Tree-width of hypergraphs and surface duality

T ree-width of h yp ergraphs and surface du alit y F r´ ed´ eric Mazoit ⋆ LaBRI Universit ´ e Bordeaux, 351 cours de la Lib´ era tion F-33405 T alence cedex, F rance Frederic.M azoit@labri.fr Abstract. In Graph Minor I I I, R ob ertson and Sey mour co njecture that W e p rove that giv en a hypergraph H on a surface of Euler genus k , the tree-width of H ∗ is at most the maximum of tw( H ) + 1 + k and t he maximum size of a hyperedge of H ∗ . 1 Preliminaries A surfac e is a connected compact 2-man yfold without bo undaries. A surface Σ can b e obtained, up to homeomor phism, by adding k ( Σ ) “cr osscaps” to the sphere. k ( Σ ) is the Euler genus or just genus of the surface. Let Σ be a sur face. A graph G = ( V , E ) on Σ is a drawing of a gra ph in Σ , i.e. each vertex v is an element of Σ , each edg e e is an op en c ur ve betw een t wo v ertices, and edges are pairwise disjoin t. W e only consider graphs up to homomorphism. A face of G is a connected co mpo nent of Σ \ G . W e denote b y V ( G ), E ( G ) and F ( G ) the vertex, edge and face sets of G . W e only consider 2-c el l graphs, i.e. graph whos e faces ar e homeo morphic to op en discs. The Euler formula link s the num b er of vertices, edges and faces of a gr aph G to the genus of the surface | V ( G ) | − | E ( G ) | + | F ( G ) | = 2 − k ( G ) . The set A ( G ) = V ( G ) ∪ E ( G ) ∪ F ( G ) of atoms of G is a partition o f Σ . Tw o A tom x a nd y of G are incident if x ∩ ¯ y o r y ∩ ¯ x is non empty , ¯ z b e ing the clos ure of z . A cut-e dge in a graph G on Σ is an edg e e s eparates G , i.e . G intersects at least tw o connected comp onents o f Σ \ ¯ e . As an example, if a pla nar g raph G has a cut-vertex u , any lo op on u that g o es “ around” a connected comp onent of G \ { u } is a cut-edg e . Let G = ( V ∪ V E , L ) b e a bipartite gra ph on Σ . The graph G can b e seen as the incidence graph of a h yp ergr aph. F o r each v e ∈ V E , merge v e and its inciden t edges into a hyp er e dge e , and call v e its c enter . Le t E b e the set of all hyp er e dges . A hyp er gr ap h on Σ is any such pair H = ( V , E ). F or brevity , w e also say e dges for hyperedg es. W e extend the notions of cut-edges, 2-cell graphs, atoms and incidence to hyper g raphs. Moreover, since they naturally corr esp ond to abstr act graphs and hypergr aphs, graph and hypergr aph on surface inherit ter minology ⋆ Researc h supp orted by the frenc h ANR-p ro ject ”Graph decompositions and algo- rithms (GRAA L) ”. from them. F o r example, we denote | e | the num b er o f vertices incident to a hyperedge e , and we denote α ( H ) the maximum size of an edge of H . No te tha t a graph on Σ is also a hypergr aph on Σ . The dual o f a hyper graph H = ( V , E ) on Σ is obtained by choosing a vertex v f for every face f of H . F or ev ery edg e e of ce n ter v e , we pick up an edge e ∗ as follows: choo se a lo ca l orien tation of the surface around v e . This lo ca l orien tation induces a cyclic o rder v 1 , f 1 , v 2 , f 2 , . . . , v d , f d of the ends o f e and of the faces incident with e (p ossibly with rep etition). The edge e ∗ is the edge obtained by “rotating” e and whose ends ar e v f 1 , . . . , v f d . A tr e e-de c omp osition of a hypergr a ph H on Σ is a pair T = ( T , ( X v ) v ∈ V ( T ) ) with T a tree a nd ( X v ) v ∈ V ( T ) a family of b ags such that: i. S v ∈ V ( T ) X v = H ; ii. ∀ x , y , z ∈ V ( T ) with y o n the path fro m x to z , X x ∩ X z ⊆ X y . The width of T is tw( T ) = max  | V ( X t ) | − 1 ; t ∈ V ( T )  and the tr e e-width t w ( H ) of H is the minimum width of one of its tree-decomp ositions . T ree-width was in tro duced by Robertso n and Seymour in connectio n with graph minors. In [RS84], they conjectured that for a planar graph G , t w ( G ) and t w( G ∗ ) differ by at most o ne . In a n unp ublished pap e r, Lapoir e [Lap96] prov es a mor e genera l result: for any hypergr aph H in an or ientable surfa c e Σ , t