then any bag W of D i con taining P ′ will yields a bag W ∪ {{ r } , X } in D , which con tains b oth P and P ′ . Similarly , if P ′ = { r } , then {{ r } , X } contains b oth P and P ′ . If P ′ ∈ P ′ , then { r ′ } P ′ ∈ E ( G [ A ′ ] / P ′ ) , and so there is a bag W of D ′ with { r ′ } , P ′ ∈ W , and therefore  W \  { r ′ }  ∪ { X } is a bag in D con taining b oth P and P ′ . (iv) P = { r } or P ′ = { r } , and assume without loss of generality that P = { r } . Then either P ′ = { X } , but this case has b een already considered, or P ′ ∈ P i for some i ∈ [ p ] . Then, an y bag W of D i con taining P ′ yields a bag W ∪ {{ r } , X } in D which contains b oth P and P ′ . BLOW-UP STRUCTURE OF GRAPHS EXCLUDING A TREE OR AN APEX-TREE AS A MINOR 11 Finally , we ha v e to show that for each connected comp onent C of G − A , there is a bag in D that contains every part of P intersecting N G ( V ( C )) . Since X ∪ { r } ⊆ A , C is either a connected comp onen t of G i − A i for some i ∈ [ p ] , or C is a connected comp onent of G ′ − A ′ . In the first case, there is a bag W of D i whose union con tains N G i ( V ( C )) , and since W ∪ {{ r } , X } is a bag of D and N G ( V ( C )) ⊆ N G i ( V ( C )) ∪ { r } ∪ X , we are done. In the second case, either r ′ ∈ N G ′ ( V ( C )) and so N G ( V ( C )) ⊆ ( N G ′ ( V ( C )) \ { r ′ } ) ∪ X , or N G ( V ( C )) = N G ′ ( V ( C )) . In b oth cases, any bag W of D ′ whose union contains N G ′ ( V ( C )) will yield a bag in D (namely  W \  { r ′ }  ∪ { X } or W ) whose union contains N G ′ ( V ( C )) . It follows that D is a path-decomp osition of G [ A ] / P , as claimed. Now, observe that D has width at most y + 2 , by construction. Therefore, the set A , the partition P , and the path- decomp osition D all together give an x -partition-path-decomp osition of ( G, S ) of width at most y + 2 suc h that the asso ciated partition contains { r } and the first bag of the asso ciated path-decomp osition contains { r } , as desired. This concludes the pro of. □ W e will also use the follo wing lemma, inspired by Lemma 23 in [ 7 ]. Lemma 20. L et G b e a gr aph, let S, R ⊆ V ( G ) , and let F b e a nonempty r o ote d for est. L et d b e the numb er of c onne cte d c omp onents of F and let p := p w( G, S ) . Assume that G do es not c ontain any ( S, R ) -r o ote d mo del of F . Then, ther e exists a set X ⊆ V ( G ) and a c onne cte d c omp onent T of F such that | X | ⩽ ( d − 1)( p + 1) , and G − X do es not c ontain any ( S, R ) -r o ote d mo del of T . Pr o of. The pro of is by induction on d . If d = 1 , then the result is true by taking X = ∅ and T = F . Next, we do the inductive step. Assume thus d ⩾ 2 , and that the lemma holds for smaller v alues of d . If for some connected comp onen t T of F , the graph G has no ( S, R ) -rooted mo del of T , then the lemma holds with X = ∅ and T . Th us, we may assume that, for every connected comp onen t T of F , there is some ( S, R ) -ro oted mo del of T in G . (Note that this implies in particular that S and R are b oth non empty .) W e ma y also assume that ev ery connected comp onen t of G con tains at least one vertex from S , since those that do not can b e freely discarded as they ha ve no impact on the existence of the set X . Consider a path-decomp osition of ( G, S ) of width at most p , with H denoting the corresp onding induced subgraph of G , and B 1 , . . . , B m the corresp onding sequence of bags. Observe that for ev ery connected comp onent C of G − V ( H ) we hav e N G ( V ( C ))  = ∅ (since ev ery connected comp onen t of G contains at least one vertex from S ) and moreov er N G ( V ( C )) ⊆ B i for some i ∈ [ m ] (by the definition of a path-decomp osition of ( G, S ) ). F or every i ∈ [ m ] , let V i b e the union of the vertex sets of the connected comp onen ts C of G − V ( H ) suc h that N G ( V ( C )) ⊆ B i . Th us, V ( H ) ∪ ( S m i =1 V i ) = V ( G ) . F or every j ∈ [ m ] , let G j := G [ S j i =1 ( B i ∪ V i )] . Observe that, by the definition of a path decomp osition of ( G, S ) , the set B j separates V ( G j ) from V ( G ) \ V ( G j ) in G . Let j ∈ [ m ] b e minimum such that G j con tains an ( S, R ) -ro oted mo del of some connected comp onen t T ′ of F . Note that j is well defined by our assumption on G , since G = G m con tains an ( S, R ) -ro oted of some connected comp onent of F . Let F ′ := F − V ( T ′ ) . Observ e that G − V ( G j ) do es not con tain an ( S, R ) -ro oted mo del of F ′ . Indeed, if it would b e the case, then G would contain an ( S, R ) -ro oted model of F , b ecause G j con tains an ( S, R ) -rooted mo del of T ′ , a contradiction. This implies that G − B j do es not contain any ( S, R ) -rooted mo del of F ′ , b ecause G j − B j do es not con tain any ( S, R ) -ro oted mo del of any tree of F ′ . 12 CLAUS, JORET, AND RAMBAUD Th us, we ma y apply the induction h yp othesis on G − B j and F ′ . Letting p ′ := p w( G − B j , S \ B j ) , this gives a connected comp onent T of F ′ and a set X ′ ⊆ V ( G − B j ) of size at most ( d − 2)( p ′ + 1) ⩽ ( d − 2)( p + 1) such that G − B j − X ′ do es not contain any ( S, R ) -ro oted mo del of T . Then, letting X := X ′ ∪ B j , it follows that | X | = | X ′ | + | B j | ⩽ ( d − 2)( p + 1) + ( p + 1) = ( d − 1)( p + 1) , and that G − X do es not contain any ( S, R ) -rooted mo del of T , as desired. This concludes the pro of. □ Next, we show an analogue of Lemma 10 (which implies Lemma 10 in the case S = V ( G ) but with a w orse b ound for the width of the partition). Lemma 21. L et T b e a t -vertex tr e e r o ote d tr e e of height h and maximum de gr e e d , wher e t ⩾ 2 and h, d ⩾ 1 . Supp ose that G is a c onne cte d gr aph with a distinguishe d set S ⊆ V ( G ) and vertex r ∈ V ( G ) such that G do es not c ontain any ( S, { r } ) -r o ote d mo del of T . Then, ther e exists a ((2 t − 1) d ) -p artition-p ath-de c omp osition of ( G, S ) of width at most 2 h − 1 such that the asso ciate d p artition c ontains { r } and the first b ag of the p ath-de c omp osition c ontains { r } . Pr o of. The pro of is b y induction on the pairs ( h, | V ( G ) | ) , in lexicographic order. If | V ( G ) | = 1 , the lemma is easily seen to hold b y taking the partition {{ r }} of V ( G ) . Supp ose thus | V ( G ) | ⩾ 2 . First, we consider the case h = 1 . Let A := S ∪ { r } and let G ′ b e the graph obtained from G b y con tracting each induced subgraph of G corresp onding to a connected comp onent of G − A in to one v ertex. Observ e that G ′ is connected, since G is connected. F or every nonnegative in teger i , let V ′ i := { v ∈ V ( G ′ ) | dist G ′ ( v , r ) = i } and V i := V ′ i ∩ A . Observe that if | V i | ⩾ d + 1 for some i ⩾ 1 , then contracting the connected subgraph of G ′ induced by V ′ 0 ∪ V ′ 1 ∪ · · · ∪ V ′ i − 1 in to a single v ertex, w e see that there is an ( S, { r } ) -ro oted mo del of T in G ′ , and th us also in G , a con tradiction. Thus, | V i | ⩽ d holds for every i ⩾ 0 . Let m ∈ N b e smallest suc h that V 2( m +1) = ∅ . Let P := {{ V 2 i ∪ V 2 i +1 } | i ∈ N , i ⩽ m } . If m  = 0 , let D i := {{ V 2 i ∪ V 2 i +1 } , { V 2 i +2 ∪ V 2 i +3 }} for every nonnegative integer i < m , and let D := { D i | i ∈ { 0 , 