A signed structure theory for oriented hypergraphs Lucas J. Rusnak - PowerPoint PPT Presentation

A signed structure theory for oriented hypergraphs Lucas J. Rusnak CombinaTexas 2016 8 May 2016 L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 1 / 33

Incidence vs Adjacency vs Edge In a graph: Incidence ⇒ adjacency ⇔ edge (sign + is implied). 1 These separate in an oriented hypergraph. 1 Incidence Matrix Magic: Generalizing the cycle space. 2 OH Matrices and Unifying Entries. 3 Weak Walk Covers and the Matrix-tree Theorem. 4 L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 2 / 33

Incidence Matrix: Graphs   − 1 0 0 − 1 1   1 − 1 0 0 0   ⇔ H G =   − 1 0 1 1 0 0 0 − 1 1 0 Minimal Dependency H ⇐ ⇒ Circle in G . L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 3 / 33

Incidence Matrix: Signed Graphs   − 1 0 0 − 1 − 1   1 1 0 0 0   ⇔ H G =   0 1 − 1 0 1 − 1 0 0 1 0 Minimal Dependency H ⇐ ⇒ Positive circle or Contrabalanced handcuff in G . L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 4 / 33

Incidence Matrix: Oriented Hypergraphs   1 1 0 0   − 1 0 1 0     ⇔ H G = 0 0 1 1     − 1 1 0 0 1 0 1 0 Minimal Dependency H ⇐ ⇒ Balanced subdivision of balanced hypercircles ( balanced ), Camion connections of disjoint floral families ( balanceable ), or ??? ( unbalanceable ). L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 5 / 33

Balanceability: Measuring Negative Circles Definitions Type Condition Note Balanced No negative circles. All graphs. Balanceable Incidence reversals result in balance. All signed graphs. Unbalanceable Not balanceable. No signed graphs. Theorem The only obstruction to balanceability is three internally-disjoint paths that begin at an edge and terminate at a vertex. Cross-theta L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 6 / 33

Hypergraphic Circle Analogs - Flowers Definition (Flower) A flower is a minimal inseparable oriented hypergraph. L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 7 / 33

The Pseudo-flower Problem Definition (Pseudo-flower) A pseudo-flower is an OH where the weak-deletion of thorns results in a flower. L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 8 / 33

Hypergraphic Path Analogs - Arteries Definition An artery is a subdivision of an edge. Theorem (R. 2013) The only* balanced minimal dependencies are balanced flowers or arterial connections of balanced pseudo-flowers. (* Up to balanced subdivision and 2 -vertex-contraction.) L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 9 / 33

Some Minimal Dependencies L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 10 / 33

Oriented Hypergraphic Matrices L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 11 / 33

Oriented Hypergraphic Matrices Incidence Matrix: H G Degree Matrix: D G Adjacency Matrix: A G Laplacian Matrix: L G : = D G − A G = H G H T G   2 1 0 1 1   − 1 − 1 1 2 0     ⇔ L G = 0 − 1 2 − 1 1     − 1 1 0 2 1 1 − 1 1 1 2 L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 12 / 33

Universal Theorems Definition A weak walk is a sequence � w = a 0 , i 1 , a 1 , i 2 , a 2 , i 3 , a 3 , ..., a n − 1 , i n , a n of vertices, edges and incidences, where { a k } is an alternating sequence of vertices and edges, and i h is an incidence containing a h − 1 and a h . Theorem (Chen, Rao, R. and Yang. 2015) � � ij = w ± ( v i , v j ; k ) . A k G � � ij = ( − 1 ) k · � L k w ± ( v i , v j ; k ) . G L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 13 / 33

Unifying Entries Theorem Let G be an oriented hypergraph. H G is the half-walk matrix. 1 D G is the strictly 1 -weak walk matrix. Called backsteps. 2 A G is the 1 -(non-weak)-walk matrix. 3 L G is negative the 1 -weak-walk matrix. 4 L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 14 / 33

