Matroids on graphs Matroids Graphs Brigitte Servatius Rigidity Worcester Polytechnic Institute Matroids on K n Geometry Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 1 of 35 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Matroids 1. Matroids Graphs Rigidity Matroids on K n Whitney [9] defined a matroid M on a set E : Geometry M = ( E, I ) Home Page E is a finite set Title Page I is a collection of subsets of E such that ◭◭ ◮◮ I1 ∅ ∈ I ; ◭ ◮ I2 If I 1 ∈ I and I 2 ⊆ I 1 , then I 2 ∈ I Page 2 of 35 I3 If I 1 and I 2 are members of I and | I 1 | < | I 2 | , then there exists an element e in I 2 − I 1 such that Go Back I 1 + e is a member of I . Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Matroids Graphs Bases Rigidity Matroids on K n Because of condition [I2], all of the maximal independent sets Geometry have the same cardinality. These maximal independent sets are Home Page called the bases of the matroid. The bases may be described directly: Let E be a finite set, a nonempty collection B of Title Page subsets of E is called a basis system for M if ◭◭ ◮◮ B1 B � = ∅ ◭ ◮ B2 For all B 1 , B 2 ∈ B , | B 1 | = | B 2 | Page 3 of 35 B3 For all B 1 , B 2 ∈ B and e 1 ∈ B 1 − B 2 , there exists e 2 ∈ B 2 − B 1 such that B 1 − e 1 + e 2 ∈ B . Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Matroids Graphs Rigidity Matroids on K n Geometry Condition [B3] is sometimes called the exchange axiom . It also has a slightly different but equivalent formulation: Home Page B3 ′ For all B 1 , B 2 ∈ B and e 2 ∈ B 2 − B 1 , there exists e 1 ∈ Title Page B 1 − B 2 such that B 1 − e 1 + e 2 ∈ B . ◭◭ ◮◮ Complements of bases also satisfy [B3], these complements are bases of the dual matroid. ◭ ◮ Every matroid M has a dual M ∗ . Page 4 of 35 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Rank Matroids Let M be a matroid on E with independent sets I and define Graphs r ( I ) , a function from the power set of E into the nonnegative Rigidity Matroids on K n integers by r ( I ) ( S ) = max {| I | : I ∈ I , I ⊆ S } . The function Geometry r = r I is called the rank function of M . In general, let E be a finite set and r a function from the power Home Page set of E into the nonnegative integers so that Title Page R1 r ( ∅ ) = 0; ◭◭ ◮◮ R2 r ( S ) ≤ | S | ; ◭ ◮ R3 if S ⊆ T then r ( S ) ≤ r ( T ); Page 5 of 35 R4 r ( S ∪ T ) + r ( S ∩ T ) ≤ r ( S ) + r ( T ); Go Back then r is called a rank function on E . If r is a rank function on E we define I ( r ) = { I ⊆ E | r ( I ) = | I |} Full Screen Condition [R4] is called the submodular inequality. Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Matroids Graphs Rigidity Cycles Matroids on K n Geometry Given a finite set E , we call a collection C a cycle system [6] Home Page for E , if the following three conditions are satisfied: Title Page Z1 If C ∈ C then C � = ∅ ◭◭ ◮◮ Z2 If C 1 and C 2 are members of C then C 1 �⊆ C 2 Z3 If C 1 and C 2 are members of C and if e is an element of ◭ ◮ C 1 ∩ C 2 then there is an element C ∈ C , such that C ⊆ Page 6 of 35 ( C 1 ∩ C 2 − e ). Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
2. Graphs Matroids Graphs A matroid is graphic if it is isomorphic to the cycle matroid on Rigidity Matroids on K n the edge set E of a graph G = ( V, E ). Non-isomorphic graphs Geometry may have the same cycle matroid, but 3-connected graphs are uniquely determined by their matroids. Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 35 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