1 , . . . , m − 1 }} . If m = 0 , let D 0 := {{ V 0 , V 1 }} and D := { D 0 } . Because G [ A ] / P is a subgraph of a path, it is easy to see that D is a path-decomp osition of G [ A ] / P of width at most 1 . If m  = 0 , for eac h connected comp onen t C of G − A , C corresp onds to a vertex v ′ of G ′ , and there exists exactly one index i b et ween 1 and m − 1 such that v ′ ∈ V ′ i . Thus, eac h v ertex of A that has a neighbor in C in G b elongs to either V i − 1 , V i or V i +1 , whic h are all con tained in a common bag of D . Moreo ver, each part of P has size at most 2 d ⩽ (2 t − 1) d . The case m = 0 is easier, and v ery similar. Thus, the lemma holds for h = 1 . No w we do the inductive step. Assume that h ⩾ 2 . Let R := N G ( r ) . Let F b e the ro oted forest obtained from T by removing its root and adding a single-vertex comp onen t. Observe that F has at most d + 1 connected comp onents, and that if G − r con tains an ( S, R ) -ro oted mo del of F , then G con tains an ( S, { r } ) -ro oted mo del of T , a con tradiction. Thus, G − r has no ( S, R ) -rooted mo del of F . Note also that G − r has no ( S, R ) -ro oted mo del of T , since such a mo del would easily give an ( S, { r } ) -ro oted mo del of T in G . Theorem 17 implies then that p w( G − r, S \ { r } ) ⩽ 2 t − 2 . It follows from the ab ov e observ ations that we ma y apply Lemma 20 with G − r, S \ { r } , R , p = 2 t − 2 and F , and thus there exists a set X ⊆ V ( G − r ) of size at most (2 t − 1) d and a tree T ′ of F such that G − r − X do es not con tain an y ( S, R ) -rooted T ′ -mo del. Observe that, b ecause h ⩾ 2 , there is a tree of F whose height is at least 1 , and thus we ma y assume that T ′ has b een chosen so that its height is at least 1 . Without loss of generality , assume that X BLOW-UP STRUCTURE OF GRAPHS EXCLUDING A TREE OR AN APEX-TREE AS A MINOR 13 is inclusion-wise minimal suc h that G − r − X do es not con tain any ( S, R ) -ro oted T ′ -mo del. Let d ′ b e the maximum degree of T ′ . Observ e that d ′ ⩽ d . W e will show that the hypotheses of Lemma 19 are satisfied with S, x = (2 t − 1) d , y = 2 h − 3 , r , and X . W e start by showing that condition (i) of Lemma 19 holds. Let v ∈ X . By the minimality of X , there exists an ( S, R ) -ro oted mo del of T ′ in ( G − r ) − ( X \{ v } ) . Because G − r − X do es not contain any ( S, R ) -ro oted mo del of T ′ , this mo del has to contain v in at least one of its branc h sets, and th us in particular there is a path in ( G − r ) − ( X \ { v } ) b et ween v and some v ertex r ′ ∈ R . Th us, in G − ( X \ { v } ) , there is a path from v to r , as desired. Next, we sho w that condition (ii) of Lemma 19 holds. Let C b e a connected comp onent of G − X − r containing a v ertex r ′ ∈ R . By h yp othesis on X , C do es not con tain any ( S, R ) - ro oted T ′ -mo del. Because the height h ′ of T ′ satisfies 1 ⩽ h ′ ⩽ h − 1 , w e may apply the induction h yp othesis to T ′ , h ′ , d ′ , C , S ∩ V ( C ) and r ′ , and it follo ws in particular that pp w((2 t − 1) d, C , S ∩ V ( C )) ⩽ ppw((2 | V ( T ′ ) | − 1) d ′ , C , S ∩ V ( C ))) ⩽ 2( h − 1) − 1 = 2 h − 3 using that d ′ ⩽ d and | V ( T ′ ) | ⩽ t . Th us, condition (ii) is v erified. Finally , we show that condition (iii) of Lemma 19 holds. Let G ′ b e a minor of G such that | V ( G ′ ) | < | V ( G ) | , and such that there exists a mo del M of G ′ in G and a vertex r ′ in G ′ whose branc h set in M contains r . Let S ′ b e the set of vertices of G ′ whose branc h set in M con tains a vertex of S . Because | V ( G ′ ) | < | V ( G ) | , and b ecause G ′ do es not