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

• n 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 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 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 Rd if and only if every generic realization of G is rigid in Rd. Hence the generic rigidity can be considered as a property

• f the underlying graph.

A graph is rigid in Rd if every/some generic realization of G is rigid in Rd. Laman provides a combinatorial characterization of rigid graphs 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.

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 (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 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 K5 together with vertex set of each of

• ther 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 K5 together with vertex set of each of

• ther 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 K5 together with vertex set of each of

• ther 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 6k-connected graph contains k edge-disjoint spanning rigid subgraphs.

Introduction Some Results

Edge-disjoint spanning rigid subgraphs

Theorem (Jord´ an, 2005) Every 6k-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 (6k − 2k|Z|)-edge-connected for every Z ⊂ V (G).

Introduction Some Results

Edge-disjoint spanning rigid subgraphs

Theorem (Jord´ an, 2005) Every 6k-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 (6k − 2k|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), eG(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), eG(π) ≥ k(|π| − 1). Corollary Every 2k-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 n0 trivial parts v1, v2, · · · , vn0. We define nZ(π) to be

1≤i≤n0 |Zi| where

Zi is the set of vertices in Z that are adjacent to vi for 1 ≤ i ≤ n0. If Z = ∅, then nZ(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 n0 trivial parts, eG−Z(π) ≥ k(3 − |Z|)n′

0 + 2kn0 − 3k − nZ(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 4k-connected and essentially 6k-connected graph G contains edge-disjoint k spanning rigid subgraphs.

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 G contains edge-disjoint k spanning rigid subgraphs. Recall: 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

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 G contains edge-disjoint k spanning rigid subgraphs. Recall: 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. Theorem (G. 2018) A simple graph G contains edge-disjoint k spanning rigid subgraphs if G is 4k-edge-connected, and G − Z is essentially (6k − 2k|Z|)-edge-connected for every Z ⊂ V (G).

Introduction Some Results

Try other sufficient conditions

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

Introduction Some Results

Try other sufficient conditions

Recall: Theorem (Lov´ asz and Yemini, 1982) Every 6-connected graph is rigid. Algebraic connectivity µ2(G) is the second smallest eigenvalue of the Laplacian matrix of a graph G.

Introduction Some Results

Try other sufficient conditions

Recall: Theorem (Lov´ asz and Yemini, 1982) Every 6-connected graph is rigid. Algebraic connectivity µ2(G) is the second smallest eigenvalue of the Laplacian matrix of a graph G. It is well known that the vertex connectivity is bounded below by the algebraic connectivity (Fiedler 1973).

Introduction Some Results

Try other sufficient conditions

Recall: Theorem (Lov´ asz and Yemini, 1982) Every 6-connected graph is rigid. Algebraic connectivity µ2(G) is the second smallest eigenvalue of the Laplacian matrix of a graph G. It is well known that the vertex connectivity is bounded below by the algebraic connectivity (Fiedler 1973). Corollary If µ2(G) ≥ 6, then G is rigid.

Introduction Some Results

New result on rigid graphs

Theorem (Cioab˘ a 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 rigid

• subgraphs. 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.

Introduction Some Results

Another application

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

Introduction Some Results

Another application

Theorem (Fiedler 1973) If µ2(G) ≥ p, then G is p-connected. Question What spectral conditions can guarantee edge-disjoint spanning p-connected subgraphs? Cioab˘ a and Wong started the problem for p = 1 and posed a conjecture.

Introduction Some Results

Another application

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

δ+1 , then

G has at least k edge-disjoint spanning trees.

Introduction Some Results

Another application

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

δ+1 , then

G has at least k edge-disjoint spanning trees. Theorem (Cioab˘ a 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-connected subgraphs.

Introduction Some Results

Thank You