Decyclings
Graph: points and lines (vertices and edges)
Cycle: a path from a vertex to itself
Forest: a graph with no cycles
Tree: a forest with all vertices connected
Removing vertices can turn a cyclic graph into
a forest (“decycling”) and a forest into a tree
7
7
3
3
3
3
2
7
In the above figure:
Originally: 5 vertices
Size of the largest induced forest: τ (G) = 4
Decycling number: ∇(G) = 1
Minimal decycling = largest induced forest
Size of the largest induced tree: t(G) = 3
All trees are forests too
Size of largest induced tree ≤ of forest
1
The graphs above extend infinitely to the left
and to the right:
Size of largest induced forest or decycling
number (or both) is infinite
Instead of size or number, we find the fraction
(“density”) of vertices
0
Density of largest induced forest: τ (G) = 34
Density of largest induced tree: t0 (G) = 34
Decycling density: ∇0 (G) = 14
3
5
6
5
4
5
3
3
2
6
3
1
1
3
6
2
0
1
1
3
2
6
3
2
3
3
3
2
We examined infinite graphs of regular
degree, particularly tessellations
3
3
3
7
7
Regular graph: d edges meet at
every vertex (d is the “degree”)
For any induced tree in the integer
lattice (such as the one above):
≤2
≤
≤2
2
≥2
≥
≥2
2
≥0
≥
≥0
0
≤4
≤
≤4
4
:: ≤≤ 44 :: 22
By extension, τ (G) ≤
0
d
2d−2
for all d
4
5
We made partial progress examining
decyclings of root lattices of rank two
We found the decycling density of the
hypercubical lattice using Cayley graphs
Applications
Logic circuit design
Logic simulation
Diagnosing circuit faults
Sequential circuit test generation
Partial scan problem for flip-flops
Future Research
We can decycle other tessellations
Tessellations in higher dimensions
Demiregular tilings (two types of vertices)
Non-Euclidean geometries, such as
tessellations of the hyperbolic plane
5
3
We established a lower bound on the
decycling density of these graphs
We calculated decycling densities for all
regular and semiregular planar tilings
Bayesian inference
Constraint satisfaction
Markov chain state space
For the square lattice, we find two induced
trees achieving the upper bound of density 23 :
the greedy tree (upper left) and a patterned
forest turned into a tree (lower left)
The hexagonal lattice (upper right) also
achieves the bound given by its degree, with
a densest induced tree of density 34
We hypothesize that the triangular lattice
(lower right) cannot achieve the bound of 35 , so
we use a clever argument taking advantage of
the fact that neighbors of a vertex are adjacent
to each other to establish a smaller upper bound
of 12 , which is achievable
3
6
1
Regular Tessellations
5
5
3
1
2
7
Artificial intelligence
6
4
3
Using a new algebraic description,
2
we proved the tree has density 3
S = {(i, j) | ij = 0 ∨ ν2 (|i|) 6= ν2 (|j|)}
This gives an upper bound on the
decycling number of the grid:
log2 (n) X n + 2i 2
∇(G) ≤ 4
i+1
2
i=0
The bound is not tight, as shown
by the figure in the box to the right
Motivation
5
0
We bounded the grid’s decycling number
We verified that the greedy tree is the
densest induced tree
7
1
3
7
2
We proved a new description for the
greedy tree in the integer lattice
Regular Graphs
Finite Grid
3
6
Summary
Results
6
Semiregular Tessellations
5
6
3
6
5
4
5
6
5
6
The greedy tree in the integer lattice
Start with the origin (labeled 0)
At each time step, add all vertices adjacent
to exactly one vertex already in the tree
Stanley and Chapman claim that this tree has
density 32 , and they seek the maximum density of an induced tree in the integer lattice
Tessellations
To the right: Semiregular tessellations 4–8–8
(left) and 4–6–12 (right) both have degree 3
and achieve the bound of 34
We can further examine root lattices and
extend our work to higher ranks
We can investigate Barton’s conjectures
about decycling densities of Cayley
graphs on 2 or 3 arbitrary generators
To the left: 3–12–12 has degree 3, but the
densest induced forest (left) and tree (right)
have a density of only 23
Right image from Reference [12]
To the right: The tessellations 3–6–3–6 (left)
and 3–4–6–4 (right) have degree 4, and both
achieve the degree bound of 23
Bibliography
To the left: To decycle 3–3–3–4–4, we first
detriangle the tiling (left) and then draw an
7
induced tree (right) of density 12
To the right: We detriangle 3–3–4–3–4 (left)
and 3–3–3–3–6 (right), also giving densest
7
induced trees of density 12
Tessellation: infinite tiling of the plane (no
overlaps, no gaps) with polygons
Regular tessellation (3 in 2D): one polygon
Semiregular tessellation (8 in 2D): many
polygons, but each vertex “looks the same”
Root Lattices
Cayley Graphs
Tilings of vectors from Lie algebra
A1 × A1 , A2 are regular tessellations
B2 and G2 contain the triangular
lattice, giving a bound
View the d-dimensional hypercubical
lattice as a Cayley graph (lower right)
Mapping to integers is regular
We achieve the decycling bound by a
clever construction (lower left)
Thus, the decycling density is 2d−2
4d−2
B2
τ 0 (G) =
0
Goal
We wish to calculate the decycling
densities of various infinite tilings,
especially of tessellations
t (G) =
G2
1
2
1
2
3+3×4=15
(3,4)
0
1
2
3
4
2+3×3=11
5
(2,3)
(4,3)
(3,3) 3+3×3=12
0
1
τ (G) ≥ 3
t0 (G) ≥ 41
4+3×3=13
(3,2)
3+3×2=9
0
1
2
3
4
5
6
7
8
9
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Approximation algorithms for the feedback vertex set
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Hyperbolic
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at
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