contain an y ( S ′ , { r ′ } ) - ro oted mo del of T —otherwise G would con tain an ( S, { r } ) -ro oted mo del of T —w e may apply the induction hypothesis. Therefore, there exists a ((2 t − 1) d ) -partition-path-decomposition of ( G ′ , S ′ ) of width at most 2 h − 1 such that the asso ciated partition contains { r ′ } and the first bag of the path-decomp osition con tains { r ′ } . Thus, condition (iii) is also verified. Therefore, w e may apply Lemma 19 as claimed, and the lemma follows. □ The pro of of Theorem 18 follows directly from applying Lemma 21 on each connected com- p onen t of the graph, by ro oting T so that its height is equal to its radius, and pic king an arbitrary v ertex r in each connected comp onent. Next, we will show Theorem 8 . Its pro of relies on Theorem 18 and a couple extra lemmas. W e start with the following lemma. W e remark that its pro of is an adaptation of a pro of idea that app ears inside the pro of of Lemma 15 in [ 7 ]. Lemma 22. L et G b e a c onne cte d gr aph, let u ∈ V ( G ) , let S := N G ( u ) and let x, y ∈ N . Supp ose that ther e exists an x -p artition-p ath-de c omp osition D of width at most y of ( G − u, S ) , with asso ciate d p artition P and asso ciate d induc e d sub gr aph H . Supp ose further that H has b e en chosen so that | V ( H ) | is minimal. Then, for e ach c onne cte d c omp onent C of G − u − V ( H ) , the gr aph G − V ( C ) is c onne cte d. Pr o of. Supp ose for contradiction that G − V ( C ) is not connected for some connected comp onent C of G − u − V ( H ) . This implies that there exists a connected comp onen t C ′ of G − V ( C ) such that V ( C ′ ) is disjoint from { u } ∪ N G ( u ) . Since G is connected, there exists an edge v w ∈ E ( G ) suc h that v ∈ V ( C ) and w ∈ V ( C ′ ) . Because C is a connected component of G − u − V ( H ) and w ∈ N G ( V ( C )) , w e hav e w ∈ { u } ∪ V ( H ) . Since w ∈ V ( C ′ ) and u ∈ V ( C ′ ) , we deduce that w ∈ V ( H ) . Let A := V ( H ) \ V ( C ′ ) . Because w ∈ V ( H ) ∩ V ( C ′ ) , w e hav e | A | < | V ( H ) | . Let P ′ := { P ∩ A | P ∈ P } \ {∅} . Let D ′ b e the path-decomp osition of G [ A ] / P ′ obtained from D by replacing 14 CLAUS, JORET, AND RAMBAUD eac h bag B by { P ∩ A | P ∈ B } \ {∅} . Our goal is to sho w that D ′ is an x -partition-path- decomp osition of ( G − u, S ) of width at most y with asso ciated subgraph G [ A ] and asso ciated partition P ′ . Since | A | < | V ( H ) | , this will contradict the c hoice of H and conclude the pro of of the lemma. The only nontrivial thing to show is that for every connected comp onent C ′′ of G − u − A , there exists a bag B ′ in D ′ suc h that the union of the sets in B ′ con tains N G − u ( V ( C ′′ )) . Let C ′′ b e a connected comp onent of G − u − A . If V ( C ′′ ) is disjoin t from V ( H ) \ A = V ( C ′ ) ∩ V ( H ) , then C ′′ is also a connected comp onent of G − u − V ( H ) . Thus, there exists a bag B ∈ D such that N G − u ( V ( C ′′ )) ⊆ S P ∈ B P . Because C ′′ is a connected comp onen t of G − u − A , we ha ve N G − u ( V ( C ′′ )) ⊆ A , and th us N G − u ( V ( C ′′ )) ⊆ S P ∈ B ( P ∩ A ) . Moreov er, { P ∩ A | P ∈ B } \ {∅} is a bag of D ′ , whic h concludes this case. W e consider now the remaining case: V ( C ′′ ) contains at least one vertex from V ( C ′ ) ∩ V ( H ) . Recall that C and C ′ are connected subgraphs of G − u − A , and that there is an edge b etw een V ( C ) and V ( C ′ ) in G . Hence, V ( C ) ∪ V ( C ′ ) induces a connected subgraph of G − u − A . Moreo v er, N G − u ( V ( C ′ )) ⊆ V ( C ) (b ecause C ′ is a connected comp onen t of G − V ( C ) ), and