Graphs with Flexible Labelings

Grasegger, Georg; Legerský, Jan; Schicho, Josef

doi:10.1007/s00454-018-0026-9

Cited by 21 publications

(55 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The idea of repetitive augmenting the graph G by edges in U(G) is to decrease the number of inequalities checking injectivity of compatible realizations -adjacent vertices must be always mapped to different points. This can be seen in Example 2.3 -the construction of a flexible labeling from a NAC-coloring described in [5] always coincides the vertices 1 and 4, or 2 and 5. But the edges {1, 4} and {2, 5} are added to the constant distance closure already in the first iteration.…”

Section: Combinatorial Toolsmentioning

confidence: 92%

“…Four general ways of constructing a proper flexible labeling are presented. The first two are known -the Dixon I construction for bipartite graphs [4] and the construction from a single NAC-coloring presented in [5]. We describe a new construction that produces an algebraic motion with two active NAC-colorings based on a certain injective embedding of vertices in R 3 .…”

Section: Construction Of Proper Flexible Labelingsmentioning

confidence: 99%

“…, L 6 , since there are NACcolorings satisfying the assumption, see Figure 5. The displayed proper flexible labelings can be obtained by the more general "zikzag" construction from [5]. Now, we present a construction assuming a special injective embedding in R 3 .…”

Section: Lemma 32 Let δ Be a Nac-coloring Of A Graph G Letmentioning

confidence: 99%

“…, 8} are isomorphic to L 1 and K 3,3 respectively. Since G 1 and G 2 satisfy the assumptions of Lemma 3.8, it is sufficient to take proper flexible labelings of G 1 and G 2 , given by Lemma 3.2 and 3.1, such that the quadrilateral (3,4,5,6) in both graphs moves as a non-degenerated rhombus, i.e., λ(3, 4) = λ(4, 5) = λ(5, 6) = λ (3,6).…”

Section: Lemma 32 Let δ Be a Nac-coloring Of A Graph G Letmentioning

confidence: 99%

See 3 more Smart Citations

Graphs with flexible labelings allowing injective realizations

Grasegger

Legerský

Schicho

2020

Discrete Mathematics

Self Cite

View full text Add to dashboard Cite

We consider realizations of a graph in the plane such that the distances between adjacent vertices satisfy the constraints given by an edge labeling. If there are infinitely many such realizations, counted modulo rigid motions, the labeling is called flexible. The existence of a flexible labeling, possibly non-generic, has been characterized combinatorially by the existence of a so called NAC-coloring. Nevertheless, the corresponding realizations are often non-injective. In this paper, we focus on flexible labelings with infinitely many injective realizations. We provide a necessary combinatorial condition on existence of such a labeling based also on NAC-colorings of the graph. By introducing new tools for the construction of such labelings, we show that the necessary condition is also sufficient up to 8 vertices, but this is not true in general for more vertices. Now, the abbreviation NAC can be explained -it stands for "No Almost Cycle". Clearly, if we swap colors of a NAC-coloring of G, we obtain another NAC-coloring of G. Definition 1.2. Let G be a graph. If δ, δ ∈ NAC G are such that δ(e) = blue ⇐⇒ δ(e) = red for all e ∈ E G , then they are called conjugated.The following definition describes the constraints on a realization in the plane given by a labeling of edges. The realizations must be counted properly, i.e., modulo rigid motions, in order to say whether the labeling is flexible.

show abstract

Section: Combinatorial Toolsmentioning

confidence: 92%

Section: Construction Of Proper Flexible Labelingsmentioning

confidence: 99%

Section: Lemma 32 Let δ Be a Nac-coloring Of A Graph G Letmentioning

confidence: 99%

Section: Lemma 32 Let δ Be a Nac-coloring Of A Graph G Letmentioning

confidence: 99%

See 2 more Smart Citations

Graphs with flexible labelings allowing injective realizations

Grasegger

Legerský

Schicho

2020

Discrete Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…The study of such graphs and their motions has a long history (see for instance [1,4,11,12,[14][15][16][17]). Recently we provided a series of results [7,8] with a deeper analysis of the existence of flexible placements. This is done via special edge colorings called NAC-colorings ("No Almost Cycles", see [7]).…”

Section: Introductionmentioning

confidence: 99%