SLIDE 1 STABLE TRACES
Sandi Klavˇ zar, Jernej Rus
Faculty of Mathematics and Physics University of Ljubljana
SLIDE 2 Motivation
Gradiˇ sar et al. (2012) presented a novel polypeptide self-assembly strategy for nanostructure design.
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SLIDE 3 Motivation
Concatenating 12 coiled-coil-forming segments in an exact order (single polypeptide chain was arranged through the 6 edges of the tetrahedron in such way that every edge was traversed exactly twice).
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SLIDE 4
Motivation
Concatenating 12 coiled-coil-forming segments in an exact order (single polypeptide chain was arranged through the 6 edges of the tetrahedron in such way that every edge was traversed exactly twice).
SLIDE 5
Motivation
Concatenating 12 coiled-coil-forming segments in an exact order (single polypeptide chain was arranged through the 6 edges of the tetrahedron in such way that every edge was traversed exactly twice).
SLIDE 6
Motivation
Concatenating 12 coiled-coil-forming segments in an exact order (single polypeptide chain was arranged through the 6 edges of the tetrahedron in such way that every edge was traversed exactly twice).
SLIDE 7
Motivation
Concatenating 12 coiled-coil-forming segments in an exact order (single polypeptide chain was arranged through the 6 edges of the tetrahedron in such way that every edge was traversed exactly twice).
SLIDE 8
Motivation
Concatenating 12 coiled-coil-forming segments in an exact order (single polypeptide chain was arranged through the 6 edges of the tetrahedron in such way that every edge was traversed exactly twice).
SLIDE 9
Motivation
Concatenating 12 coiled-coil-forming segments in an exact order (single polypeptide chain was arranged through the 6 edges of the tetrahedron in such way that every edge was traversed exactly twice).
SLIDE 10
Motivation
Concatenating 12 coiled-coil-forming segments in an exact order (single polypeptide chain was arranged through the 6 edges of the tetrahedron in such way that every edge was traversed exactly twice).
SLIDE 11
Motivation
Concatenating 12 coiled-coil-forming segments in an exact order (single polypeptide chain was arranged through the 6 edges of the tetrahedron in such way that every edge was traversed exactly twice).
SLIDE 12
Motivation
Concatenating 12 coiled-coil-forming segments in an exact order (single polypeptide chain was arranged through the 6 edges of the tetrahedron in such way that every edge was traversed exactly twice).
SLIDE 13
Motivation
Concatenating 12 coiled-coil-forming segments in an exact order (single polypeptide chain was arranged through the 6 edges of the tetrahedron in such way that every edge was traversed exactly twice).
SLIDE 14
Motivation
Concatenating 12 coiled-coil-forming segments in an exact order (single polypeptide chain was arranged through the 6 edges of the tetrahedron in such way that every edge was traversed exactly twice).
SLIDE 15
Motivation
Concatenating 12 coiled-coil-forming segments in an exact order (single polypeptide chain was arranged through the 6 edges of the tetrahedron in such way that every edge was traversed exactly twice).
SLIDE 16 Motivation
Concatenating 12 coiled-coil-forming segments in an exact order (single polypeptide chain was arranged through the 6 edges of the tetrahedron in such way that every edge was traversed exactly twice).
