Large-girth roots of graphs
Anna Adamaszek, Michal Adamaszek
DIMAP, University of Warwick, UK
STACS 6.03.2010
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Powers and roots of graphs
Definition
If H is a graph, its r -th power G = H r is the graph on the same
vertex set such that two distinct vertices are adjacent in G if their
distance in H is at most r . We call H the r -th root of G .
1
1
2
6
2
6
3
5
3
5
4
H
Anna Adamaszek, Michal Adamaszek
4
H2
Large-girth roots of graphs
Powers and roots of graphs
Definition
If H is a graph, its r -th power G = H r is the graph on the same
vertex set such that two distinct vertices are adjacent in G if their
distance in H is at most r . We call H the r -th root of G .
1
1
2
6
2
6
3
5
3
5
4
H
Anna Adamaszek, Michal Adamaszek
4
H2
Large-girth roots of graphs
Problems related to graph roots
Given a graph G , we can ask:
Does G have an r -th root?
Is the r -th root of G unique?
How to compute the r -th root?
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Problems related to graph roots
Given a graph G , we can ask:
Does G have an r -th root?
Is the r -th root of G unique?
How to compute the r -th root?
Theorem (R. Motwani, M. Sudan ’94)
It is NP-complete to decide if G has a square root.
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Problems related to graph roots
Given a graph G , we can ask:
Does G have an r -th root?
Is the r -th root of G unique?
How to compute the r -th root?
Theorem (R. Motwani, M. Sudan ’94)
It is NP-complete to decide if G has a square root.
Also: r -th root (Van Bang Le, Ngoc Tuy Nguyen, 2009)
But: there is a linear time algorithm to find some r -th tree
root of a graph (non-unique).
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Roots with large girth
The square root of (C6 )2 is not unique.
1
1
2
6
3
5
1
2
6
3
5
6
,
−→
4
2
3
4
5
4
The same result holds for the r -th root of (C2r +2 )r .
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Roots with large girth
The square root of (C6 )2 is not unique.
1
1
2
6
3
5
1
2
6
3
5
2
6
,
−→
4
3
5
4
4
The same result holds for the r -th root of (C2r +2 )r .
The square root of (C7 )2 is unique.
1
1
2
2
7
3
6
4
5
−→
7
3
6
4
5
The same result holds for the r -th root of (C2r +3 )r .
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Roots with large girth and Levenshtein’s conjecture
Definition
The girth of a graph is the length of its shortest cycle.
Conjecture (Levenshtein ’08)
A graph has at most one r -th root with girth at least 2r + 3 and
no leaves.
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Roots with large girth and Levenshtein’s conjecture
Definition
The girth of a graph is the length of its shortest cycle.
Conjecture (Levenshtein ’08)
A graph has at most one r -th root with girth at least 2r + 3 and
no leaves.
“no leaves” restriction necessary
value 2r + 3 best possible
proved for girth at least 2r + 2d(r − 1)/4e + 1
proof yields an exponential time reconstruction algorithm
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Large girth and r =2
[Farzad et al., STACS ’09]
Deciding if G has a square root H
of girth at least 4 — NP-complete,
of girth at least 6 — constructive polynomial,
of girth at least 5 — ???.
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Large girth and r =2
[Farzad et al., STACS ’09]
Deciding if G has a square root H
of girth at least 4 — NP-complete,
of girth at least 6 — constructive polynomial,
of girth at least 5 — ???.
Remark. 2 · 2 + 3 = 7
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Large girth and r =2
[Farzad et al., STACS ’09]
Deciding if G has a square root H
of girth at least 4 — NP-complete,
of girth at least 6 — constructive polynomial,
of girth at least 5 — ???.
Remark. 2 · 2 + 3 = 7
Conjecture (Farzad et al., STACS ’09)
Deciding if a graph G has an r -th root H with girth ≥ 3r − 1 is
polynomial.
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Our results, arbitrary r
Polynomial time reconstruction of all r -th roots of G with
girth ≥ 2r + 3 and no leaves.
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Our results, arbitrary r
Polynomial time reconstruction of all r -th roots of G with
girth ≥ 2r + 3 and no leaves.
Deciding if G has any r -th root with girth ≥ 2r + 3 in
polynomial time, constructively.
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Our results, arbitrary r
Polynomial time reconstruction of all r -th roots of G with
girth ≥ 2r + 3 and no leaves.
Deciding if G has any r -th root with girth ≥ 2r + 3 in
polynomial time, constructively.
Deciding if G has an r -th root with girth ≥ r + 2 is
NP-complete.
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Roots with no leaves
Task: We know Bx = Br (x) for for each vertex x ∈ H. What can
we say about H?
