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Cycles and Bicycles Cycles and Bicycles Peter F. STADLER Institut f¨ur Theoretische Chemie, Universit¨at Wien Santa Fe Institute, New Mexico http://www.tbi.univie.ac.at/~studla Bled, Slovenia, January 2002 1 Undirected Graphs We consider cycles in simple undirected or directed graphs G(V; E). E : : : edge set V : : : vertex set In the directed case we distinguish cycles : : : without orientation circuits : : : following the orientation of the edges. U ⊆ E . jEj-dimension vector (indexed by the edges): A cycle (=subgraph with even vertex degrees) is an edge-disjoint union of elementary cycles. TWO-CONNECTED GRAPHS only: every edge is contained in a cycle. 2 Vector Spaces of Edges 1 if e 2 U Ue = 0 if e 2= U Incidence matrix H of G: 1 if x 2 e Hxe = 0 if x 2= e All cycles satisfy HU = 0 over GF(2) = (f0; 1g; ⊕; ·) Cycle space C = vector space spanned by cycles Dimension: dimC = γ(G) = jEj − jV j + components(G) 3 Bases of a Vector Space A set fx1; : : : ; xLg of vectors is linearly independent if the linear equation λ1x1 + λ2x2 + : : : λlxl = 0 has no solution except λ1 = λ2 = · · · = λL = 0. A basis of a vector space is a maximal set of linearly independent vectors. Each vector x can be written as a linear combination of the basis elements B = fy1; y2; : : : ; yng: x = λ1y1 + λ2y2 + · · · + λnyn 4 Cycle Bases Kirchhoff basis: graph ) spanning tree ) cycles C(T; e) for all e 2= T . A basis of B is a Kirchhoff basis (or a strictly fundamental basis) of C if there is a spanning tree T such that B = fC(T; e)je 2 EnT g. 5 Fundamental Cycle Bases A collection of ν(G) cycles in G is called fundamental if there is an ordering of these cycles such that Cj n (C1 [ C2 [ · · · [ Cj−1) 6= ; for 2 ≤ j ≤ ν(G) Strictly Fundamental implies fundamental but not vice versa. 6 Ear Decomposition Every two-connected graph has an ear decomposition. Each ear decomposition defines a basis of the cycle space C. 7 Minimal Length Cycle Bases Length jCj of cycle C = number of edges Length of a cycle basis `(B) = C2B jCj. P Relevant Cycles (Plotkin '71, Vismara '97); C is contained in a minimal cycle basis () C cannot be written as a ⊕-sum of shorter cycles 8 Some Counter examples • Not every MCB is strictly fundamental (Horton, Deo) • Not every MCB is fundamental (examples are quite complicated) • The MCB of a planar graph not necessarily consists of faces 8 7 6 1 2 5 3 4 9 Who cares about MCBs? Chemical Ring Perception (SSSR). O O O O O O O O 10 Analysis of chemical reaction networks. SO3 SO2 SO + SO + O+ 2 O(1D) SO + O 2 S+ O O 2 S NaO + S2 NaS 2 NaO 2 NaS Na + NaO Na Na2 Na2O Na2S NaO 3 + Na2O + Na2S + Na2 Reaction network of Io's athmosphere 11 Matroid Property The cycles of G form a matroid =) A minimal cycle basis is obtained from the set of all cycles by a greedy procedure: 1. Sort set C of cycles by length B ; 2. while(C 6= ;) C C n fCg if B [ fCg independent: B B [ fCg. Problem: exponentially many cycles. Necessary conditions: elementary (all vertices have degree 2) short (isometric) for all vertices x; y in C, the cycle C contains a shortest paths between x and y. 12 Horton's Polynomial Time Algorithm A cycle is edge short if C contains an edge e = fx; yg and a vertex z such that C = fx; yg [ P (x; z) [ P (y; z) where P (x; z) and P (y; z) are shortest paths. If C is relevant then it is edge-short (Horton'87). Construct (at most) jEj × jV j edge-short cycles. Horton showed that even if P (x; z) is not unique one may choose any shortest path, i.e., the jEj × jV j cycles contain a minimal cycle basis. Alternative trick [Hartvigsen'94]: small perturbation of edge length to make minimum weight cycle basis unique. 13 Graph Operations The length of minimal cycle bases does not behave \well" under many simple graph operations: G1 G2 G1 has ν(G1) = 3 and `(G1) = 38. Deletion of a single edge leads to G2 with ν(G2) = 2 but `(G2) = 44. Similar: other graph minor operations. 