Sandi Klavˇ zar, Jernej Rus (FMF) Stable Traces
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SLIDE 17 Mathematical model
Polyhedron P, composed from a single polymer chain, can be represented with the graph G(P):
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SLIDE 18 Mathematical model
Polyhedron P, composed from a single polymer chain, can be represented with the graph G(P): vertices are the endpoints of segments,
Sandi Klavˇ zar, Jernej Rus (FMF) Stable Traces
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SLIDE 19 Mathematical model
Polyhedron P, composed from a single polymer chain, can be represented with the graph G(P): vertices are the endpoints of segments, two vertices being adjacent if there is a segment connecting them,
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SLIDE 20 Mathematical model
Polyhedron P, composed from a single polymer chain, can be represented with the graph G(P): vertices are the endpoints of segments, two vertices being adjacent if there is a segment connecting them, every edge of G(P) corresponds to a coiled-coil dimer – two segments are associated with a fixed edge of G(P),
Sandi Klavˇ zar, Jernej Rus (FMF) Stable Traces
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SLIDE 21 Mathematical model
Polyhedron P, composed from a single polymer chain, can be represented with the graph G(P): vertices are the endpoints of segments, two vertices being adjacent if there is a segment connecting them, every edge of G(P) corresponds to a coiled-coil dimer – two segments are associated with a fixed edge of G(P), sequence of coiled-coil segments corresponds to a double trace in G(P).
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SLIDE 22 Double traces
Definition A double trace in a graph G is a circuit which traverses every edge exactly twice.
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SLIDE 23 Double traces
Definition A double trace in a graph G is a circuit which traverses every edge exactly twice. Theorem Every graph G has a double trace.
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SLIDE 24 Retracing
A double trace contains a retracing if it has an immediate succession of e by its parallel copy.
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SLIDE 25 Retracing
A double trace contains a retracing if it has an immediate succession of e by its parallel copy. v e
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SLIDE 26 Proper traces
Definition A proper trace is a double trace that has no retracing.
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SLIDE 27 Proper traces
Definition A proper trace is a double trace that has no retracing. Theorem (Sabidussi, 1977) G admits a proper trace if and only if δ(G) ≥ 2.
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SLIDE 28 Repetition
v a vertex of a G and u and w two different neighbors of v. Then double trace contains a repetition through v if the vertex sequence u → v → w appears twice in any direction (u → v → w or w → v → u).
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SLIDE 29 Repetition
v a vertex of a G and u and w two different neighbors of v. Then double trace contains a repetition through v if the vertex sequence u → v → w appears twice in any direction (u → v → w or w → v → u). v
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SLIDE 30 Repetition
v a vertex of a G and u and w two different neighbors of v. Then double trace contains a repetition through v if the vertex sequence u → v → w appears twice in any direction (u → v → w or w → v → u). v v
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SLIDE 31 Graphs that admit stable traces
Definition Stable trace is double trace without retracings and repetitions through its vertices.
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SLIDE 32 Graphs that admit stable traces
Definition Stable trace is double trace without retracings and repetitions through its vertices. Theorem G admits a stable trace if and only if δ(G) ≥ 3.
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SLIDE 33 Idea of proof
(⇐ =) If G is cubic proper and stable traces are equivalent,
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SLIDE 34 Idea of proof
(⇐ =) If G is cubic proper and stable traces are equivalent, inductions on ∆(G) (and number of vertices v: dG(v) = ∆(G)),
Sandi Klavˇ zar, Jernej Rus (FMF) Stable Traces
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SLIDE 35 Idea of proof
(⇐ =) If G is cubic proper and stable traces are equivalent, inductions on ∆(G) (and number of vertices v: dG(v) = ∆(G)), construction of new graph G ′,
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SLIDE 36 Idea of proof
G v v1 v2 . . . v∆
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SLIDE 37 Idea of proof
G v v1 v2 . . . v∆ G ′ v′ v′′ v1 v2 . . . v⌈ ∆
2 ⌉ v⌈ ∆ 2 ⌉+1
. . . v∆
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SLIDE 38 Idea of proof
(⇐ =) If G is cubic proper and stable traces are equivalent, inductions on ∆(G) (and number of vertices v: dG(v) = ∆(G)), construction of new graph G ′,
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SLIDE 39 Idea of proof
(⇐ =) If G is cubic proper and stable traces are equivalent, inductions on ∆(G) (and number of vertices v: dG(v) = ∆(G)), construction of new graph G ′, G ′ admits stable trace T ′,
Sandi Klavˇ zar, Jernej Rus (FMF) Stable Traces
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SLIDE 40 Idea of proof
(⇐ =) If G is cubic proper and stable traces are equivalent, inductions on ∆(G) (and number of vertices v: dG(v) = ∆(G)), construction of new graph G ′, G ′ admits stable trace T ′, T ′ without v′v′′ and v′′v′ is stable trace in G.