Assumption: H has girth ≥ 2r + 3 and no leaves.
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Roots with no leaves
Task: We know Bx = Br (x) for for each vertex x ∈ H. What can
we say about H?
Assumption: H has girth ≥ 2r + 3 and no leaves.
Bx is a tree,
x
all leaves at level r .
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Roots with no leaves
Task: We know Bx = Br (x) for for each vertex x ∈ H. What can
we say about H?
Assumption: H has girth ≥ 2r + 3 and no leaves.
Bx is a tree,
x
all leaves at level r .
Technique: start locally
assume xy is an edge in H,
in one step compute all the
neighbours for one vertex
using information from Bv .
Test all initial edges.
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Roots with no leaves
Task: We know Bx = Br (x) for for each vertex x ∈ H. What can
we say about H?
Assumption: H has girth ≥ 2r + 3 and no leaves.
Bx is a tree,
x
all leaves at level r .
Technique: start locally
assume xy is an edge in H,
in one step compute all the
neighbours for one vertex
using information from Bv .
Test all initial edges.
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Roots with no leaves
Task: We know Bx = Br (x) for for each vertex x ∈ H. What can
we say about H?
Assumption: H has girth ≥ 2r + 3 and no leaves.
Bx is a tree,
x
all leaves at level r .
Technique: start locally
assume xy is an edge in H,
in one step compute all the
neighbours for one vertex
using information from Bv .
Test all initial edges.
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Roots with no leaves
Task: We know Bx = Br (x) for for each vertex x ∈ H. What can
we say about H?
Assumption: H has girth ≥ 2r + 3 and no leaves.
Bx is a tree,
x
all leaves at level r .
Technique: start locally
assume xy is an edge in H,
in one step compute all the
neighbours for one vertex
using information from Bv .
Test all initial edges.
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
One step — expanding neighbourhoods
y
x
Nx
r
Bx ∩ By
r−1
Bx \ By
By \ Bx
Bx ∪ By is a tree
the leaves are Bx \ By and By \ Bx
goal: find Nx
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
One step — expanding neighbourhoods
x
r
y
Bx ∩ By
r−1
By \ Bx
The set R =
S
v ∈By \Bx
Bv is marked in red
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
One step — expanding neighbourhoods
x
r
y
Bx ∩ By
r−1
By \ Bx
R=
S
v ∈By \Bx
Bv
G = Bx ∩ By \ R \ {x}
Nx ⊆G
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
One step — expanding neighbourhoods
x
r
y
Bx ∩ By
r−1
By \ Bx
S
R = v ∈By \Bx Bv
G = Bx ∩ By \ R \ {x}
S
B = Bx ∩ By ∩ v ∈G Bv
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
One step — expanding neighbourhoods
x
r
y
Bx ∩ By
r−1
By \ Bx
S
R = v ∈By \Bx Bv
G = Bx ∩ By \ R \ {x}
S
B = Bx ∩ By ∩ v ∈G Bv
T
Nx = Bx ∩ By ∩ v ∈B Bv \ {x} — neighbours of x
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
General recognition problem
Task: We know Bx = Br (x) for for each vertex x ∈ H. What can
we say about H?
Assumption: H has girth ≥ 2r + 3.
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
General recognition problem
Task: We know Bx = Br (x) for for each vertex x ∈ H. What can
we say about H?
Assumption: H has girth ≥ 2r + 3.
H looks like:
core(H)
trees
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Recognizing core and tree vertices
Idea:
If v is ”close to the bottom” of the tree then Bv ⊆ Bparent(v ) .
If v is in the core, Bv is not contained in any Bw .
w
v
Bv ⊆ Bw
∀w Bv * Bw
v
We can find all tree vertices.
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Reconstructing the graph
Task: We know Bx = Br (x) for for each vertex x ∈ H. What can
we say about H?
Assumption: H has girth ≥ 2r + 3 and no leaves.
Split the vertices into “core” and “tree” vertices.
Find all r -th roots of the core (if Levenshtein’s conjecture
holds — only one!)
Attach trees to each core:
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Reconstructing the graph
Task: We know Bx = Br (x) for for each vertex x ∈ H. What can
we say about H?
Assumption: H has girth ≥ 2r + 3 and no leaves.
Split the vertices into “core” and “tree” vertices.
Find all r -th roots of the core (if Levenshtein’s conjecture
holds — only one!)
Attach trees to each core:
core(H)
r
x
determine the partition of “tree vertices”
into trees,
determine the depths of tree vertices,
for each tree solve a tree r -th root
problem with additional depth restrictions.
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs
Open problems
Levenshtein’s conjecture
close the gap for polynomial time reconstruction
Anna Adamaszek, Michal Adamaszek
Large-girth roots of graphs