An Inductive Construction for Plane Laman Graphs via Vertex Splitting

Fekete, Zoltán; Jordán, Tibor; Whiteley, Walter

doi:10.1007/978-3-540-30140-0_28

Cited by 12 publications

(35 citation statements)

References 10 publications

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“…The first approach is a constructive one, and it can be used for plane (2, 2)-tight graphs and plane co-Laman graphs. We find a special construction sequence for each graph in these two classes (similar to the Henneberg moves [14] for (2, 3)-tight graphs, also see [11,21,23]), and we show that this construction sequence can be modified into a construction sequence for a CCA-representation. The second approach is structural.…”

Section: Introductionmentioning

confidence: 69%

“…; see Fekete et al [11]). The duals of the (2, 4)-tight graph are exactly the 4-regular plane graphs.…”

Section: Introductionmentioning

confidence: 93%

“…Henneberg moves H 1 and H 2 were introduced by Henneberg [14], moves E 3 and V 2 2 were defined by Whiteley [23], E 3 also appears in [11] under the name vertex-splitting, and the move V 4 was introduced by Nixon and Owen [21].…”

Section: Contact Representations From Henneberg Movesmentioning

confidence: 99%

“…Fekete et al [11] claimed the following without a proof on generating Laman-plus-one graphs by E 3 moves starting from K 4 . For the sake of completeness we prove for the claim in Appendix B. Lemma 4 (Fekete et al [11]). Every plane Laman-plus-one graph can be generated by E 3 moves starting from K 4 .…”

Section: Contact Representations From Henneberg Movesmentioning

confidence: 99%

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Contact Graphs of Circular Arcs

Alam

Eppstein

Kaufmann

et al. 2015

Lecture Notes in Computer Science

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We study representations of graphs by contacts of circular arcs, CCA-representations for short, where the vertices are interiordisjoint circular arcs in the plane and each edge is realized by an endpoint of one arc touching the interior of another. A graph is (2, k)-sparse if every s-vertex subgraph has at most 2s − k edges, and (2, k)-tight if in addition it has exactly 2n − k edges, where n is the number of vertices. Every graph with a CCA-representation is planar and (2, 0)-sparse, and it follows from known results that for k ≥ 3 every (2, k)-sparse graph has a CCA-representation. Hence the question of CCA-representability is open for (2, k)-sparse graphs with 0 ≤ k ≤ 2. We partially answer this question by computing CCA-representations for several subclasses of planar (2, 0)-sparse graphs. Next, we study CCA-representations in which each arc has an empty convex hull. We show that every plane graph of maximum degree 4 has such a representation, but that finding such a representation for a plane (2, 0)-tight graph with maximum degree 5 is NP-complete. Finally, we describe a simple algorithm for representing plane (2, 0)-sparse graphs with wedges, where each vertex is represented with a sequence of two circular arcs (straight-line segments).

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Section: Introductionmentioning

confidence: 69%

“…; see Fekete et al [11]). The duals of the (2, 4)-tight graph are exactly the 4-regular plane graphs.…”

Section: Introductionmentioning

confidence: 93%

Section: Contact Representations From Henneberg Movesmentioning

confidence: 99%

Section: Contact Representations From Henneberg Movesmentioning

confidence: 99%

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Contact Graphs of Circular Arcs

Alam

Eppstein

Kaufmann

et al. 2015

Lecture Notes in Computer Science

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“…A detailed treatment of rigidity and flexibility can be found for example in [14,27,31,34], and techniques from rigidity theory have been applied to motion planning [29], molecular conformations [31], sensor and network topologies [4] or formations of multiple vehicles [26]. For other interesting works related to rigidity, see [2,7,9,15,16,19].…”

Section: Introductionmentioning

confidence: 99%