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SLIDE 41 Parallel double traces
Edge e is a parallel edge if it is traversed twice in the same direction.
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SLIDE 42 Parallel double traces
Edge e is a parallel edge if it is traversed twice in the same direction. If every edge of T is parallel, T is parallel double trace.
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SLIDE 43 Parallel double traces
Edge e is a parallel edge if it is traversed twice in the same direction. If every edge of T is parallel, T is parallel double trace. Proposition G admits a parallel double trace (parallel proper trace) if and only if G is Eulerian.
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SLIDE 44 Antiparallel double traces
If e is not traversed in the same direction, e is an antiparallel edge.
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SLIDE 45 Antiparallel double traces
If e is not traversed in the same direction, e is an antiparallel edge. T is an antiparallel double trace if every edge of T is antiparallel.
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SLIDE 46 Antiparallel double traces
If e is not traversed in the same direction, e is an antiparallel edge. T is an antiparallel double trace if every edge of T is antiparallel. Proposition Every graph admits an antiparallel double trace.
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SLIDE 47 Antiparallel double traces
If e is not traversed in the same direction, e is an antiparallel edge. T is an antiparallel double trace if every edge of T is antiparallel. Proposition Every graph admits an antiparallel double trace. Theorem (Thomassen, 1990) G admits an antiparallel proper trace if and only if δ(G) > 1 and G has a spanning tree T such that each connected component of G − E(T) either has an even number of edges or contains a vertex v, dG(v) ≥ 4.
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SLIDE 48 Open problem 1
Problem Characterize graphs that admit parallel (antiparallel) stable traces.
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SLIDE 49 Numerical results
Enumeration results for six polyhedra: the tetrahedron, 4-pyramid, 3-bipyramid, octahedron, 3-prism and 3-cube.
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SLIDE 50 Numerical results
Enumeration results for six polyhedra: the tetrahedron, 4-pyramid, 3-bipyramid, octahedron, 3-prism and 3-cube. Only non-equivalent traces are counted:
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SLIDE 51 Numerical results
Enumeration results for six polyhedra: the tetrahedron, 4-pyramid, 3-bipyramid, octahedron, 3-prism and 3-cube. Only non-equivalent traces are counted:
reversion,
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SLIDE 52 Numerical results
Enumeration results for six polyhedra: the tetrahedron, 4-pyramid, 3-bipyramid, octahedron, 3-prism and 3-cube. Only non-equivalent traces are counted:
reversion, shifting,
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SLIDE 53 Numerical results
Enumeration results for six polyhedra: the tetrahedron, 4-pyramid, 3-bipyramid, octahedron, 3-prism and 3-cube. Only non-equivalent traces are counted:
reversion, shifting, applying a permutation induced by an automorphisms of G,
Sandi Klavˇ zar, Jernej Rus (FMF) Stable Traces
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SLIDE 54 Numerical results
Enumeration results for six polyhedra: the tetrahedron, 4-pyramid, 3-bipyramid, octahedron, 3-prism and 3-cube. Only non-equivalent traces are counted:
reversion, shifting, applying a permutation induced by an automorphisms of G, using any combination of the previous three operations.
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SLIDE 55 Numerical results
graph PT aPT pPT ST aST pST tetrahedron 3 3 4-pyramid 101 5 82 5 3-bipyramid 925 24 470
53372 668 1352 22246 275 3-prism 25 2 25 2 3-cube 40 40
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SLIDE 56 Open problem 2
Problem Analytically enumerate ((anti)parallel) proper and stable traces in graphs, in particular in polyhedra.
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SLIDE 57 Thank you for your attention!
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