Co-rigid sets in redundantly rigid augmentations Andr´ as Mih´ alyk´ o with Tibor Jord´ an and Csaba Kir´ aly Department of Operations Research, E¨ otv¨ os Lor´ and University Budapest, Hungary Geometric constraint systems: rigidity, flexibility and applications, Lancaster 14th June 2019
Redundant rigid augmentation ( k , ℓ )-redundant augmentation Redundantly rigid subgraph ( k , ℓ )-redundant subgraph Introduction Rigidity of graphs Definition A G = ( V , E ) graph minimally rigid (or Laman), if i ( X ) ≤ 2 | X | − 3 ∀ X ⊂ V , | X | ≥ 2 and | E | = 2 | V | − 3. Definition A rigid graph is redundantly rigid , if its every edge is redundant, which is if we leave out any edge, it still remains rigid. Theorem [Jackson and Jord´ an, 2005] A graph of at least 4 vertices is globally rigid ⇔ redundantly rigid and 3-connected. Andr´ as Mih´ alyk´ o GCS, 2019 Co-rigid sets in redundantly rigid augmentations 2 / 20
Redundant rigid augmentation ( k , ℓ )-redundant augmentation Redundantly rigid subgraph ( k , ℓ )-redundant subgraph Introduction Rigidity of graphs Definition A G = ( V , E ) graph minimally rigid (or Laman), if i ( X ) ≤ 2 | X | − 3 ∀ X ⊂ V , | X | ≥ 2 and | E | = 2 | V | − 3. Definition A rigid graph is redundantly rigid , if its every edge is redundant, which is if we leave out any edge, it still remains rigid. Theorem [Jackson and Jord´ an, 2005] A graph of at least 4 vertices is globally rigid ⇔ redundantly rigid and 3-connected. Andr´ as Mih´ alyk´ o GCS, 2019 Co-rigid sets in redundantly rigid augmentations 2 / 20
Redundant rigid augmentation ( k , ℓ )-redundant augmentation Redundantly rigid subgraph ( k , ℓ )-redundant subgraph Introduction Rigidity of graphs Definition A G = ( V , E ) graph minimally rigid (or Laman), if i ( X ) ≤ 2 | X | − 3 ∀ X ⊂ V , | X | ≥ 2 and | E | = 2 | V | − 3. Definition A rigid graph is redundantly rigid , if its every edge is redundant, which is if we leave out any edge, it still remains rigid. Theorem [Jackson and Jord´ an, 2005] A graph of at least 4 vertices is globally rigid ⇔ redundantly rigid and 3-connected. Andr´ as Mih´ alyk´ o GCS, 2019 Co-rigid sets in redundantly rigid augmentations 2 / 20
Redundant rigid augmentation ( k , ℓ )-redundant augmentation Redundantly rigid subgraph ( k , ℓ )-redundant subgraph Problem statement Problem [Garc´ ıa and Tejel, 2011] Let L be a rigid graph. What is the minimal number of edges we need to add to L that augments it to redundantly rigid? Theorem [Garc´ ıa and Tejel, 2011] This problem is NP-hard, but if L is Laman, the optimal edgeset can be found polynomially. O ( n 2 ) algorithm, but no theorem Andr´ as Mih´ alyk´ o GCS, 2019 Co-rigid sets in redundantly rigid augmentations 3 / 20
Redundant rigid augmentation ( k , ℓ )-redundant augmentation Redundantly rigid subgraph ( k , ℓ )-redundant subgraph Problem statement Problem [Garc´ ıa and Tejel, 2011] Let L be a rigid graph. What is the minimal number of edges we need to add to L that augments it to redundantly rigid? Theorem [Garc´ ıa and Tejel, 2011] This problem is NP-hard, but if L is Laman, the optimal edgeset can be found polynomially. O ( n 2 ) algorithm, but no theorem Andr´ as Mih´ alyk´ o GCS, 2019 Co-rigid sets in redundantly rigid augmentations 3 / 20
Redundant rigid augmentation ( k , ℓ )-redundant augmentation Redundantly rigid subgraph ( k , ℓ )-redundant subgraph Andr´ as Mih´ alyk´ o GCS, 2019 Co-rigid sets in redundantly rigid augmentations 4 / 20
Redundant rigid augmentation ( k , ℓ )-redundant augmentation Redundantly rigid subgraph ( k , ℓ )-redundant subgraph Andr´ as Mih´ alyk´ o GCS, 2019 Co-rigid sets in redundantly rigid augmentations 4 / 20
Redundant rigid augmentation ( k , ℓ )-redundant augmentation Redundantly rigid subgraph ( k , ℓ )-redundant subgraph Andr´ as Mih´ alyk´ o GCS, 2019 Co-rigid sets in redundantly rigid augmentations 4 / 20
Redundant rigid augmentation ( k , ℓ )-redundant augmentation Redundantly rigid subgraph ( k , ℓ )-redundant subgraph Co-rigid sets Add a uv edge to a graph G . L ( uv ) = { ij | G + uv − ij is rigid } . L ( uv ) is the set of generated edges by uv . Lemma If G is Laman, L ( uv ) = ∩{ L | u , v ∈ L , L Laman subgraph } Definition Let L = ( V , E ) be a Laman graph. C ⊂ V is called co-rigid , if V − C is rigid. Equivalently, | C | < | V | − 1 and e ( C ) = 2 | C | . If L + H is redundantly rigid, H must touch every co-rigid sets. Lemma Let L be a Laman graph, and H an edgeset so that L + H is � � |C| redundantly rigid. | H | ≥ { | C is set of disjoint co-rigid sets } . 2 Andr´ as Mih´ alyk´ o GCS, 2019 Co-rigid sets in redundantly rigid augmentations 5 / 20
