Delaunay Graphs For Various Geometric Objects

Delaunay Graphs For Various Geometric
Objects
Akanksha Agrawal
Faculty Advisor: Prof. Sathish Govindarajan
Computer Science and Automation
Indian Institute of Science, Bangalore, India

Delaunay Graphs
Delaunay Graph for the family of circles (Delaunay Triangulation)
p3
p4
p1
p2

Delaunay Graphs
Given a set of n points P ⊂ R2, the Delaunay graph of P for a
family of geometric objects C is a graph deﬁned as follows: the
vertex set is P and two points p, p′ ∈ P are connected by an edge
if and only if there exists some C ∈ C containing p, p′ but no other
point of P.

Our Results
Vertex Cover on Delaunay Graphs

Our Results
Circles (Delaunay Triangulations)
Hardness on graphs realizable as
Delaunay Triangulation
Hardness of vertex cover on
maximal planar graphs

Our Results
Hardness of vertex cover on
maximal planar graphs
Axis-parallel slabs
Hardness
Fixed Parameter Tractable
algorithm on graphs with
maximum degree at most four

Related Works (Triangulations)
◮ Many graph theoretic properties such has hamiltonicity
[Dillencourt, 1996], colorability [Diks et al., 2002],
dominating sets [King and Pelsmajer, 2010], etc have been
widely studied on triangulations.

◮ Realizability of a (near)triangulated graph as a Delaunay
triangulation has been studied in
[Dillencourt and Smith, 1996],[Hiroshima et al., 2000],etc.

◮ Realizability of a (near)triangulated graph as a Delaunay
triangulation has been studied in
[Dillencourt and Smith, 1996],[Hiroshima et al., 2000],etc.
◮ [Alam et al., 2012] gave a polynomial time algorithm for the
realization of an outer-planar (near)triangulated graph as a
Delaunay triangulation.

Related Works(Delaunay graphs for other
geometric objects)
◮ Conﬂict free coloring on the Delaunay graph of axis-parallel
rectangle is well studied [Ackerman and Pinchasi, 2013,
Chan, 2012, Chen et al., 2008].

geometric objects)
Chan, 2012, Chen et al., 2008].
◮ [Even et al., 2002] studies the conﬂict free coloring on
various objects like circles, axis-parallel rectangles, axis-parallel
regular hexagon, etc.

geometric objects)
Chan, 2012, Chen et al., 2008].
◮ [Even et al., 2002] studies the conﬂict free coloring on
various objects like circles, axis-parallel rectangles, axis-parallel
regular hexagon, etc.
◮ Delaunay graphs for axis parallel equilateral triangles has been
studied in [Babu et al., 2013].

Outline
Axis-parallel slabs
Hardness
Fixed Parameter Tractable
algorithm on graphs with
maximum degree at most four

Triangulation
A 2-connected plane graph in which all the interior faces are
triangles.

Delaunay Realizable Triangulations
Chord in a triangulation:

Delaunay Realizable Triangulations
Non-facial triangle in a triangulation:

Hardness Of Vertex Cover On Delaunay
Realizable Triangulations
Vertex Cover on 3-connected triangle-free planar graphs is
NP-complete [Uehara, 1996].

Vertex Cover on 3-connected triangle-free planar graphs is
NP-complete [Uehara, 1996].
Triangulation with no chord or non-facial triangle is Delaunay
Realizable [Dillencourt and Smith, 1996].

Triangulating a face:
v0
v1 v2
v3

Triangulating a face:
v0
v1 v2
v3
u1
u2
u0
u3

Theorem:
Vertex Cover on Delaunay Realizable Triangulations is
NP-complete.

Axis-Parallel Slabs
1
2
3
4
−1
−2
−3
−4
1 2 3 4−1−2−3−4

Axis-Parallel Slabs
1
2
3
4
−1
−2
−3
−4
1 2 3 4−1−2−3−4
1
2
3
4
−1
−2
−3
−4
1 2 3 4−1−2−3−4

Delaunay Graph Of Axis-Parallel Slabs
Example:

Braid Graph
A graph is a Braid Graph if the edges of the graph is the union of
two Hamiltonian paths.