14 Cartesian and Strong Graph Products Given two non-empty graphs G = (VG; EG) and H = (VH ; EH ): Cartesian product GH: Vertex set VG × VH Edges: (x1; x2)(y1; y2) is an edge in EGH iff either x2 = y2 and x1y1 2 EG or if x1 = y1 and x2y2 2 EH Direct product G × H: Vertex set VG × VH Edges: (x1; x2)(y1; y2) is an edge if x1y1 2 EG and x2y2 2 EH Strong product G H: Vertex set VG × VH Edges: those of the direct product and those of the Cartesian product 15 Relevant Cycles in Product Graphs G = (VG; EG) and H = (VH; EH) be two non-empty graphs, TG and TH spanning trees, BG and BH cycle bases of G and H. Hammack's Basis for Cartesian Products (1999): H1 = fe fje 2 TG; f 2 THg y H2 = fC jC 2 BG; y 2 VHg x H3 = f Cjx 2 VG; C 2 BHg ∗ B = H1 [ H2 [ H3 ∗ B is i general not minimal even if BG and BH are minimal. • Counterexample: C5K2 Hammack basis: 4 squares and 2 pentagons. Minimal length basis: 5 squares and 1 pentagon. 16 Hammack's basis is minimal if BG and BH consist of triangles and squares only. Consider a triangle-free graph for the moment. Idea. Start with the Hammack basis and replace as many cycles in the fibres as possible by squares from C = fe fje 2 EG; f 2 EHg. Lemma. For all C 2 BG and all x; y 2 VH there is a collection of squares in C such that Cx = Cy ⊕ squares. It is hence sufficient to have one copy BG and one copy of BH in one G and one H-fibre. The rest of the basis can be completed from C. 17 To show that there are no relevant cycles that are not contained in C or a fibre we consider the following procedure: Set δ(x) = sum of distance of x 2 GH from two fixed fibres. Define for any cycle C ∗ in H: deg ∗ (x) δ(C∗) = C δ(x); 2 xX2VC∗ We show that we keep adding squares from C to C∗ until we arrive at δ(C(k)) = 0 and jC(k)j ≤ jC∗j. Since C∗ is either strictly shorter than C∗ or it is the edge-disjoint union of a cycle in xH and a cycle in Gy C∗ cannot be relevant. It remains to show that it is impossible to replace any further basis cycle by squares from C. (not hard) For graphs with triangles: retain triangles in each fibre and use the longer basis cycles in a single fibre only. 18 Total Basis Length in Iterated Products `(GH) = `(G) + `(H)+ 3 tG(jVHj − 1) + 3tH(jVGj − 1) + 4 (jEGj − tG)(jVHj − 1) + jEHj − tH)(jVGj − 1)− (jVHj − 1)(jVGj − 1) Substitute Gn = GGn−1 ' Gn−1G; G1 = G and set a = jEj=jV j and τ = tG=jV j, where tG is the number of triangles in the MCB of G. `(Gn+1) = `(G) + `(Gn) + 3τ jV j(jV nj − 1) + njV jn(jV j − 1) + 4(a − τ) jV j(jV jn − 1)+ njV jn(jV j − 1) − (jV jn − 1)(jV j − 1) : Dividing by ν(Gn+1) and setting ξ = 1=V eventually yields: τ a − τ L1 = lim Ln = 3 + 4 : n!1 a a 19 Graphs with a Unique MCB Outerplanar graphs Two-connected o.p. graphs have a Hamiltonian Cycle H and each chord separates G into 2 two-connected outerplanar graphs G1 and G2. Take an edge e in H. The shortest cycle C through e contains at least one chord, hence C is a member of the MCB. Split C along the chords and repeat. 20 RNA Secondary Structures outerplanar. \loops" = cycles of the unique MCB. interior base pair interior base pair closing base pair G G A A 5 G U G 5 5 C 3 C U 3 3 C A C G U closing base pair closing base pair stacking pair hairpin loop interior loop closing base pair C A interior base pairs G 5 C G A A G 5 3 C A U 3 interior base U C pair A A closing base pair bulge multiple loop Pseudoknots: MCB usually not unique. U A A3' C G C C 5' A 360 G G C A U G C G U pk 1 G C 2 C G G C C 10 U A G C 350 U U A pk A U 160 C Ψ C A 3 U C G C C C 2108b G 8a 200 A U C A G G C C G A U G C G C U A C G C A U G C G G G Ψ 9 A C m5U U A 340 G C C U G G G G U U G C G U 12 G A U C G A C G G 220 A A U G C C G G A C U 230 20 G U 330 A U A G U U A A 240 U A A U A U C A A G C 2a U A C G 190 U A G G U 10a A U G A U G C G G A 250 A G 150 30 C A 280 A U G A C G G C C A U G U A U C A A G U G U G C 7 G G U A U 320 10b A U 160C G U A pk C A C G A U C 2 C G U G U A C G C 2b C G pk A U A A G 4 A A 290 190 A G C 6c C G G C A G A 270 G C 10c C A G 150 G C 3 + 4 40 G C A C G C C A U G U G C G C C U A A U G C C A G C possibilities 2c G U A G C 170 G 310 C G 170 6b A G C C A U C A U A U G C A U U U 2d G C A U U A U A C C A 7 G C G 11a C G C A U 6a A A C C U A C 50 C G 300 G U 140 G G A A U G A C A U 90 100 G C A 3 G C 180 G G U U C G C A A A C G A C G A C C U C A G G C A G A A C G G 140 C A U G A A U 110 U G A A A A G 70 A C U A C A U U U 180 U 5a A U G 60 G A C 130 A C U C G G G G C A C U C 4 G G A A C U C C C A 5b G C G A A C A pk U A 1 80 U U G 120 A A tmRNA from E.coli with its pseudoknots 22 Exchangability of Relevant Cycles Set R of relevant cycles of an undirected graph can be computed efficiently by Vismara's algortithm (1997).
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