Rigidity in the Euclidean plane Xiaofeng Gu (University of West - PowerPoint PPT Presentation

Rigidity in the Euclidean plane Xiaofeng Gu (University of West Georgia) 31st Cumberland Conference on Combinatorics, Graph Theory and Computing UCF May 19, 2019

Introduction Some Results Background Rigidity, arising in discrete geometry, is the property of a structure that does not flex. A d -dimensional framework is a pair ( G, p ) , where G ( V, E ) is a graph and p is a map from V to R d . Roughly speaking, it is a straight line realization of G in R d . Two frameworks ( G, p ) and ( G, q ) are equivalent if || p ( u ) − p ( v ) || = || q ( u ) − q ( v ) || holds for every edge uv ∈ E , where || · || denotes the Euclidean norm in R d . Two frameworks ( G, p ) and ( G, q ) are congruent if || p ( u ) − p ( v ) || = || q ( u ) − q ( v ) || holds for every pair u, v ∈ V . The framework ( G, p ) is rigid if there exists an ε > 0 such that if ( G, p ) is equivalent to ( G, q ) and || p ( u ) − q ( u ) || < ε for every u ∈ V , then ( G, p ) is congruent to ( G, q ) .

Introduction Some Results Background (cont...) A generic realization of G is rigid in R d if and only if every generic realization of G is rigid in R d . Hence the generic rigidity can be considered as a property of the underlying graph. A graph is rigid in R d if every/some generic realization of G is rigid in R d . Laman provides a combinatorial characterization of rigid graphs in R 2 .

Introduction Some Results Rigid graphs Let G = ( V, E ) be a graph. A graph G is sparse if | E ( X ) | ≤ 2 | X | − 3 for all X ⊆ V ( G ) with | X | ≥ 2 .

Introduction Some Results Rigid graphs Let G = ( V, E ) be a graph. A graph G is sparse if | E ( X ) | ≤ 2 | X | − 3 for all X ⊆ V ( G ) with | X | ≥ 2 . If in addition | E ( G ) | = 2 | V ( G ) | − 3 , then G is minimally rigid.

Introduction Some Results Rigid graphs Let G = ( V, E ) be a graph. A graph G is sparse if | E ( X ) | ≤ 2 | X | − 3 for all X ⊆ V ( G ) with | X | ≥ 2 . If in addition | E ( G ) | = 2 | V ( G ) | − 3 , then G is minimally rigid. A minimally rigid graph is also called a Laman graph.

Introduction Some Results Rigid graphs Let G = ( V, E ) be a graph. A graph G is sparse if | E ( X ) | ≤ 2 | X | − 3 for all X ⊆ V ( G ) with | X | ≥ 2 . If in addition | E ( G ) | = 2 | V ( G ) | − 3 , then G is minimally rigid. A minimally rigid graph is also called a Laman graph. A graph G is rigid if G contains a spanning minimally rigid subgraph.

Introduction Some Results Rigid graphs Let G = ( V, E ) be a graph. A graph G is sparse if | E ( X ) | ≤ 2 | X | − 3 for all X ⊆ V ( G ) with | X | ≥ 2 . If in addition | E ( G ) | = 2 | V ( G ) | − 3 , then G is minimally rigid. A minimally rigid graph is also called a Laman graph. A graph G is rigid if G contains a spanning minimally rigid subgraph. Every rigid graph with at least 3 vertices is 2-connected.

Introduction Some Results Rigid graphs Let G = ( V, E ) be a graph. A graph G is sparse if | E ( X ) | ≤ 2 | X | − 3 for all X ⊆ V ( G ) with | X | ≥ 2 . If in addition | E ( G ) | = 2 | V ( G ) | − 3 , then G is minimally rigid. A minimally rigid graph is also called a Laman graph. A graph G is rigid if G contains a spanning minimally rigid subgraph. Every rigid graph with at least 3 vertices is 2-connected. Combinatorial rigidity theory.

Introduction Some Results Examples of minimally rigid graphs Recall: A graph G is minimally rigid if | E ( G ) | = 2 | V ( G ) | − 3 and | E ( X ) | ≤ 2 | X | − 3 for all X ⊆ V ( G ) with | X | ≥ 2 . v v | V ( G ) | = 2 | V ( G ) | = 3 | V ( G ) | = 4 | V ( G ) | = 5 Extension operations: 1. Add a new vertex v and two edges. 2. Subdivide an edge by a new vertex v and add a new edge.

Introduction Some Results Non-rigid example Constructed by Lov´ asz and Yemini (1982):

Introduction Some Results An alternative definition (Lov´ asz and Yemini, 1982) A graph G is rigid if � X ∈G (2 | X | − 3) ≥ 2 | V | − 3 for every collection G of sets of V ( G ) such that { E ( X ) , X ∈ G} partitions E ( G ) , where E ( X ) is the edge set of the subgraph of G induced by X .

Introduction Some Results An alternative definition (Lov´ asz and Yemini, 1982) A graph G is rigid if � X ∈G (2 | X | − 3) ≥ 2 | V | − 3 for every collection G of sets of V ( G ) such that { E ( X ) , X ∈ G} partitions E ( G ) , where E ( X ) is the edge set of the subgraph of G induced by X . In the example, choose a collection G in this way: the vertex set of each K 5 together with vertex set of each of other single edges.