N G − u ( V ( C )) ⊆ V ( H ) ⊆ V ( C ′ ) ∪ A (b ecause C is a connected comp onent of G − u − V ( H ) ). Therefore, N G − u ( V ( C ) ∪ V ( C ′ )) ⊆ A . This implies that V ( C ) ∪ V ( C ′ ) induces a connected comp onen t of G − u − A . Because V ( C ′′ ) in tersects V ( C ′ ) , we deduce that V ( C ) ∪ V ( C ′ ) = V ( C ′′ ) . In particular, N G − u ( V ( C ′′ )) ⊆ A ∩ ( N G − u ( V ( C )) ∪ N G − u ( V ( C ′ ))) ⊆ N G − u ( V ( C )) ∩ A. Since D is an x -partition-path-decomp osition of ( G − u, S ) , and b ecause C is a connected comp onen t of G − u − V ( H ) , there is a bag B in D such that N G − u ( V ( C )) ⊆ S P ∈ B P . Therefore, w e conclude that N G − u ( V ( C ′′ )) ⊆ N G − u ( V ( C )) ∩ A ⊆ S P ∈ B ( P ∩ A ) . Since { P ∩ A | P ∈ B } \ {∅} is a bag in D ′ , this concludes the pro of that D ′ is an x -partition-path- decomp osition of ( G − u, S ) of width at most y . □ No w we prov e the following technical lemma, which directly implies Theorem 8 . Lemma 23. F or every tr e e T with t ⩾ 2 vertic es, r adius h , and maximum de gr e e d , for every gr aph G not c ontaining T + as a minor, and for every vertex u ∈ V ( G ) , ther e exists a p artition P of V ( G ) of width at most 2( t − 1) d such that { u } ∈ P , and G/ P has a tr e e-de c omp osition of width at most 4 h − 1 such that e ach b ag c ontaining { u } has size at most 2 h + 1 . Pr o of. The pro of is by induction on | V ( G ) | . The base case | V ( G ) | = 1 is trivial. Now supp ose | V ( G ) | > 1 . No w we do the inductiv e case. If G is not connected, w e may apply the induction hypothesis on each connected comp onent of G . So we ma y assume that G is connected. Let S := N G ( u ) . If G − u contains an S -ro oted mo del of T , then this mo del together with { u } yields a mo del of T + in G , a con tradiction. So G − u has no S -ro oted mo del of T , and so b y Theorem 18 , there exists a (2 t − 1) d -partition-path-decomp osition of ( G − u, S ) of width at most 2 h − 1 . Let H , P H and D H b e, resp ectively , the induced subgraph of G − u , the partition of V ( G − u ) and the path-decomp osition asso ciated to this decomp osition. F or the purp ose of this pro of, it will b e con v enient to see the path-decomp osition not as a sequence of bags (as in previous pro ofs) but as a tree-decomp osition where the tree indexing the decomp osition is a path; we let P denote this path. W e choose suc h a tuple ( H , P H , D H , P ) with | V ( H ) | minimum. Let C 1 , C 2 , . . . , C m b e the connected comp onents of G − u − V ( H ) . By assumption, for each i ∈ { 1 , . . . , m } , there exists a bag B i in D H suc h that the union of the sets in B i con tains N G − u ( V ( C i )) , and even BLOW-UP STRUCTURE OF GRAPHS EXCLUDING A TREE OR AN APEX-TREE AS A MINOR 15 N G ( V ( C i )) since N G ( u ) ⊆ V ( H ) . W e remark that B i and B j ma y p ossibly refer to the same bag for i  = j . Let i ∈ { 1 , . . . , m } . By the minimality of | V ( H ) | and by Lemma 22 , G − V ( C i ) is connected. Let G ′ i b e the graph obtained from G b y contracting the connected subgraph induced by G − V ( C i ) in to one vertex u i . Th us, V ( G ′ i ) = V ( C i ) ∪ { u i } and G ′ i is a minor of G , and in particular G ′ i has no T + minor. Since G is connected and | V ( G ) | ⩾ 2 , it follo ws that S and V ( H ) are not empt y , and hence | V ( G ′ i ) | < | V ( G i ) | . Hence, by the induction h yp othesis, there exists a partition P i of V ( G ′ i ) of width at most (2 t − 1) d and such that { u i } ∈ P i , and G ′ i / P i has a tree-decomp osition D i of width at most 4 h − 1 with