Overview
![](static/img/octo.png)
Stack number
A book embedding is an embedding of a graph to a collection of books, that is, half-planes having the same line as their boundary. The vertices of the graph lie on this boundary line, called the spine, and the edges stay within a single half-plane. The book thickness of a graph (also called page number, stack number and fixed outerthickness) is the smallest possible number of half-planes for any book embedding of the graph. Stack layouts were introduced by Ollmann in 1973.
graph class | lower bound | reference | upper bound | reference |
---|---|---|---|---|
tree | 1 | gen-ref-BK79 | 1 | gen-ref-BK79 |
outerplanar | 1 | gen-ref-BK79 | 1 | gen-ref-BK79 |
series-parallel | 2 | gen-ref-RM95 | 2 | gen-ref-RM95 |
planar 3-tree | 3 | gen-ref-H84 | 3 | gen-ref-H84 |
Halin | 2 | gen-ref-G95 | 2 | gen-ref-G95 |
planar Laman | 3 | [dot] | 4 | gen-ref-Y89 |
planar bipartite | 2 | [img] | 2 | gen-ref-FMP95 gen-ref-O98 |
planar max-degree ≤ 4 | 2 | [img] | 2 | gen-ref-BGR16 gen-ref-G73 |
planar 3-connected max-degree ≤ 5 | 2 | [img] | 2 | gen-ref-HK19 |
planar max-degree ≤ 5 | 2 | [img] | 3 | gen-ref-GY19 |
planar max-degree ≥ 7 | 3 | [dot] [img] | 4 | gen-ref-Y89 |
maximal planar max-degree ≤ 6 | 2 | [img] | 2 | gen-ref-G73 |
planar | 4 | [img] gen-ref-BKKPRU20 | 4 | gen-ref-Y89 |
genus g | 4 | [img] gen-ref-BKKPRU20 | O(√g) | gen-ref-M94 |
proper minor-closed | 4 | [img] gen-ref-BKKPRU20 | O(1) | gen-ref-B03 |
1-planar | 4 | gen-ref-ABK15 gen-ref-BKZ15 | 10 | gen-ref-F20 gen-ref-F23 |
k-planar | Ω(√ k) | gen-ref-ABK15 gen-ref-BLGGMR20 | O(k log n) | gen-ref-DF18 |
pathwidth k | k | gen-ref-TY02 | k | gen-ref-TY02 |
treewidth k | k + 1 | gen-ref-VDG09 | k + 1 | gen-ref-GH01 |
complete Kn | ⌈n / 2⌉ | gen-ref-BK79 | ⌈n / 2⌉ | gen-ref-BK79 |
complete bipartite Kn,m | min(n, m) | gen-ref-BK79 gen-ref-SPE13 | min(n, m) | gen-ref-BK79 gen-ref-ENO97 |
Queue number
In a queue layout, the vertices of a graph are restricted to a line and the edges are drawn at different half-planes delimited by this line, called queues. The task is to find a linear order of the vertices along the underlying line and a corresponding assignment of the edges of the graph to the queues, so that no two independent edges of the same queues are nested. The queue number of a graph is the smallest number of queues that are required by any queue layout of the graph. Queue layouts were defined by Heath and Rosenberg in 1992.