Introduction Some Results An alternative definition (Lov´ asz and Yemini, 1982) A graph G is rigid if � X ∈G (2 | X | − 3) ≥ 2 | V | − 3 for every collection G of sets of V ( G ) such that { E ( X ) , X ∈ G} partitions E ( G ) , where E ( X ) is the edge set of the subgraph of G induced by X . In the example, choose a collection G in this way: the vertex set of each K 5 together with vertex set of each of other single edges. X ∈G (2 | X | − 3) = 8(2 · 5 − 3) + 20(2 · 2 − 3) = 76 �

Introduction Some Results An alternative definition (Lov´ asz and Yemini, 1982) A graph G is rigid if � X ∈G (2 | X | − 3) ≥ 2 | V | − 3 for every collection G of sets of V ( G ) such that { E ( X ) , X ∈ G} partitions E ( G ) , where E ( X ) is the edge set of the subgraph of G induced by X . In the example, choose a collection G in this way: the vertex set of each K 5 together with vertex set of each of other single edges. X ∈G (2 | X | − 3) = 8(2 · 5 − 3) + 20(2 · 2 − 3) = 76 � However, 2 | V | − 3 = 2 · 40 − 3 = 77 .

Introduction Some Results Known results on rigid graphs Theorem (Lov´ asz and Yemini, 1982) Every 6 -connected graph is rigid.

Introduction Some Results Known results on rigid graphs Theorem (Lov´ asz and Yemini, 1982) Every 6 -connected graph is rigid. There exist 5-connected non-rigid graphs.

Introduction Some Results Known results on rigid graphs Theorem (Lov´ asz and Yemini, 1982) Every 6 -connected graph is rigid. There exist 5-connected non-rigid graphs. Theorem (Jackson, Servatius and Servatius, 2007) Every 4 -connected essentially 6-connected graph is rigid.

Introduction Some Results Known results on rigid graphs Theorem (Lov´ asz and Yemini, 1982) Every 6 -connected graph is rigid. There exist 5-connected non-rigid graphs. Theorem (Jackson, Servatius and Servatius, 2007) Every 4 -connected essentially 6-connected graph is rigid. Theorem (Jackson and Jord´ an, 2009) If a simple graph G is 6-edge-connected, G − v is 4-edge-connected for every vertex v , and G − u − v is 2-edge-connected for every pair of vertices u, v , then G is rigid.

Introduction Some Results Edge-disjoint spanning rigid subgraphs Theorem (Jord´ an, 2005) Every 6 k -connected graph contains k edge-disjoint spanning rigid subgraphs.

Introduction Some Results Edge-disjoint spanning rigid subgraphs Theorem (Jord´ an, 2005) Every 6 k -connected graph contains k edge-disjoint spanning rigid subgraphs. Theorem (Cheriyan, Durand de Gevigney, Szigeti, 2014) A simple graph G contains edge-disjoint k spanning rigid subgraphs if G − Z is (6 k − 2 k | Z | ) -edge-connected for every Z ⊂ V ( G ) .

Introduction Some Results Edge-disjoint spanning rigid subgraphs Theorem (Jord´ an, 2005) Every 6 k -connected graph contains k edge-disjoint spanning rigid subgraphs. Theorem (Cheriyan, Durand de Gevigney, Szigeti, 2014) A simple graph G contains edge-disjoint k spanning rigid subgraphs if G − Z is (6 k − 2 k | Z | ) -edge-connected for every Z ⊂ V ( G ) . Kriesell conjectured that there exists a (smallest) integer f ( p ) such that every f ( p ) -connected graph G a spanning tree T such that G − E ( T ) is p -connected.

Introduction Some Results Spanning Trees Packing Theorem Theorem (Nash-Williams and Tutte, 1961, independently) A graph G has k edge-disjoint spanning trees if and only if for any partition π of V ( G ) , e G ( π ) ≥ k ( | π | − 1) .

Introduction Some Results Spanning Trees Packing Theorem Theorem (Nash-Williams and Tutte, 1961, independently) A graph G has k edge-disjoint spanning trees if and only if for any partition π of V ( G ) , e G ( π ) ≥ k ( | π | − 1) . Corollary Every 2 k -edge-connected graph contains a packing of k spanning trees.

Introduction Some Results Partition condition for edge-disjoint spanning rigid subgraphs Let Z ⊂ V ( G ) and π be a partition of V ( G − Z ) with n 0 trivial parts v 1 , v 2 , · · · , v n 0 . We define n Z ( π ) to be � 1 ≤ i ≤ n 0 | Z i | where Z i is the set of vertices in Z that are adjacent to v i for 1 ≤ i ≤ n 0 . If Z = ∅ , then n Z ( π ) = 0 . Theorem (G. 2018) A simple graph G contains k edge-disjoint spanning rigid subgraphs if for any partition π of V ( G − Z ) with n 0 trivial parts, e G − Z ( π ) ≥ k (3 − | Z | ) n ′ 0 + 2 kn 0 − 3 k − n Z ( π ) for every Z ⊂ V ( G ) .

Introduction Some Results More sufficient conditions Recall: Theorem (Jackson, Servatius and Servatius, 2007) Every 4 -connected essentially 6-connected graph is rigid.

Introduction Some Results More sufficient conditions Recall: Theorem (Jackson, Servatius and Servatius, 2007) Every 4 -connected essentially 6-connected graph is rigid. Theorem (G. 2018) Every 4 k -connected and essentially 6 k -connected graph G contains edge-disjoint k spanning rigid subgraphs.