Weak Walk Covers L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 15 / 33

Stirling Covers Example Representations of some permutations via Stirling covers. When do they exist in a graph? Include any "missing" adjacencies/backsteps. Consider their sign to be 0. A weak walk contributor of G is a labeling of a Stirling cover. L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 16 / 33

Some Contributors L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 17 / 33

Activation Classes Example L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 18 / 33

Signing Contributors and Activation Classes Definition The sign of a contributor c is defined by sgn ( c ) = ( − 1 ) pc ( c ) ( 0 ) zc ( c ) . Theorem Given a graph G, for every contribution class C we have ∑ sgn ( c ) = 0 . c ∈ C L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 19 / 33

Contributors are Cycle Cover Analogs Theorem For an oriented graph G, det ( L G ) = ∑ sgn ( c ) = 0 c ∈ C L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 20 / 33

The Matrix-tree Theorem L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 21 / 33

Cutting the Contributor Posets Definitions M 0 ( v r , v r ; C ) Maximal element(s) in C where v r is in no active circle. m 1 ( v r , v k ; C ) Minimal element in class C where v r is in an active circle. L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 22 / 33

Relative Signings Definitions Let ( c ; v ; w ) be contributor c with each adjacency or non-zero 1 backstep v r i → v k i removed (zero backsteps are not removed). The sign of a contributor c with respect to ( v ; w ) is defined as 2 sgn ( c ; v ; w ) = ( − 1 ) pc ( c ; v ; w ) ( 0 ) zc ( c ; v ; w ) ( − 1 ) pl ( c ; v ; w ) ( 0 ) zl ( c ; v ; w ) Define C = � = 0 ( v ; w ) as the set of all non-zero contributors in the classes 3 that all have the same sign within a single class. C × ( v ; w ) : = { ( c ; v ; w ) | c ∈ C = ∅ ( v ; w ) (identification of contributors 4 along the admissible exceptions). L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 23 / 33

Moving to Weak Walks Example (A v 1 -cut) The only members in a C = � = 0 ( v 1 ; v k ) are both contributors of the third class. The top member of the fourth class is not a member of C = � = 0 ( v 1 ; v 4 ) , but is a member of C = � = 0 ( v 4 ; v 2 ) since the adjacency v 4 → v 2 does not exist in G . L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 24 / 33

The Matrix-tree Theorem Theorem The number of spanning trees in a graph G, T ( G ) , is: ∑ T ( G ) = ε ( v ; w ) sgn ( c ; v ; w ) . c ∈ C × ( v ; w ) L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 25 / 33

The Matrix-tree Theorem L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 26 / 33

All Minors Matrix-tree Theorem Theorem Let ( L G ; v ; w ) be the minor determined by removing the rows of v and the columns of w , then ∑ det ( L G ; v ; w ) = ε ( v ; w ) sgn ( c ; v ; w ) c ∈ C × ( v ; w ) L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 27 / 33

The Signed Graph AMMTT Theorem Let ( L G ; v ; w ) be the minor determined by removing the rows of v and the columns of w , then ∑ det ( L G ; v ; w ) = ε ( v ; w ) sgn ( c ; v ; w ) c ∈ C × ( v ; w ) L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 28 / 33

Cracking Hypergraphs L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 29 / 33

Hyperedges and Stirling covers Definition A closed vertex-cotrail is called a cirque. Cirque Augmentation L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 30 / 33

The Cirque Order and Extending Cuts Multiple M 0 ( v r , v r ; C ) and m 1 ( v r , v k ; C ) elements. L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 31 / 33

All Minors Matrix-tree Theorem Theorem Let ( L G ; v ; w ) be the minor determined by removing the rows of v and the columns of w , then ∑ det ( L G ; v ; w ) = ε ( v ; w ) sgn ( c ; v ; w ) c ∈ C × ( v ; w ) Combine the cirque order by augmentation and activation order. (Cirque order first.) L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 32 / 33

The End! L. Rusnak (CombinaTexas 2016) Oriented Hypergraphs 8 May 2016 33 / 33