asso ciated tree Y i suc h that each bag containing { u i } has size at most 2 h + 1 . Let D ′ i b e the tree-decomp osition of the graph induced in G b y V ( C i ) ∪ S Q ∈ B i Q obtained from D i b y replacing every bag B containing { u i } b y  B \  { u i }  ∪ B i . (W e keep the same tree asso ciated to the tree-decomposition.) F or every bag W ′ of D ′ i , either W ′ is a bag in D i and so has size at most 4 h , or W ′ =  W \  { u i  }  ∪ B i for some bag W of D i with { u i } ∈ W . But then, | W | ⩽ 2 h + 1 , and so | W ′ | ⩽ (2 h + 1) − 1 + | B i | ⩽ 4 h . Hence, D ′ i has width at most 4 h − 1 . No w define P := P H ∪ ( S m i =1 P i ) \  S m i =1  { u i }  , and let D and Y b e the tree-decomp osition of G/ P and the tree asso ciated to D , obtained from D H , P , the D ′ i ’s and the Y i ’s in the follo wing w a y: (i) Add { u } to all the bags of D H . (ii) F or each i ∈ { 1 , . . . , m } , link one vertex of Y i corresp onding to a bag of D i con taining { u i } to a v ertex of P whose corresp onding bag is B i . This gives the tree Y . Let us now sho w that P and D hav e the prop erties stated in the lemma. By construction, P is a partition of V ( G ) , and { u } ∈ P . Because every part of P is either { u } , or a part of P H , or a part of P i for some i ∈ { 1 , . . . , m } , w e hav e by induction that eac h part of P has size at most (2 t − 1) d . All the bags W of D containing { u } are of the form W = W H ∪  { u }} for some bag W H of D H , and so | W | ⩽ 2 h + 1 . Every other bag of D is a bag of D ′ i for some i ∈ { 1 , . . . , m } , and so has size at most 4 h since D ′ i has width at most 4 h − 1 . The only parts of P that p oten tially app ear in bags coming from multiples decomp ositions among D H and the D ′ i ’s are bags that app ear in D H , so it is easy to c heck that, by construction, for every part of P , the vertices of Y corresponding to bags of D con taining this part span a subtree of Y . Hence, D is a tree-decomp osition of G/ P , as claimed. This pro v es that P and D ha ve the desired prop erties, and concludes the pro of of the lemma. □ 5. Open questions Some of the b ounds w e hav e provided are not known to b e tight: (i) The b ound of t − 2 on the size of the clique in the blow-up in Theorem 3 is close to optimal but there migh t still b e a small ro om for improv emen t; we are only aw are of a Ω( t h ) lo wer b ound, as explained in the introduction. (ii) The b ound of (2 t − 1) d on the size of the clique in the blow-up in Theorem 8 is likely not optimal. W e exp ect that O ( t ) b ound should hold in this setting as w ell. Again, we only kno w of a low er b ound of Ω( t h ) . 16 CLAUS, JORET, AND RAMBAUD (iii) W e do not know if the b ound of 4 h − 1 on the treewidth in Theorem 8 is optimal, but w e know that it should b e at least 2 h , as mentioned in the introduction. W e exp ect the lo w er b ound to b e closer to the truth. References [1] Daniel Biensto ck, Neil Rob ertson, Paul D. Seymour, and Robin Thomas. Quickly exclud- ing a forest. Journal of Combinatorial The ory, Series B , 52(2):274–283, 1991. [2] Quentin Claus, Jędrzej Ho dor, Gwenaël Joret, and Pat Morin. Excluding an ap ex-forest or a fan as quic kly as p ossible. arXiv pr eprint , 2026. . [3] Reinhard Diestel. Graph minors I.: A short pro of of the path-width theorem. Combina- torics, Pr ob ability and Computing , 4(1):27–30, 1995. [4] Vida Dujmo vić, Rob ert Hic kingbotham, Gwenaël Joret, Piotr Micek, P at Morin, and Da vid R W o od. The excluded tree minor theorem revisited. Combinatorics, Pr ob ability and Computing , 33(1):85–90, 