Equivalence:

Equivalence:
0
1
2
3
4
5
6
7
8
9
10
11
12
0 1 2 3 4 5 6 7 8 9 10 11 12
p1
p2
p3
p4
p5

Equivalence:
0
1
2
3
4
5
6
7
8
9
10
11
12
0 1 2 3 4 5 6 7 8 9 10 11 12
p1
p2
p3
p4
p5
p1
p2 p3
p4
p5

Hardness Of Vertex Cover On Braid Graphs
The Vertex Cover problem is NP-complete on cubic 2-connected
planar graphs [Mohar, 2001] 1
1
[Mohar, 2001] B. Mohar (2001).Face covers and the genus problem for
apex graphs.Journal of Combinatorial Theory, Series B, 82(1):102 117.

Goal: Hardness of vertex cover on Braid Graphs
VC is hard on
cubic 2-connected graphs

VC is hard on
Has Hardness
No apparent
structure

VC is hard on
Has Hardness
No apparent
structure
4-regular graphs
can be decomposed into
two sets of cycles

VC is hard on
Has Hardness
No apparent
structure
4-regular graphs
can be decomposed into
two sets of cycles
Hardness not immediate
Has structure:
cycles can be “stitched”
into paths

A 4-regular graph has two 2-factoring.

Example Of 2-factoring
2-factoring
H H′

H H′
C C′

H H′
C C′
SH SH′

H H′
C C′
SH SH′
˜v
˜u
˜u
˜v

Breaking Cycles: Only One Cycle
v
v1
v2
H
v
v3
v4
H′

v
v1
v2
H
v
v3
v4
H′
v1 v2
v3 v4
v′ v′′
W

Gadget W
a
v1 v3
b v2v4
v
′′
y
y′
v′
x
x′
w1
w
w2 w3
w4 w5
w6

Two Paths Covering All Vertices Of W
v2
bv4
a
v1 v3
v
′′
y′
v′
x
x′
w1
w
w2
w4
w5
w3
w6
y

Two Paths Covering All Vertices Of W
v2
bv4
a
v1 v3
v
′′
y′
v′
x
x′
w1
w
w2
w4
w5
w3
w6
y
v2
bv4
a
v1 v3
v
′′
y
y′
v′
x
x′
w1
w
w2
w4
w3
w5
w6

Vertex Cover For W
a b
v1 v3 v4 v2
w

Vertex Cover For W (at least one of a or b
included)
a b
v1 v3 v4 v2
w

Requirement Of Artiﬁcial Path
H′ C1
Ck
u′
u′′
v
v4
v3
C′
j
C′
l
u′
u′′
vv1
v2
Ci
Ck
C1H

Requirement Of Artiﬁcial Path
H′ C1
Ck
u′
u′′
v
v4
v3
C′
j
C′
l
u′
u′′
vv1
v2
Ci
Ck
C1H
W
˜W
u′ u′′
u1 u2
u4u3
v1 v2
v3 v4
˜v′
˜v′′

˜v′ ˜v
H H′
C C′
˜v
˜v′

Stitching Cycles To Braid Graph
H′
H
C
C′
˜v′
˜u′
˜v′′
˜u′′
˜u′′
˜u′
˜v′ ˜v′′

Hardness of Vertex Cover On Braid Graphs
Theorem:
The Vertex Cover problem on a Braid Graph is NP-complete.

Fixed Parameter Tractable Algorithm
◮ A parameterized problem is Fixed Parameter Tractable
(FPT) if it admits an algorithm with running time
O(f (k) · |x|c ) on input x and parameter k, where f is an
arbitrary function depending only on k and c is some constant.

◮ The FPT Algorithm that we propose is a Branching based
algorithm.

◮ The FPT Algorithm that we propose is a Branching based
algorithm.
◮ The algorithm involves exhaustive case analysis.

Preprocessing Rules
◮ Elimination of simplicial vertices.

Preprocessing Rules
◮ Tactfully folding a degree-2 vertex [Chen et al., 1999].

Preprocessing Rules
k sized vertex cover k − 1 sized vertex cover

Preprocessing Rules
k sized vertex cover k − 1 sized vertex cover
Degree > 4

The branch vectors and the running times
Scenario Cases Branch Vector c
Degree Two (not easily foldable) (2, 5) 1.2365
Degree Three Edge in N(v) (3, 3) 1.2599
CommonNeighborBranch
(3,4) 1.2207
(4, 8, 4) 1.2465
Scenario A
Case 1
(2, 5) 1.2365
(7, 4, 5) 1.2365
(7, 9, 5, 5) 1.2498
(2, 10, 6) 1.2530
(7, 4, 10, 6) 1.2475
(7, 9, 5, 10, 6) 1.2575
Case 2 (I)
(4, 7, 5) 1.2365
(9, 5, 7, 5) 1.2498
(4, 7, 10, 6) 1.2475
(9, 5, 7, 10, 6) 1.2575
Case 2 (II)
(4, 5, 6) 1.2498
(4, 10, 6, 6) 1.2590
Scenario B
(2, 5) 1.2365
(2, 6, 10) 1.2530
Scenario C
Case 1 (2, 10, 6) 1.2530
Case 2 (8, 3, 8, 7) 1.2631
Case 3
(7,3,5) 1.2637
(5, 7, 7, 6) 1.2519
(10, 6, 7, 7, 6) 1.2592