M is co-graphic if M ∗ is graphic. M is graphic as well as co-graphic if and only if G is planar. Map duality (geometric duality) agrees with matroid duality. Matroids The facial cycles generate the cycle space. Graphs Rigidity Matroids on K n Geometry Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 35 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Matroids Graphs Rigidity Matroids on K n Geometry Euler’s formula Home Page If G ( V, E ) is planar and connected, its cycle matroid has rank Title Page | V | − 1, its co-cycle matroid has rank | F | − 1, so | V | − 1 + | F | − 1 = | E | , i.e. ◭◭ ◮◮ ◭ ◮ | V | − | E | + | F | = 2 Page 9 of 35 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
3. Rigidity framework (in m -space) a triple ( V, E, − → Matroids p ), Graphs Rigidity ( V, E ) is a graph Matroids on K n Geometry → − p : V − → R m Home Page Title Page rigid framework ◭◭ ◮◮ if all solutions to the corresponding system of quadratic equa- tions of length constraints for the edges in some neighborhood ◭ ◮ of the original solution (as a point in mn -space) come from Page 10 of 35 congruent frameworks. Go Back 1 Full Screen Close 2 Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Rigidity Matrix Jacobian of the system Matroids 1 Graphs Rigidity Matroids on K n 2 3 4 5 Geometry Home Page 6 Title Page p 1 =(0 , 1) p 2 =( − 2 , 0) p 3 =( − 1 , 0) p 4 =(1 , 0) p 5 =(2 , 0) p 6 =(0 , − 1) ◭◭ ◮◮ ( − 2 , − 1) ( 2 , 1) (1 , 2) ( − 1 , − 1) ( 1 , 1) (1 , 3) ◭ ◮ ( 1 , − 1) ( − 1 , 1) (1 , 4) Page 11 of 35 ( 2 , − 1) ( − 2 , 1) (1 , 5) ( − 2 , 1) ( − 2 , 1) ( 2 , − 1) (2 , 6) Go Back ( − 1 , 1) ( 1 , − 1) (3 , 6) Full Screen ( 1 , 1) ( − 1 , − 1) (4 , 6) ( 2 , 1) ( − 2 , − 1) (5 , 6) Close ( − 1 , 0) ( 1 , 0) ( 2 , 1) (2 , 3) ( − 1 , 0) ( 1 , 0) Quit (4 , 5) • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
1 2 3 4 5 Matroids Graphs Rigidity 6 Matroids on K n Geometry Home Page Title Page p 1 ,x p 2 ,x p 3 ,x p 4 ,x p 5 ,x p 6 ,x p 1 ,y p 2 ,y p 3 ,y p 4 ,y p 5 ,y p 6 ,y − 2 2 − 1 1 ◭◭ ◮◮ (1 , 2) − 1 1 − 1 1 (1 , 3) ◭ ◮ − 1 − 1 1 1 (1 , 4) − 2 − 1 2 1 Page 12 of 35 (1 , 5) − 2 − 2 1 − 1 2 1 (2 , 6) Go Back − 1 1 1 − 1 (3 , 6) 1 − 1 1 − 1 (4 , 6) Full Screen 2 − 2 1 − 1 (5 , 6) Close − 1 1 2 0 0 1 (2 , 3) − 1 1 0 0 (4 , 5) Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
1 2 3 4 5 Matroids Graphs Rigidity 6 Matroids on K n Geometry Home Page Title Page p 1 ,x p 2 ,x p 3 ,x p 4 ,x p 5 ,x p 6 ,x p 1 ,y p 2 ,y p 3 ,y p 4 ,y p 5 ,y p 6 ,y 1 1 − 1 − 1 ◭◭ ◮◮ (1 , 2) 1 1 − 1 − 1 (1 , 3) ◭ ◮ − 1 − 1 1 1 (1 , 4) − 1 − 1 1 1 Page 13 of 35 (1 , 5) − 1 − 1 − 1 1 1 1 (2 , 6) Go Back 1 1 − 1 − 1 (3 , 6) 1 1 − 1 − 1 (4 , 6) Full Screen 1 1 − 1 − 1 (5 , 6) Close 1 1 1 − 1 − 1 − 1 (2 , 3) 1 1 − 1 − 1 (4 , 5) Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
1 2 3 4 5 Matroids Graphs Rigidity 6 Matroids on K n Geometry Home Page Title Page p 1 ,x p 2 ,x p 3 ,x p 4 ,x p 5 ,x p 6 ,x p 1 ,y p 2 ,y p 3 ,y p 4 ,y p 5 ,y p 6 ,y 1 1 1 1 ◭◭ ◮◮ (1 , 2) 1 1 1 1 (1 , 3) ◭ ◮ 1 1 1 1 (1 , 4) 1 1 1 1 Page 14 of 35 (1 , 5) 1 1 1 1 1 1 (2 , 6) Go Back 1 1 1 1 (3 , 6) 1 1 1 1 (4 , 6) Full Screen 1 1 1 1 (5 , 6) Close 1 1 1 1 1 1 (2 , 3) 1 1 1 1 (4 , 5) Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Kirchhoff’s matrix-tree theorem Let A be the incidence matrix of a graph G on n vertices. The determinant of an ( n − 1) × ( n − 1) minor of A T A (the Laplacian matrix of G ) counts the number of spanning trees in Matroids G . Graphs Rigidity Matroids on K n 1 Geometry Home Page 2 3 4 5 Title Page ◭◭ ◮◮ 6 ◭ ◮ Page 15 of 35 4 1 1 1 1 0 Go Back 3 1 0 0 1 1 3 1 0 0 1 1 3 0 0 1 Full Screen 1 1 3 0 0 1 A T A = det 0 0 3 1 1 = 192 1 0 0 3 1 1 Close 0 0 1 3 1 1 0 0 1 3 1 1 1 1 1 4 Quit 0 1 1 1 1 4 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
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