2024. . [5] Vida Dujmović, Rob ert Hickingbotham, Jędrzej Ho dor, Gwenaël Joret, Hoang La, Piotr Micek, P at Morin, Clémen t Rambaud, and Da vid R. W o o d. The grid-minor theorem revisited. Combinatoric a , 45(62), 2025. . [6] Vida Dujmo vić, Gwenaël Joret, Piotr Micek, and P at Morin. Tigh t b ound for the Erdős– Pósa prop ert y of tree minors. Combinatorics, Pr ob ability and Computing , 34(2):321–325, 2025. . [7] Jędrzej Ho dor, Hoang La, Piotr Micek, and Clément Rambaud. Quic kly excluding an ap ex- forest. SIAM Journal on Discr ete Mathematics , 40(1):282–307, 2026. . [8] Alexander Leaf and Paul D. Seymour. T ree-width and planar minors. Journal of Combi- natorial The ory, Series B , 111:38–53, 2015. [9] Chun-Hung Liu and Y oungho Y oo. T ree-width of a graph excluding an ap ex-forest or a wheel as a minor. arXiv pr eprint , 2025. . [10] Neil Rob ertson and Paul D. Seymour. Graph minors. I. excluding a forest. Journal of Combinatorial The ory, Series B , 35(1):39–61, 1983. [11] Neil Rob ertson and Paul D. Seymour. Graph minors. V. excluding a planar graph. Journal of Combinatorial The ory, Series B , 41(1):92–114, 1986. [12] Paul D. Seymour and Robin Thomas. Graph searching and a min-max theorem for tree- width. Journal of Combinatorial The ory, Series B , 58(1):22–33, 1993. Appendix A. Shor t proof of Theorem 5 Let us emphasize again that this short pro of is heavily based on the pro ofs in [ 8 ], in particular the pro ofs of statements 4.3 and 4.4 in that pap er: It is essen tially a copy of these pro ofs with some parts remov ed, and with some easy adaptations. T o emphasize the similarities, we chose to sta y as close as p ossible to the presentation in [ 8 ]. First, we need to recall some definitions. F or k ∈ N , a br amble of order k in a graph G is a set B of non-empty connected subgraphs of G , such that (i) every tw o members B 1 , B 2 ∈ B touch, that is, either V ( B 1 ) ∩ V ( B 2 )  = ∅ , or there is an edge of G with one end in V ( B 1 ) and the other in V ( B 2 ) , (ii) for every X ⊆ V ( G ) with | X | < k , there exists B ∈ B with X ∩ V ( B ) = ∅ . It was prov ed in [ 12 ] that for every p ositiv e integer k , a graph has treewidth at least k − 1 if and only if it has a bramble of order k . BLOW-UP STRUCTURE OF GRAPHS EXCLUDING A TREE OR AN APEX-TREE AS A MINOR 17 A sep ar ation of G is a pair ( A, B ) of subsets of V ( G ) such that A ∪ B = V ( G ) and every edge of G is either contained in A or contained in B . The or der of ( A, B ) is | A ∩ B | . F or X , Y ⊆ V ( G ) , an X – Y p ath is a path in G that is either a one-v ertex path with the v ertex in X ∩ Y or a path with one endp oint in X and the other endp oin t in Y such that no internal v ertices are in X ∪ Y . If ( A, B ) is a separation of G , we sa y that ( A, B ) left-c ontains a mo del { W v ⊆ V ( G ) | v ∈ V ( H ) } of H if | A ∩ B | = | V ( H ) | , for ev ery v ∈ V ( H ) , W v ⊆ A , and | W v ∩ A ∩ B | = 1 . If such a mo del exists w e sa y that ( A, B ) left-c ontains H . W e also need Menger’s Theorem: Theorem 24 (Menger’s Theorem) . L et G b e a gr aph and X, Y ⊆ V ( G ) . Ther e exists a sep ar ation ( A, B ) of G such that X ⊆ A , Y ⊆ B , and ther e exists | A ∩ B | p airwise disjoint X – Y p aths. W e will show the following lemma, whic h directly implies Theorem 5 (by letting G [ B ] − A mo del the ap ex vertex): Lemma 25. L et w ⩾ 1 b e an inte ger, let T b e a tr e e with | V ( T ) | = w , and let G b e a gr aph with tr e ewidth at le ast w . Then ther e is a sep ar ation ( A, B ) of G such that (i) | A ∩ B | = w , (ii) G [ B ] − A is c onne cte d and every vertex