Rigidity in the Euclidean plane

Xiaofeng Gu(University of West Georgia)

31st Cumberland Conferenceon Combinatorics, Graph Theory and Computing

UCF

May 19, 2019

Introduction Some Results

Background

Rigidity, arising in discrete geometry, is the property of astructure 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 Rd. Roughly speaking,it is a straight line realization of G in Rd.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 Rd.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 suchthat if (G, p) is equivalent to (G, q) and ||p(u)− q(u)|| < εfor every u ∈ V , then (G, p) is congruent to (G, q).

Introduction Some Results

Background (cont...)

A generic realization of G is rigid in Rd if and only if everygeneric realization of G is rigid in Rd.Hence the generic rigidity can be considered as a propertyof the underlying graph.A graph is rigid in Rd if every/some generic realization of Gis rigid in Rd.Laman provides a combinatorial characterization of rigidgraphs in R2.

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 minimallyrigid.A minimally rigid graph is also called a Laman graph.A graph G is rigid if G contains a spanning minimally rigidsubgraph.Every rigid graph with at least 3 vertices is 2-connected.Combinatorial rigidity theory.

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 minimallyrigid.

A minimally rigid graph is also called a Laman graph.A graph G is rigid if G contains a spanning minimally rigidsubgraph.Every rigid graph with at least 3 vertices is 2-connected.Combinatorial rigidity theory.

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 minimallyrigid.A minimally rigid graph is also called a Laman graph.

A graph G is rigid if G contains a spanning minimally rigidsubgraph.Every rigid graph with at least 3 vertices is 2-connected.Combinatorial rigidity theory.

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 minimallyrigid.A minimally rigid graph is also called a Laman graph.A graph G is rigid if G contains a spanning minimally rigidsubgraph.

Every rigid graph with at least 3 vertices is 2-connected.Combinatorial rigidity theory.

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 minimallyrigid.A minimally rigid graph is also called a Laman graph.A graph G is rigid if G contains a spanning minimally rigidsubgraph.Every rigid graph with at least 3 vertices is 2-connected.

Combinatorial rigidity theory.

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 minimallyrigid.A minimally rigid graph is also called a Laman graph.A graph G is rigid if G contains a spanning minimally rigidsubgraph.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 (G)| = 2 |V (G)| = 3

v

|V (G)| = 4

v

|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 Lovasz and Yemini (1982):

Introduction Some Results

An alternative definition

(Lovasz 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 thesubgraph of G induced by X.

In the example, choose a collection G in this way: thevertex set of each K5 together with vertex set of each ofother 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

An alternative definition

(Lovasz 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 thesubgraph of G induced by X.In the example, choose a collection G in this way: thevertex set of each K5 together with vertex set of each ofother 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

An alternative definition

(Lovasz 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 thesubgraph of G induced by X.In the example, choose a collection G in this way: thevertex set of each K5 together with vertex set of each ofother 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

An alternative definition

(Lovasz 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 thesubgraph of G induced by X.In the example, choose a collection G in this way: thevertex set of each K5 together with vertex set of each ofother 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 (Lovasz 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 Jordan, 2009)If a simple graph G is 6-edge-connected, G− v is4-edge-connected for every vertex v, and G− u− v is2-edge-connected for every pair of vertices u, v, then G isrigid.

Introduction Some Results

Known results on rigid graphs

Theorem (Lovasz 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 Jordan, 2009)If a simple graph G is 6-edge-connected, G− v is4-edge-connected for every vertex v, and G− u− v is2-edge-connected for every pair of vertices u, v, then G isrigid.

Introduction Some Results

Known results on rigid graphs

Theorem (Lovasz 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 Jordan, 2009)If a simple graph G is 6-edge-connected, G− v is4-edge-connected for every vertex v, and G− u− v is2-edge-connected for every pair of vertices u, v, then G isrigid.

Introduction Some Results

Known results on rigid graphs

Theorem (Lovasz 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 Jordan, 2009)If a simple graph G is 6-edge-connected, G− v is4-edge-connected for every vertex v, and G− u− v is2-edge-connected for every pair of vertices u, v, then G isrigid.

Introduction Some Results

Edge-disjoint spanning rigid subgraphs

Theorem (Jordan, 2005)Every 6k-connected graph contains k edge-disjointspanning rigid subgraphs.

Theorem (Cheriyan, Durand de Gevigney, Szigeti, 2014) Asimple graph G contains edge-disjoint k spanning rigidsubgraphs if G− Z is (6k − 2k|Z|)-edge-connected forevery Z ⊂ V (G).Kriesell conjectured that there exists a (smallest) integerf(p) such that every f(p)-connected graph G a spanningtree T such that G− E(T ) is p-connected.

Introduction Some Results

Edge-disjoint spanning rigid subgraphs

Theorem (Jordan, 2005)Every 6k-connected graph contains k edge-disjointspanning rigid subgraphs.Theorem (Cheriyan, Durand de Gevigney, Szigeti, 2014) Asimple graph G contains edge-disjoint k spanning rigidsubgraphs if G− Z is (6k − 2k|Z|)-edge-connected forevery Z ⊂ V (G).