One of the branching Rules
Scenario B: Degree 3 vertex v, with neighbors u, w, x and
d(u) = 4, d(w) = d(x) = 3 and a vertex t adjacent to w and x.
v
u
w
x
t
Branch on u

v
u
w
x
t
Branch on u
u ∈ S
Delete u

v
u
w
x
t
Branch on u
u ∈ S
Delete u
|N(w) ∪ N(x)| ≤ 4

v
u
w
x
t
Branch on u
u ∈ S
Delete u
|N(w) ∪ N(x)| ≤ 4
fold vertex v

v
u
w
x
t
Branch on u
u ∈ S
Delete u
|N(w) ∪ N(x)| ≤ 4
fold vertex v
drop of 2 in k

v
u
w
x
t
Branch on u
u ∈ S
Delete u
|N(w) ∪ N(x)| ≤ 4
fold vertex v
drop of 2 in k
u
/∈
S
|N(u)| = 4

v
u
w
x
t
Branch on u
u ∈ S
Delete u
|N(w) ∪ N(x)| ≤ 4
fold vertex v
drop of 2 in k
u
/∈
S
|N(u)| = 4
drop of 4 in k

v
u
w
x
t
Branch on u
u ∈ S
Delete u
|N(w) ∪ N(x)| ≤ 4
fold vertex v
drop of 2 in k
u
/∈
S
|N(u)| = 4
drop of 4 in k
w foldable?

v
u
w
x
t
Branch on u
u ∈ S
Delete u
|N(w) ∪ N(x)| ≤ 4
fold vertex v
drop of 2 in k
u
/∈
S
|N(u)| = 4
drop of 4 in k
w foldable?
fold w

v
u
w
x
t
Branch on u
u ∈ S
Delete u
|N(w) ∪ N(x)| ≤ 4
fold vertex v
drop of 2 in k
u
/∈
S
|N(u)| = 4
drop of 4 in k
w foldable?
fold w
drop of 1 in k

v
u
w
x
t
Branch on u
u ∈ S
Delete u
|N(w) ∪ N(x)| ≤ 4
fold vertex v
drop of 2 in k
u
/∈
S
|N(u)| = 4
drop of 4 in k
w foldable?
fold w
drop of 1 in k
Branch on w

v
u
w
x
t
Branch on u
u ∈ S
Delete u
|N(w) ∪ N(x)| ≤ 4
fold vertex v
drop of 2 in k
u
/∈
S
|N(u)| = 4
drop of 4 in k
w foldable?
fold w
drop of 1 in k
Branch on w
w
∈ S
include
second neighborhood
of w in S

v
u
w
x
t
Branch on u
u ∈ S
Delete u
|N(w) ∪ N(x)| ≤ 4
fold vertex v
drop of 2 in k
u
/∈
S
|N(u)| = 4
drop of 4 in k
w foldable?
fold w
drop of 1 in k
Branch on w
w
∈ S
include
second neighborhood
of w in S
drop of at least 6 in k

v
u
w
x
t
Branch on u
u ∈ S
Delete u
|N(w) ∪ N(x)| ≤ 4
fold vertex v
drop of 2 in k
u
/∈
S
|N(u)| = 4
drop of 4 in k
w foldable?
fold w
drop of 1 in k
Branch on w
w
∈ S
include
second neighborhood
of w in S
w
/∈
S
include
neighbors of
w in S

v
u
w
x
t
Branch on u
u ∈ S
Delete u
|N(w) ∪ N(x)| ≤ 4
fold vertex v
drop of 2 in k
u
/∈
S
|N(u)| = 4
drop of 4 in k
w foldable?
fold w
drop of 1 in k
Branch on w
w
∈ S
include
second neighborhood
of w in S
w
/∈
S
include
neighbors of
w in S
drop of 2 in k

Theorem:
The Vertex Cover problem on graphs that have maximum degree at
most four can be solved in O⋆(1.2637k ) worst-case running time. 2
2
Given f : N → N, O⋆
(f (k)) is O⋆
(f (k) · p(n)), where p(·) is some
polynomial function. That is, the O⋆
notation suppresses polynomial factors in
the running-time expression.