Redundant rigid augmentation ( k , ℓ )-redundant augmentation Redundantly rigid subgraph ( k , ℓ )-redundant subgraph Co-rigid sets Add a uv edge to a graph G . L ( uv ) = { ij | G + uv − ij is rigid } . L ( uv ) is the set of generated edges by uv . Lemma If G is Laman, L ( uv ) = ∩{ L | u , v ∈ L , L Laman subgraph } Definition Let L = ( V , E ) be a Laman graph. C ⊂ V is called co-rigid , if V − C is rigid. Equivalently, | C | < | V | − 1 and e ( C ) = 2 | C | . If L + H is redundantly rigid, H must touch every co-rigid sets. Lemma Let L be a Laman graph, and H an edgeset so that L + H is � � |C| redundantly rigid. | H | ≥ { | C is set of disjoint co-rigid sets } . 2 Andr´ as Mih´ alyk´ o GCS, 2019 Co-rigid sets in redundantly rigid augmentations 5 / 20
Redundant rigid augmentation ( k , ℓ )-redundant augmentation Redundantly rigid subgraph ( k , ℓ )-redundant subgraph Co-rigid sets Add a uv edge to a graph G . L ( uv ) = { ij | G + uv − ij is rigid } . L ( uv ) is the set of generated edges by uv . Lemma If G is Laman, L ( uv ) = ∩{ L | u , v ∈ L , L Laman subgraph } Definition Let L = ( V , E ) be a Laman graph. C ⊂ V is called co-rigid , if V − C is rigid. Equivalently, | C | < | V | − 1 and e ( C ) = 2 | C | . If L + H is redundantly rigid, H must touch every co-rigid sets. Lemma Let L be a Laman graph, and H an edgeset so that L + H is � � |C| redundantly rigid. | H | ≥ { | C is set of disjoint co-rigid sets } . 2 Andr´ as Mih´ alyk´ o GCS, 2019 Co-rigid sets in redundantly rigid augmentations 5 / 20
Redundant rigid augmentation ( k , ℓ )-redundant augmentation Redundantly rigid subgraph ( k , ℓ )-redundant subgraph Andr´ as Mih´ alyk´ o GCS, 2019 Co-rigid sets in redundantly rigid augmentations 6 / 20
Redundant rigid augmentation ( k , ℓ )-redundant augmentation Redundantly rigid subgraph ( k , ℓ )-redundant subgraph Andr´ as Mih´ alyk´ o GCS, 2019 Co-rigid sets in redundantly rigid augmentations 7 / 20
Redundant rigid augmentation ( k , ℓ )-redundant augmentation Redundantly rigid subgraph ( k , ℓ )-redundant subgraph Structure of minimal co-rigid sets Lemma [Jord´ an, 2014] Let L be a Laman graph and C be the family of inclusion-wise minimal co-rigid sets of L . The sets of C are pairwise disjoint or ∃{ u , v } that C ∩ { u , v } � = ∅ ∀ C ∈ C . Moreover, this u and v are not neighbouring. If there exists such a { u , v } pair, we can augment to redundantly rigid with one edge between u and v . Suppose that the sets of C are pairwise disjoint. Representative vertices of the minimal co-rigid sets: i 1 , ..., i |C| Lemma L ( i 1 i 2 ) ⊃ C 1 ∪ N ( C 1 ) ∪ C 2 ∪ N ( C 2 ) Andr´ as Mih´ alyk´ o GCS, 2019 Co-rigid sets in redundantly rigid augmentations 8 / 20
Redundant rigid augmentation ( k , ℓ )-redundant augmentation Redundantly rigid subgraph ( k , ℓ )-redundant subgraph Structure of minimal co-rigid sets Lemma [Jord´ an, 2014] Let L be a Laman graph and C be the family of inclusion-wise minimal co-rigid sets of L . The sets of C are pairwise disjoint or ∃{ u , v } that C ∩ { u , v } � = ∅ ∀ C ∈ C . Moreover, this u and v are not neighbouring. If there exists such a { u , v } pair, we can augment to redundantly rigid with one edge between u and v . Suppose that the sets of C are pairwise disjoint. Representative vertices of the minimal co-rigid sets: i 1 , ..., i |C| Lemma L ( i 1 i 2 ) ⊃ C 1 ∪ N ( C 1 ) ∪ C 2 ∪ N ( C 2 ) Andr´ as Mih´ alyk´ o GCS, 2019 Co-rigid sets in redundantly rigid augmentations 8 / 20
Redundant rigid augmentation ( k , ℓ )-redundant augmentation Redundantly rigid subgraph ( k , ℓ )-redundant subgraph Lemma Let L be a Laman graph with minimal co-rigid sets C . For any connected graph H on the representative vertices, L + H is redundantly rigid. Lemma (reduction) Let L be a Laman graph and H an edge set on its representative vertices for which L + H is redundantly rigid. If there exists a vertex v with d H ( v ) ≥ 3, there is an edge set H ′ on the same vertices for which | H ′ | < | H | and L + H ′ is redundant too. Andr´ as Mih´ alyk´ o GCS, 2019 Co-rigid sets in redundantly rigid augmentations 9 / 20
Recommend
More recommend