Kriesell conjectured that there exists a (smallest) integerf(p) such that every f(p)-connected graph G a spanningtree T such that G− E(T ) is p-connected.

Introduction Some Results

Edge-disjoint spanning rigid subgraphs

Theorem (Jordan, 2005)Every 6k-connected graph contains k edge-disjointspanning rigid subgraphs.Theorem (Cheriyan, Durand de Gevigney, Szigeti, 2014) Asimple graph G contains edge-disjoint k spanning rigidsubgraphs if G− Z is (6k − 2k|Z|)-edge-connected forevery Z ⊂ V (G).Kriesell conjectured that there exists a (smallest) integerf(p) such that every f(p)-connected graph G a spanningtree 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 iffor any partition π of V (G), eG(π) ≥ k(|π| − 1).

CorollaryEvery 2k-edge-connected graph contains a packing of kspanning trees.

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 iffor any partition π of V (G), eG(π) ≥ k(|π| − 1).

CorollaryEvery 2k-edge-connected graph contains a packing of kspanning trees.

Introduction Some Results

Partition condition for edge-disjoint spanning rigidsubgraphs

Let Z ⊂ V (G) and π be a partition of V (G− Z) with n0 trivialparts v1, v2, · · · , vn0 . We define nZ(π) to be

∑1≤i≤n0

|Zi| whereZi is the set of vertices in Z that are adjacent to vi for1 ≤ i ≤ n0. If Z = ∅, then nZ(π) = 0.Theorem (G. 2018)A simple graph G contains k edge-disjoint spanning rigidsubgraphs if for any partition π of V (G− Z) with n0 trivial parts,eG−Z(π) ≥ k(3− |Z|)n′0 + 2kn0 − 3k − nZ(π) for everyZ ⊂ V (G).

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 4k-connected and essentially 6k-connected graph Gcontains edge-disjoint k spanning rigid subgraphs.Recall: Theorem (Jackson and Jordan, 2009)If a simple graph G is 6-edge-connected, G− v is4-edge-connected for every vertex v, and G− u− v is2-edge-connected for every pair of vertices u, v, then G isrigid.Theorem (G. 2018)A simple graph G contains edge-disjoint k spanning rigidsubgraphs if G is 4k-edge-connected, and G− Z isessentially (6k − 2k|Z|)-edge-connected for everyZ ⊂ V (G).

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 4k-connected and essentially 6k-connected graph Gcontains edge-disjoint k spanning rigid subgraphs.

Recall: Theorem (Jackson and Jordan, 2009)If a simple graph G is 6-edge-connected, G− v is4-edge-connected for every vertex v, and G− u− v is2-edge-connected for every pair of vertices u, v, then G isrigid.Theorem (G. 2018)A simple graph G contains edge-disjoint k spanning rigidsubgraphs if G is 4k-edge-connected, and G− Z isessentially (6k − 2k|Z|)-edge-connected for everyZ ⊂ V (G).

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 4k-connected and essentially 6k-connected graph Gcontains edge-disjoint k spanning rigid subgraphs.Recall: Theorem (Jackson and Jordan, 2009)If a simple graph G is 6-edge-connected, G− v is4-edge-connected for every vertex v, and G− u− v is2-edge-connected for every pair of vertices u, v, then G isrigid.

Theorem (G. 2018)A simple graph G contains edge-disjoint k spanning rigidsubgraphs if G is 4k-edge-connected, and G− Z isessentially (6k − 2k|Z|)-edge-connected for everyZ ⊂ V (G).

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 4k-connected and essentially 6k-connected graph Gcontains edge-disjoint k spanning rigid subgraphs.Recall: Theorem (Jackson and Jordan, 2009)If a simple graph G is 6-edge-connected, G− v is4-edge-connected for every vertex v, and G− u− v is2-edge-connected for every pair of vertices u, v, then G isrigid.Theorem (G. 2018)A simple graph G contains edge-disjoint k spanning rigidsubgraphs if G is 4k-edge-connected, and G− Z isessentially (6k − 2k|Z|)-edge-connected for everyZ ⊂ V (G).

Introduction Some Results

Try other sufficient conditions

Recall: Theorem (Lovasz and Yemini, 1982)Every 6-connected graph is rigid.

Algebraic connectivity µ2(G) is the second smallesteigenvalue of the Laplacian matrix of a graph G.It is well known that the vertex connectivity is boundedbelow by the algebraic connectivity (Fiedler 1973).CorollaryIf µ2(G) ≥ 6, then G is rigid.

Introduction Some Results

Try other sufficient conditions

Recall: Theorem (Lovasz and Yemini, 1982)Every 6-connected graph is rigid.Algebraic connectivity µ2(G) is the second smallesteigenvalue of the Laplacian matrix of a graph G.

