Sparsity Counts in Group-labeled Graphs and Rigidity Rintaro Ikeshita1 Shin-ichi Tanigawa2 1University of Tokyo 2CWI & Kyoto University July 15, 2015 1 / 23 I Examples I k = ` = 1: forest I k = 1; ` = 0: pseudoforest I k = `: Decomposability into edge-disjoint k forests (Nash-Williams) I ` ≤ k: Decomposability into edge-disjoint k − ` pseudoforests and ` forests I general k; `: Rigidity of graphs and scene analysis (k; `)-sparsity def I A finite undirected graph G = (V ; E) is (k; `)-sparse , jF j ≤ kjV (F )j − ` for every F ⊆ E with kjV (F )j − ` ≥ 1. 2 / 23 (k; `)-sparsity def I A finite undirected graph G = (V ; E) is (k; `)-sparse , jF j ≤ kjV (F )j − ` for every F ⊆ E with kjV (F )j − ` ≥ 1. I Examples I k = ` = 1: forest I k = 1; ` = 0: pseudoforest I k = `: Decomposability into edge-disjoint k forests (Nash-Williams) I ` ≤ k: Decomposability into edge-disjoint k − ` pseudoforests and ` forests I general k; `: Rigidity of graphs and scene analysis 2 / 23 I Examples I k = ` = 1: graphic matroid I k = 1; ` = 0: bicircular matroid I ` ≤ k: union of k − ` copies of bicircular matroid and ` copies of graphic matroid I k = 2; ` = 3: generic 2-rigidity matroid (Laman70) Count Matroids I Suppose ` ≤ 2k − 1. Then Mk;`(G) = (E; Ik;`) forms a matroid, called the (k; `)-count matroid, where Ik;` = fI ⊆ E : I is (k; `)-sparseg: 3 / 23 Count Matroids I Suppose ` ≤ 2k − 1. Then Mk;`(G) = (E; Ik;`) forms a matroid, called the (k; `)-count matroid, where Ik;` = fI ⊆ E : I is (k; `)-sparseg: I Examples I k = ` = 1: graphic matroid I k = 1; ` = 0: bicircular matroid I ` ≤ k: union of k − ` copies of bicircular matroid and ` copies of graphic matroid I k = 2; ` = 3: generic 2-rigidity matroid (Laman70) 3 / 23 Group-labeled Graphs I A group-labeled graph (Γ-labeled graph) (G; ) is a directed finite graph whose edges are labeled invertibly from a group Γ. I G: the underlying directed graph I : E(G) ! Γ + + a b + a2b + a + a2 + Γ = {−, +} 1Γ 2 2 Γ = {1Γ, a, a , ab, a b} − 1Γ 4 / 23 I I ⊆ E is independent iff ( 0 if F is balanced jF j ≤ jV (F )j − 1 + (F ⊆ I ) 1 otherwise I What would be (k; `)-sparsity in group-labeled graphs?? Frame Matroids I In the frame matroid (or, Dowling geometry, bias matroid) of (G; ), I ⊆ E is independent iff each connected component of I contains no cycle or just one cycle, which is unbalanced if exists. − − − − − − − − − + unbalanced balanced I Examples I graphic matroid I bicircular matroid I signed graphic matroid I 1-dimensional locally completability matroid!! 5 / 23 I What would be (k; `)-sparsity in group-labeled graphs?? Frame Matroids I In the frame matroid (or, Dowling geometry, bias matroid) of (G; ), I ⊆ E is independent iff each connected component of I contains no cycle or just one cycle, which is unbalanced if exists. − − − − − − − − − + unbalanced balanced I Examples I graphic matroid I bicircular matroid I signed graphic matroid I 1-dimensional locally completability matroid!! I I ⊆ E is independent iff ( 0 if F is balanced jF j ≤ jV (F )j − 1 + (F ⊆ I ) 1 otherwise 5 / 23 Frame Matroids I In the frame matroid (or, Dowling geometry, bias matroid) of (G; ), I ⊆ E is independent iff each connected component of I contains no cycle or just one cycle, which is unbalanced if exists. − − − − − − − − − + unbalanced balanced I Examples I graphic matroid I bicircular matroid I signed graphic matroid I 1-dimensional locally completability matroid!! I I ⊆ E is independent iff ( 0 if F is balanced jF j ≤ jV (F )j − 1 + (F ⊆ I ) 1 otherwise I What would be (k; `)-sparsity in group-labeled graphs?? 5 / 23 Motivation: Rigidity of Symmetric Frameworks I Characterization of the infinitesimal rigidity of symmetric bar-joint frameworks I A framework (G; p) is Γ-symmetric if I Γ acts on G through θ :Γ ! Aut(G) and I p(θ(γ)v) = γp(v) for any γ 2 Γ and v 2 V I Extension of classical theorem | count conditions on the underlying quotient group-labeled graphs (Malestein-Theran2010,2012, Ross2011, ...) 