w ( H ∗ ) ≤ ma x(t w ( H ) + 1 + k ( Σ ) , α ( H ∗ ) − 1 ). Nev ertheles s, his pro of is r ather long and tec hnical. L ater, Bouchitt ´ e et al. [BMT03] gav e an easier proo f for planar graphs. Here we generalise s Lap oire’s result to ar bitrary sur faces while being less technical. T o a void technicalities, w e supp ose that H is connected, con tains at least t wo edges, has no p ending vertices (i.e. vertices incident with only one edge ) and no cut-edge. 2 P-trees and d uality F rom no w on, H = ( V , E ) is a hypergr a ph on a surface Σ . The b or der of a partition µ of E is the set o f vertices δ ( µ ) that are incide nt with edges in a t least t wo parts of µ , a nd the bo rder of X ⊆ E is the b order of the partition { X , E \ X } . A pa rtition µ = { X 1 , . . . , X p } of E is c onne cte d if there is a c onne cting p artition { V 1 , X 1 , F 1 , . . . , V p , X p , F p } of A ( H ) \ δ ( µ ) so that each V i ∪ X i ∪ F i is connec ted in Σ . A p-tr e e of H is a tree T whose internal node s hav e degree three a nd whose leav es are lab elled with the edges of H in a bijective w ay . Removing a n internal no de v of T results in a partition µ v of E . La b elling each int erna l no de v of T with δ ( µ v ), turns T int o a tree-deco mpo sition. The t r e e-width of a p-tree is its tr e e-width , seen as a tree-decomp osition. A p-tree is c onne cte d if all its nodes partitions are connected. Let { A, B } be a connected bipartition of H and { V A , A, F A , V B , B , F B } a corres p o nding connecting partition. W e define a c ontr acte d h yp erg raph H / A a s follows. Consider the incidence graph G H ( V ∪ V E , L ) of H , and identify the edges in A with their centers. By adding edg es trough faces in F A , we ca n make G H [ A ∪ V A ] connected. W e then contract A ∪ V A int o a single edg e center v A . T o make the r esulting g raph bipar tite, we remove a ll v A -lo ops. When removing a lo op e incident to only one face F , the new face F ∪ e is not a disc but a cr osscap. Since the bor der of F ∪ e is a lo op, we can “cut” Σ along this lo op and replace F ∪ e by an o p e n disc while decreasing the genus o f the sur face. The obtained graph is the bipartite gr a ph of H / A . A connected partition { A, B } is non tr ivial if neither H/ A nor H / B are equal to H . W e need the following folklore lemma: Lemma 1. F or any c onne cte d bip artition { A, B } of H , tw( H ) ≤ max  t w ( H/ A ) , t w ( H/ B )  . If δ ( { A, B } ) b elongs to a b ag of an optimal t r e e-de c omp osition, then t w ( H ) = max  t w ( H/ A ) , tw( H /B )  . Let S b e a set of vertices of H . An S - bridge is a minimal subset X of E with the prop erty that δ ( X ) ⊆ S . There are tw o kind of S - bridges: sing letons containing an e dg e whos e ends all b elo ng to S and sets E C containing a ll the edges incident to at leas t o ne vertex in C , a connec ted co mpo nent of G \ S . The S -bridge s pa rtition E . W e define the abstract gr aph G /S whose vertices ar e the S -bridge s and in w hich { X , Y } is an e dg e if there is a face incident with both an edge in X a nd an edge in Y . A k ey fact is that any bipar tition { A, B } of V ( G /S ) such that G /S [ A ] a nd G /S [ B ] is connected corr esp onds to the connected bipartition {∪ A, ∪ B } . Prop ositi on 1 . Ther e exists a c onne cte d p-tr e e T of H with tw( T ) = tw( H ) . Pr o of. By induction on | E | , if | E | ≤ 3, s inc e H has no cut-edge, the only p-tree is connected and optimal. W e can suppo se that | E | ≥ 4. W e claim that there exists a c o nnected non trivia l bipartitio n { A, B } of E whose b order is contained in a ba g of an optimal tree-deco mpo sition of H . Two cases aris e: – If the trivial one v ertex tree-decomp osition whos e bag is H is optimal, we consider the graph G /V . Since they are in bijection with the edges of H , and since H has no cut edge, G /V has a t least four vertices and no cut vertex. There thus exists a bipartition { A, B } of V ( G /V ) with | A | , | B | ≥ 2, G /V [ A ] and G /V [ B ] connected which gives a connected non trivial bipartition of E . – Otherwise, there exists a s eparator S contained in a bag of an o ptimal tree- decomp osition of H . Let C and D b e t wo connected compo nent of H \ S , and S C and S C their corres po nding S -bridg es. Since H con tains no p ending vertex, | S C | , | S D | ≥ 2 . Let x and y b e the vertices o f G /S corres p o nding to S C and S D . T a ke a spa nning tree of G /S . Removing an edge b etw een x and y leads to a connected non-tr iv ial bipartition of E , which finis hes the pro of of the claim. Since { A, B } is connected, e A and e B are r esp ectively no t cut-edg es in H / A and H/ B . By induction, there exists connected p-trees T A and T B of optimal width of H / A and H /B . B y removing the leaves lab elled e A and e B and adding an e dge b etw een their resp ective neighbour, w e obtain from T A ⊔ T B a p-tree of H whic h is connected. Its width is ma x(t w ( T / A ) , tw( T /B )) which is equal, by Lemma 1 to tw( H ). ⊓ ⊔ Because o f the natural bijection b etw een E ( H ) and E ( H ∗ ), a p-tree T of H also corres po nds to a p-tree T ∗ of H ∗ . Prop ositi on 2 . F or any c onne cte d p-tr e e T of H , t w ( T ∗ ) ≤ max(tw( T ) + 1 + k ( Σ ) , α ( H ∗ ) − 1 ) . Pr o of. Let v be a v ertex of T lab elled X v in T and X ∗ v in T ∗ . If v is a leaf, then X ∗ v = { e ∗ } and | X ∗ v | − 1 ≤ max(t w ( T ) + 1 + k ( Σ ) , α ( H ∗ ) − 1). Otherwise, let { A, B , C } b e the E -partition asso ciated to v . The lab el of v in T and T ∗ is resp ectively X v = δ ( { A, B , C } ) and X ∗ v , the s et o f faces incident with edges in at least t wo par ts among A , B and C . As for the pr o of of P rop osition 1, since { A, B , C } is connected, we may contract A (and B a nd C ). B ut since we now ca re ab out the fa c es of H , we have to b e mor e careful. W e wan t an upp er b ound o n | X ∗ v | , we may th us add but not remov e fa c e s to X ∗ v . So a dding edges to make G H [ A ∪ V A ] connected is O K, but we canno t r emov e a lo op e on say v A incident with tw o fac e s in X ∗ v . Instead, we cut Σ along e and fill the holes with op en discs. While doing so, we r emov ed e , we cut v A in t wo siblings , and we decrease d the genus of Σ . After c ontracting A , B and C , we obtain a bipar tite gr aph G v on Σ ′ that has | X v | + 3 + s v ertices with s the num b er of sibling s , a t least | X ∗ v | faces and with k ( Σ ′ ) ≤ k ( Σ ) − s . Since G v is bipartite and faces in X ∗ v are incident with at lea s t 4 edges, 2 | E ( G v ) | = 4 | F 4 | + 6 | F 6 | + · · · ≥ 4 | F ( G v ) | with F 2 k the set of 2 k -g ones faces of G v , and th us | E ( G v ) | ≥ 2 | F ( G v ) | . If we apply E uler’s for mula to G v on Σ ′ , w e obtain: | X v | + 3 + s − | E ( G v ) | + | F ( G v ) | = 2 − k ( Σ ′ ) ≥ 2 − k ( Σ ) + s . Adding this to | E ( G v ) | ≥ 2 | F ( G v ) | , we get | X v | + 1 + k ( Σ ) ≥ | F ( G v ) | ≥ | X ∗ v | which pro ves that | X ∗ v | − 1 ≤ max(t w ( T ) + 1 + k ( Σ ) , α ( H ∗ ) − 1), and thus t w ( T ∗ ) ≤ max(t w ( T ) + 1 + k ( Σ ) , α ( H ∗ ) − 1). ⊓ ⊔ Let us now pr ov e the main theorem. Theorem 1. F or any hyp er gr aph H on a su rfac e Σ , t w ( H ∗ ) ≤ max  t w ( H ) + 1 + k ( Σ ) , α ( H ∗ ) − 1  . Pr o of. By Prop osition 1, let T be a connected p-tree of H suc h that t w( T ) = t w ( H ). By P rop osition 2, t w ( T ∗ ) ≤ max(t w ( T ) + 1 + k ( Σ ) , α ( H ∗ ) − 1 ). Since t w ( H ∗ ) ≤ tw( T ∗ ), we deduce, tw( H ∗ ) ≤ max(t w( H ) + 1 + k ( Σ ) , α ( H ∗ ) − 1). ⊓ ⊔ References BMT03. V. Bouc hitt´ e, F. Mazoit, and I. T o dinca. Chordal embeddings of planar graphs. Discr ete M athematics , 273:85– 102, 2003. Lap96. D. Lap oire. T reewidth and duality for planar hypergraphs. Man uscript http://www .labri.fr/perso /lapoire/papers/dual planar treewidth.ps , 1996. RS84. N. R ob ertson and P . D. Seymour. Graph Mi nors. I I I. Planar T ree-Width. Journal of Combinatorial The ory Series B , 36(1):49–64 , 1984.