Open Problems
◮ Better FPT algorithms for vertex cover on maximum degree 4
graphs.

Open Problems
graphs.
◮ Exploit the structure of Braid graphs to design an algorithm
with better running time for computing an optimal vertex
cover on them.

Open Problems
graphs.
cover on them.
◮ Exploit the structure of planar embedding of a Delaunay
Triangulation to compute optimal vertex cover on them.

Open Problems
graphs.
cover on them.
◮ Exploit the structure of planar embedding of a Delaunay
Triangulation to compute optimal vertex cover on them.
◮ Dillencourt [Dillencourt, 1990] showed that Delaunay
triangulations have a perfect matching. It remains an open
direction whether we can obtain above-guarantee FPT
algorithm for Delaunay triangulations.

Acknowledgement
Special thanks to Neeldhara Misra and Saurabh Ray for helpful
discussions...

Bibliography I
Ackerman, E. and Pinchasi, R. (2013).
On coloring points with respect to rectangles.
Journal of Combinatorial Theory, Series A, 120(4):811 – 815.
Alam, A., Rivin, I., and Streinu, I. (2012).
Outerplanar graphs and delaunay triangulations.
In Computation, Physics and Beyond, pages 320–329. Springer.
Babu, J., Biniaz, A., Maheshwari, A., and Smid, M. H. M. (2013).
Fixed-orientation equilateral triangle matching of point sets.
In Ghosh, S. K. and Tokuyama, T., editors, WALCOM, volume 7748 of
Lecture Notes in Computer Science, pages 17–28. Springer.
Chan, T. M. (2012).
Conﬂict-free coloring of points with respect to rectangles and
approximation algorithms for discrete independent set.
In Proceedings of the Twenty-eighth Annual Symposium on
Computational Geometry, SoCG ’12, pages 293–302, New York, NY,
USA. ACM.

Bibliography II
Chen, J., Kanj, I. A., and Jia, W. (1999).
Vertex cover: Further observations and further improvements.
In Widmayer, P., Neyer, G., and Eidenbenz, S., editors, Graph-Theoretic
Concepts in Computer Science, volume 1665 of Lecture Notes in
Computer Science, pages 313–324. Springer Berlin Heidelberg.
Chen, X., Pach, J., Szegedy, M., and Tardos, G. (2008).
Delaunay graphs of point sets in the plane with respect to axis-parallel
rectangles.
In Proceedings of the nineteenth annual ACM-SIAM symposium on
Discrete algorithms, SODA ’08, pages 94–101, Philadelphia, PA, USA.
Society for Industrial and Applied Mathematics.
Diks, K., Kowalik, L., and Kurowski, M. (2002).
A new 3-color criterion for planar graphs.
In Graph-Theoretic Concepts in Computer Science, pages 138–149.
Springer.

Bibliography III
Dillencourt, M. B. (1990).
Toughness and delaunay triangulations.
Discrete & Computational Geometry, 5(1):575–601.
Dillencourt, M. B. (1996).
Finding hamiltonian cycles in delaunay triangulations is np-complete.
Discrete Applied Mathematics, 64(3):207 – 217.
Dillencourt, M. B. and Smith, W. D. (1996).
Graph-theoretical conditions for inscribability and delaunay realizability.
Discrete Mathematics, 161(13):63 – 77.
Even, G., Lotker, Z., Ron, D., and Smorodinsky, S. (2002).
Conﬂict-free colorings of simple geometric regions with applications to
frequency assignment in cellular networks.
In Proceedings of the 43rd Symposium on Foundations of Computer
Science, FOCS ’02, pages 691–700, Washington, DC, USA. IEEE
Computer Society.

Bibliography IV
Hiroshima, T., Miyamoto, Y., and Sugihara, K. (2000).
Another proof of polynomial-time recognizability of delaunay graphs.
IEICE Transactions on Fundamentals of Electronics, Communications and
Computer Sciences, 83(4):627–638.
King, E. L. and Pelsmajer, M. J. (2010).
Dominating sets in plane triangulations.
Discrete Mathematics, 310(1718):2221 – 2230.
Mohar, B. (2001).
Face covers and the genus problem for apex graphs.
Journal of Combinatorial Theory, Series B, 82(1):102 – 117.
Uehara, R. (1996).
NP-completeness of the problems on a restricted graph.
Technical Report TWCU-M-0004, Tokyo Woman’s Christian University.

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