Ribbon graphs and their minors Iain Moffatt Royal Holloway, - PowerPoint PPT Presentation

Ribbon graphs and their minors Iain Moffatt Royal Holloway, University of London British Combinatorial Conference, 9 th July 2015

Graph minors 1 Embedded graphs Graph minors Ribbon graph minors edge deletion Excluded minors H is a minor of G if it is Matroids obtained by edge deletion, vertex deletion edge contraction & vertex edge contraction deletion. Robertson-Seymour Theorem ◮ In any infinite collection of graphs, one graph is a minor of another. ◮ Every minor-closed family of graphs is characterised by a finite set of excluded minors. ◮ G can be embedded in R 2 ⇐ ⇒ no K 5 - or K 3 , 3 -minor. ◮ G can be embedded in R P 2 ⇐ ⇒ none of 35 minors. ◮ G can be embedded in surface Σ ⇐ ⇒ none of finite list of minors. 10

Cellularly embedded graphs G is cellularly embedded if it is 2 Embedded graphs drawn on a surface Σ such that Ribbon graph minors ◮ edges don’t cross, Excluded minors ◮ faces are discs. Matroids contraction deletion not cell. embedded 10

Ribbon graphs Ribbon graphs describe cellularly embedded graphs. Embedded graphs e i g h b o l e t e f a c e e n u r h d e s a k o t o 3 d Ribbon graph minors Excluded minors e g s T a s p i n l u e f a c e k e i n Matroids Ribbon graph A “topological graph” with ◮ discs for vertices, ◮ ribbons for edges. Considered up to homeomorphisms that preserve vertex-edge structure and cyclic order at vertices. = = 10

Ribbon graph minors Edge and vertex deletion Embedded graphs 4 Ribbon graph minors e vertex deletion Excluded minors d g e d Matroids e l e t i o n Edge contraction G / e G T o contract e = ( u , v ) : non-loop ◮ attach a disc to each ∂ -cpt. of n.-o. loop v ∪ e ∪ u ◮ remove o. loop v ∪ e ∪ u 10

Ribbon graph minors Embedded graphs R.-S. theory for embedded graphs? 5 Ribbon graph minors ◮ Claim: the “correct” minors for embedded graphs. Excluded minors ◮ Conjecture: Every minor-closed family of ribbon Matroids graphs is characterised by a finite set of excluded minors. ◮ But wait, is this not just a special case of Robertson-Seymour? ◮ The two types of minor are incompatible. ◮ Contracting loops seems too hard. Can we just delete loops like in the graph case? ◮ No, e.g., 10

Excluded minor characterisations Proposition Embedded graphs Ribbon graph minors G is orientable ⇐ ⇒ no 6 Excluded minors -minor Matroids � 2 × genus if orientable Euler genus : γ ( G ) := genus if non-orientable Theorem G is of Euler genus ≤ n ⇐ ⇒ no minor in ◮ n odd: { G | γ ( G ) = n + 1 , G = � [ 1 vert., 1 ∂ -cpt ] } ◮ n even: { G | ( γ ( G ) = n + 1 , G = � [ 1 vert., 1 ∂ -cpt ]) or ( γ ( G ) = n + 2 , orient, G = � [ 1 vert., 1 ∂ -cpt ]) } Corollary Orientable G is of genus ≤ n ⇐ ⇒ no minor in { G | ( γ ( G ) = n + 2 , orient G = � [ 1 vert., 1 ∂ -cpt ]) } 10

Excluded minor characterisations Embedded graphs Knots & links can be represented by ribbon graphs Ribbon graph (Dasbach, Futer, Kalfagianni, Lin, Stoltzfus, ’05): minors 7 Excluded minors 1 1 3 3 Matroids 2 1 3 2 2 5 5 4 4 4 6 6 8 6 8 8 8 5 7 7 7 Theorem G represents link diagram ⇐ ⇒ no minor isomorphic to , , 10

Excluded minor characterisations Partial dual – form the dual w.r.t. only some edges. Embedded graphs Ribbon graph minors 8 Excluded minors G ∗ = G { 1 , 2 , 3 } Matroids G { 1 } G { 1 , 2 } G Theorem Partial dual of plane graph ⇐ ⇒ no minor isomorphic to , , Theorem Partial dual of R P 2 graph ⇐ ⇒ no minor isomorphic to , , 10

A connection with matroids ← → Graph minors Matroid minors Embedded graphs (Robertson, Seymour) (Geelen, Gerards, Whittle) Ribbon graph minors Excluded minors matroids (via bases) delta-matroids 9 Matroids M = ( E , B ) M = ( E , F ) ◮ B � = ∅ , subsets of E ◮ F � = ∅ , subsets of E ◮ B satisfies SEA ◮ F satisfies SEA ∗ ◮ X , Y ∈ B = ⇒ | X | = | Y | ◮ X , Y ∈ F = ⇒ | X | = | Y | ∆ -matroid (quasi-trees) Graphic matroid (trees) 1 1 2 2 3 3 M G = ( E , {{ 2 } , { 3 }} ) D G = ( E , {{ 1 , 2 , 3 }{ 2 } , { 3 }} ) ∗ ∀ X , Y ∈ F , u ∈ X △ Y = ⇒ ∃ v ∈ X △ Y s.t. { u , v }△ X ∈ F . 10

Embedded graphs Ribbon graph minors Excluded minors Thank you! 10 Matroids ◮ I. Moffatt, Ribbon graph minors and low-genus partial duals , Annals Combin., to appear. arXiv:1502.00269 ◮ I. Moffatt, Excluded minors and the graphs of knots , J. Graph Theory, to appear. arXiv:1311.2160 ◮ C. Chun, I. Moffatt, S. Noble and R. Rueckriemen, Matroids, Delta-matroids and Embedded Graphs , preprint. arXiv:1403.0920 10