It is well known that the vertex connectivity is boundedbelow by the algebraic connectivity (Fiedler 1973).CorollaryIf µ2(G) ≥ 6, then G is rigid.

Introduction Some Results

Try other sufficient conditions

Recall: Theorem (Lovasz and Yemini, 1982)Every 6-connected graph is rigid.Algebraic connectivity µ2(G) is the second smallesteigenvalue of the Laplacian matrix of a graph G.It is well known that the vertex connectivity is boundedbelow by the algebraic connectivity (Fiedler 1973).

CorollaryIf µ2(G) ≥ 6, then G is rigid.

Introduction Some Results

Try other sufficient conditions

Recall: Theorem (Lovasz and Yemini, 1982)Every 6-connected graph is rigid.Algebraic connectivity µ2(G) is the second smallesteigenvalue of the Laplacian matrix of a graph G.It is well known that the vertex connectivity is boundedbelow by the algebraic connectivity (Fiedler 1973).CorollaryIf µ2(G) ≥ 6, then G is rigid.

Introduction Some Results

New result on rigid graphs

Theorem (Cioaba and G. 2019+)Let G be a graph with minimum degree δ ≥ 6k. If

µ2(G) > 2 +2k − 1

δ − 1,

then G has at least k edge-disjoint spanning rigidsubgraphs. In particular, if µ2(G) > 2 + 1

δ−1 , then G is rigid.

Introduction Some Results

Another application

Theorem (Fiedler 1973)If µ2(G) ≥ p, then G is p-connected.

Question What spectral conditions can guaranteeedge-disjoint spanning p-connected subgraphs?Cioaba and Wong started the problem for p = 1 and poseda conjecture.Theorem (Liu, Hong, G., Lai 2014)Let G be a simple graph with δ ≥ 2k. If µ2(G) > 2k−1

δ+1 , thenG has at least k edge-disjoint spanning trees.Theorem (Cioaba and G. 2019+)Let G be a simple graph with δ ≥ 6k. If µ2(G) > 2 + 2k−1

δ−1 ,then G has at least k edge-disjoint spanning 2-connectedsubgraphs.

Introduction Some Results

Another application

Theorem (Fiedler 1973)If µ2(G) ≥ p, then G is p-connected.Question What spectral conditions can guaranteeedge-disjoint spanning p-connected subgraphs?

Cioaba and Wong started the problem for p = 1 and poseda conjecture.Theorem (Liu, Hong, G., Lai 2014)Let G be a simple graph with δ ≥ 2k. If µ2(G) > 2k−1

δ+1 , thenG has at least k edge-disjoint spanning trees.Theorem (Cioaba and G. 2019+)Let G be a simple graph with δ ≥ 6k. If µ2(G) > 2 + 2k−1

δ−1 ,then G has at least k edge-disjoint spanning 2-connectedsubgraphs.

Introduction Some Results

Another application

Theorem (Fiedler 1973)If µ2(G) ≥ p, then G is p-connected.Question What spectral conditions can guaranteeedge-disjoint spanning p-connected subgraphs?Cioaba and Wong started the problem for p = 1 and poseda conjecture.

Theorem (Liu, Hong, G., Lai 2014)Let G be a simple graph with δ ≥ 2k. If µ2(G) > 2k−1

δ+1 , thenG has at least k edge-disjoint spanning trees.Theorem (Cioaba and G. 2019+)Let G be a simple graph with δ ≥ 6k. If µ2(G) > 2 + 2k−1

δ−1 ,then G has at least k edge-disjoint spanning 2-connectedsubgraphs.

Introduction Some Results

Another application

Theorem (Fiedler 1973)If µ2(G) ≥ p, then G is p-connected.Question What spectral conditions can guaranteeedge-disjoint spanning p-connected subgraphs?Cioaba and Wong started the problem for p = 1 and poseda conjecture.Theorem (Liu, Hong, G., Lai 2014)Let G be a simple graph with δ ≥ 2k. If µ2(G) > 2k−1

δ+1 , thenG has at least k edge-disjoint spanning trees.

Theorem (Cioaba and G. 2019+)Let G be a simple graph with δ ≥ 6k. If µ2(G) > 2 + 2k−1

δ−1 ,then G has at least k edge-disjoint spanning 2-connectedsubgraphs.

Introduction Some Results

Another application

Theorem (Fiedler 1973)If µ2(G) ≥ p, then G is p-connected.Question What spectral conditions can guaranteeedge-disjoint spanning p-connected subgraphs?Cioaba and Wong started the problem for p = 1 and poseda conjecture.Theorem (Liu, Hong, G., Lai 2014)Let G be a simple graph with δ ≥ 2k. If µ2(G) > 2k−1

δ+1 , thenG has at least k edge-disjoint spanning trees.Theorem (Cioaba and G. 2019+)Let G be a simple graph with δ ≥ 6k. If µ2(G) > 2 + 2k−1

δ−1 ,then G has at least k edge-disjoint spanning 2-connectedsubgraphs.

Introduction Some Results

Thank You