in B ∩ A has a neighb or in B \ A , and (iii) ( A, B ) left-c ontains T . Pr o of. Cho ose a vertex t 1 of T , and n um b er the other vertices t 2 , . . . , t w in suc h a w a y that for 2 ⩽ i ⩽ w , t i is adjacent to one of t 1 , . . . , t i − 1 . F or 1 ⩽ i ⩽ w , let T i b e the subtree of T induced by t 1 , . . . , t i . No w let G b e a graph with treewidth at least w ; we know that it has a bram ble B of order at least w + 1 . Cho ose B maximal; thus if C is a connected subgraph of G including a mem b er of B , then C ∈ B (because otherwise it could b e added to B , contrary to maximality). F or each X ⊆ V ( G ) with | X | ⩽ w , there is therefore a unique comp onen t of G − X that b elongs to B ; let its vertex set b e β ( X ) . Cho ose v ∈ β ( ∅ ) ; then β ( { v } ) ⊆ β ( ∅ ) , and the separation ( V ( G ) \ ( β ( ∅ ) \ { v } ) , β ( ∅ )) left- con tains T 1 . Consequen tly we may choose a separation ( A, B ) of G with the follo wing prop er- ties: (i) ( A, B ) has order at least one, and at most w ; sa y order k where 1 ⩽ k ⩽ w , (ii) ( A, B ) left-contains T k , (iii) β ( A ∩ B ) ⊆ B , (iv) there is no separation ( A ′ , B ′ ) of G of order strictly less than k , with A ⊆ A ′ and B ′ ⊆ B , and suc h that β ( A ′ ∩ B ′ ) ⊆ B ′ (v) sub ject to these conditions, | A | − | B | is maximum. Claim. Ther e is no sep ar ation ( A ′ , B ′ ) of G of or der k with A ⊆ A ′ , B ′ ⊆ B and ( A, B )  = ( A ′ , B ′ ) such that β ( A ′ ∩ B ′ ) ⊆ B ′ . Pr o of of the claim. Supp ose that there is suc h a separation ( A ′ , B ′ ) . F rom the optimalit y of ( A, B ) , ( A ′ , B ′ ) do es not left-contain T k . Consequently there do not exist k v ertex-disjoin t ( A ∩ B ) – ( A ′ ∩ B ′ ) paths; and so b y Menger’s Theorem there is a separation ( C , D ) of order less than k , with A ⊆ C and B ′ ⊆ D . Since β ( C ∩ D ) touc hes β ( A ′ ∩ B ′ ) , and β ( A ′ ∩ B ′ ) ⊆ B ′ ⊆ D , it follo ws that β ( C ∩ D ) ⊆ D . But this contradicts the fourth condition ab o ve. ♢ 18 CLAUS, JORET, AND RAMBAUD Claim. G [ B ] − A is c onne cte d, and every vertex of B ∩ A has a neighb or in B \ A . Pr o of of the claim. Now β ( A ∩ B ) ⊆ B \ A and hence is the v ertex set of a connected comp onent of G [ B ] − A . Let D := ( A ∩ B ) ∪ β ( A ∩ B ) , and let C := V ( G ) \ β ( A ∩ B ) ; then ( C, D ) is a separation of G satisfying the first four conditions ab ov e. F rom the optimality of ( A, B ) it follo ws that ( A, B ) = ( C, D ) , and in particular, β ( A ∩ B ) = B \ A . This prov es the first assertion. F or the second assertion, supp ose some v ertex v ∈ A ∩ B has no neighbor in B \ A . Then ( A, B − { v } ) is a separation, and since β ( A ∩ ( B \ { v } )) touc hes β ( A ∩ B ) , and hence is con tained in B \ { v } , this contradicts the fourth condition ab ov e. ♢ Claim. k = w Pr o of of the claim. Arguing b y contradiction, supp ose that k < w . Let { W j | 1 ⩽ i ⩽ k } b e a mo del of T k in A , where for every j ∈ { 1 , . . . , k } , W j is the branch set of t j , and con tains a unique v ertex v j of A ∩ B . Let i ∈ { 1 , . . . , k } b e suc h that t k +1 is adjacent in T to t i . By our second claim, v i has a neighbor in B \ A , say v k +1 . Let A ′ := A ∪ { v k +1 } ; then ( A ′ , B ) is a separation of G , and it left-contains T k +1 (with the mo del { W j | 1 ⩽ i ⩽ k } ∪ {{ v k +1 }} ). Moreo v er, since β ( A ′ ∩ B ) touches β ( A ∩ B ) , it is a subset of B ; and ( A ′ , B ) satisfies the fourth condition ab o ve b ecause of our first claim. But this con tradicts the optimalit y of ( A, B ) , and hence pro ves this claim. ♢ This concludes the pro of of the lemma. □