6 / 23 I R1(G; p) = M1;1(G) for any 1-d realization (G; p) I Theorem(Laman 1970) G can be realized as an infinitesimally rigid framework in R2 iff G contains a spanning subgraph G 0 with 2jV (G)j − 3 edges satisfying the following count condition: jF j ≤ 2jV (F )j − 3 ; 6= 8F ⊆ E(G 0) I R2(G; p) = M2;3(G) for any generic 2-d realization (G; p) Characterization of Rigid Graphs 1 I G can be realized as an infinitesimally rigid framework in R iff G is connected iff G contains a spanning subgraph G 0 with jV (G)j − 1 edges satisfying the following count condition: jF j ≤ jV (F )j − 1 ; 6= 8F ⊆ E(G 0) 7 / 23 I Theorem(Laman 1970) G can be realized as an infinitesimally rigid framework in R2 iff G contains a spanning subgraph G 0 with 2jV (G)j − 3 edges satisfying the following count condition: jF j ≤ 2jV (F )j − 3 ; 6= 8F ⊆ E(G 0) I R2(G; p) = M2;3(G) for any generic 2-d realization (G; p) Characterization of Rigid Graphs 1 I G can be realized as an infinitesimally rigid framework in R iff G is connected iff G contains a spanning subgraph G 0 with jV (G)j − 1 edges satisfying the following count condition: jF j ≤ jV (F )j − 1 ; 6= 8F ⊆ E(G 0) I R1(G; p) = M1;1(G) for any 1-d realization (G; p) 7 / 23 Characterization of Rigid Graphs 1 I G can be realized as an infinitesimally rigid framework in R iff G is connected iff G contains a spanning subgraph G 0 with jV (G)j − 1 edges satisfying the following count condition: jF j ≤ jV (F )j − 1 ; 6= 8F ⊆ E(G 0) I R1(G; p) = M1;1(G) for any 1-d realization (G; p) I Theorem(Laman 1970) G can be realized as an infinitesimally rigid framework in R2 iff G contains a spanning subgraph G 0 with 2jV (G)j − 3 edges satisfying the following count condition: jF j ≤ 2jV (F )j − 3 ; 6= 8F ⊆ E(G 0) I R2(G; p) = M2;3(G) for any generic 2-d realization (G; p) 7 / 23 One Approach (Schulze-T13) I Compute the block-diagonalization of rigidity matrix R(G; p) 0 1 R0(G; p) 0 > B .. C T R(G; p)S := @ . A 0 Rr (G; p) I Its "sparsity pattern" is carried over (??): I In each block, each row and each column are associated with an edge orbit and a vertex orbit, respectively. d I The zero-nonzero pattern of each block is the same as I (G=Γ) ⊗ R , where I (G=Γ) denotes the incidence matrix of G=Γ. I Characterize the rank of each block in terms of combinatorial conditions of the underlying quotient graph G=Γ 8 / 23 1-dimensional case I (G; p): Z2-symmetric framework with Z2 = {−; +g 1 I p : V ! R with p((−)v) = −p(v) for each v 2 V I R (G; p) 0 T >R(G; p)S := 0 0 R1(G; p) I R1(G; p) = I (G=Z2) I The row-independence of R0(G; p) cannot be described by G=Z2... 9 / 23 Quotient Group-labled Graphs I Quotient group labeled graph (G=Γ; ) I G=Γ: a directed quotient graph I : E(G=Γ) ! Γ encoding the covering map 1 C32 1 C33 3 2 C31 C 2 C id 3 C22 3 C C1 C23 C3 id 2 (G=Γ; ) C2 C21 G 10 / 23 I , independent in the frame matroid of (G=Z2; ). I , 8F ⊆ E=Z2, ( 0 if F is balanced jF j ≤ jV (F )j − 1 + 1 if F is unbalanced 1-dimensional case I (G; p): Z2-symmetric framework 1 I p : V ! R with p((−)v) = −p(v) for each v 2 V I R (G; p) 0 T >R(G; p)S := 0 0 R1(G; p) I R1(G; p) = I (G=Z2) I R0(G; p) = I (G=Z2; ) I where I (G=Z2; ) is row independent iff each connected component contains no cycle or just one cycle, which is unbalanced if exists 11 / 23 I , 8F ⊆ E=Z2, ( 0 if F is balanced jF j ≤ jV (F )j − 1 + 1 if F is unbalanced 1-dimensional case I (G; p): Z2-symmetric framework 1 I p : V ! R with p((−)v) = −p(v) for each v 2 V I R (G; p) 0 T >R(G; p)S := 0 0 R1(G; p) I R1(G; p) = I (G=Z2) I R0(G; p) = I (G=Z2; ) I where I (G=Z2; ) is row independent iff each connected component contains no cycle or just one cycle, which is unbalanced if exists I , independent in the frame matroid of (G=Z2; ). 11 / 23 1-dimensional case I (G; p): Z2-symmetric framework 1 I p : V ! R with p((−)v) = −p(v) for each v 2 V I R (G; p) 0 T >R(G; p)S := 0 0 R1(G; p) I R1(G; p) = I (G=Z2) I R0(G; p) = I (G=Z2; ) I where I (G=Z2; ) is row independent iff each connected component contains no cycle or just one cycle, which is unbalanced if exists I , independent in the frame matroid of (G=Z2; ). I , 8F ⊆ E=Z2, ( 0 if F is balanced jF j ≤ jV (F )j − 1 + 1 if F is unbalanced 11 / 23 2-dimensional reflection symmetry (Schulze-T 13) I (G; p): 2-dimensional framework with reflection symmetry 2 I p : V ! R is "regular" under the reflection symmetry I R (G; p) 0 T >R(G; p)S := 0 0 R1(G; p) I Ri (G; p) is row-independent iff 8F ⊆ E=Γ, ( 0 if F is balanced jF j ≤ 2jV (F )j − 3 + 2 − i if F is unbalanced I A similar count works for grid-like frameworks (Kitson) − + + + + + + − 12 / 23 I Examples of counts in rigidity with symmetry: 8 0 hF i is trivial <> v jF j ≤ 2jV (F )|−3+ 2 hF iv is nontrivial and cyclic (F ⊆ E=Γ) :>3 otherwise Given an integer i, 8 0 hF iv is trivial > <>1 hF i ' jF j ≤ 2jV (F )|−3+ v Z2 (F ⊆ E=Γ) >2 hF iv ' Zk for some k 2 S(n; i) > :3 otherwise where S(n; i) = fk : n ≡ 0 (mod k) and i 2 {−1; 0; 1g (mod k)g n f2g.