graph class | lower bound | reference | upper bound | reference |
---|---|---|---|---|
tree | 1 | gen-ref-HR92 | 1 | gen-ref-HR92 |
outerplanar | 2 | gen-ref-HR92 | 2 | gen-ref-HLR92 |
outerplanar bipartite | 1 | gen-ref-BDDEW18 | 1 | gen-ref-BDDEW18 |
series-parallel | 3 | gen-ref-W17 | 3 | gen-ref-RM95 |
planar 3-tree | 4 | gen-ref-ABGKP18 | 5 | gen-ref-ABGKP18 |
Halin | 2 | gen-ref-G95 | 2 | gen-ref-BDDEW18 |
X-Tree | 2 | gen-ref-HR92 | 2 | gen-ref-HR92 |
planar bipartite | 3 | gen-ref-FKMPR23 | 28 | gen-ref-FKMPR23 |
planar max-degree Δ=3 | 2 | [img] | O(Δ2) | gen-ref-BFGMMRU19 gen-ref-DMW19 |
planar | 4 | gen-ref-ABGKP18 | 42 | gen-ref-DJMMUW19 gen-ref-BGR23 |
genus g | 4 | gen-ref-ABGKP18 | O(g) | gen-ref-DJMMUW19 |
proper minor-closed | 4 | gen-ref-ABGKP18 | O(1) | gen-ref-DJMMUW19 |
1-planar | 4 | gen-ref-ABGKP18 | 82 | gen-ref-DMW20 gen-ref-BDHK22 |
k-planar | Ω(√ k) | gen-ref-ABGKP18 | O(f(k)) | gen-ref-DJMMUW19 gen-ref-DMW20 |
pathwidth k | k | gen-ref-W02 | k | gen-ref-W02 |
treewidth k | k + 1 | gen-ref-W17 | 2k - 1 | gen-ref-W17 |
complete Kn | ⌊n / 2⌋ | gen-ref-HR92 | ⌊n / 2⌋ | gen-ref-HR92 |
complete bipartite Kn,m | min(⌈n / 2⌉, ⌈m / 2⌉) | gen-ref-HR92 | min(⌈n / 2⌉, ⌈m / 2⌉) | gen-ref-HR92 |
Track number
A track layout of a graph is a partition of its vertices into sequences, called tracks, such that the vertices in each sequence form an independent set and the edges between each two pairs of tracks form a non-crossing set. Track layouts were introduced by Dujmović et al., although similar structures are implicit in several previous works studying low-volume three-dimensional graph drawings.
graph class | lower bound | reference | upper bound | reference |
---|---|---|---|---|
tree | 3 | gen-ref-FLW06 | 3 | gen-ref-FLW06 |
outerplanar | 5 | [img] | 5 | gen-ref-DPW04 |
outerplanar bipartite | 3 | [img] | 3 | gen-ref-FLW06 gen-ref-BDDEW18 |
series-parallel | 7 | gen-ref-DMW05 gen-ref-M20 | 15 | gen-ref-GLM03 |
planar 3-tree | 8 | gen-ref-P19 | 25 | gen-ref-P19 |
Halin | 5 | [img] | 6 | gen-ref-BDDEW18 |
X-Tree | 5 | [img] | 5 | gen-ref-P19 |
planar bipartite | 5 | [dot] | 225 | gen-ref-P19 |
planar | 8 | gen-ref-P19 | 225 | gen-ref-P19 |
treewidth k | 1 ⁄ 2 (k + 1)(k + 2) + 1 | gen-ref-DMW05 gen-ref-M20 | (k + 1)(2k+1-2)k | gen-ref-W17 |
Directed graphs
Stack and queue layouts can be extended to directed acyclic graphs (DAGs) by requiring that the vertex order is a topological
order of the DAG. It was first discussed by Heath et al. Similar layouts are studied for
posets (partially ordered sets), or equivalently, DAGs without transitive edges.
For stack layouts of DAGs, gen-ref-HPT99 and gen-ref-HP99 show that directed trees and unicyclic
DAGs have stack numbers 1 and 2, respectively;
gen-ref-MS09 proves that N-free upward planar DAGs, which contain series-parallel digraphs, have
stack number 2.
gen-ref-AL05 investigate stack layouts of bipartite and complete multipartite ordered sets.
A study in gen-ref-FFR13 gave several conditions under which upward planar
triangulations have a constant stack number: (i) maximal
upward planar 3-trees and (ii) planar triangulations with a bounded (directed) diameter.
Later gen-ref-NP21 and gen-ref-BLMN22 extend the classes of DAGs that have bounded stack number.
A sub-linear bound on the stack number of upward planar graphs is known (gen-ref-JMU22a) and a (very large)
constant upper bound on the stack number of (non-upward) outerplanar DAGs (gen-ref-JMU22b).
For the lower bounds, there exist upward outerplanar DAGs with stack number 4, upward planar 3-tree DAGs and
planar posets with stack number 5 (gen-ref-NP21, gen-ref-JMU22a), and
(non-upward) 2-tree DAGs with unbounded stack number (gen-ref-JMU22b).
References
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