Finding S uc u e(s) in G aphs o Bounded Pa h-Wid h
Oli e Bach le and I ene Hein ich
Depa men o Ma hema ics
TU Kaise slau e n
and
Depa men o Ma hema ics
TU Da ms ad
Fu u e Resea ch in Combina o ial Op imiza ion, 2022
Mo i a ion
Conjec u e
E e y g aph in Gsa is ies p ope y π.
Now: Find such “ ame” s uc u es and collec hem in a se U.
Wan : E e y cubic g aph o pa h-wid h a mos khas a subg aph in U.
⇒The conjec u e holds o all cubic g aphs o pa h-wid h a mos k.
Ques ion: Does e e y cubic g aph o pa h-wid h a mos kcon ain a
subg aph in U?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 1 / 14
Mo i a ion
Conjec u e
E e y g aph in Gsa is ies p ope y π.
Now: Find such “ ame” s uc u es and collec hem in a se U.
Wan : E e y cubic g aph o pa h-wid h a mos khas a subg aph in U.
⇒The conjec u e holds o all cubic g aphs o pa h-wid h a mos k.
Ques ion: Does e e y cubic g aph o pa h-wid h a mos kcon ain a
subg aph in U?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 1 / 14
Mo i a ion
Conjec u e
E e y g aph in G
cubic g aph
sa is ies p ope y π.
Now: Find such “ ame” s uc u es and collec hem in a se U.
Wan : E e y cubic g aph o pa h-wid h a mos khas a subg aph in U.
⇒The conjec u e holds o all cubic g aphs o pa h-wid h a mos k.
Ques ion: Does e e y cubic g aph o pa h-wid h a mos kcon ain a
subg aph in U?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 1 / 14
Mo i a ion
Conjec u e
E e y g aph in G
cubic g aph
sa is ies p ope y π.
Now: Find such “ ame” s uc u es and collec hem in a se U.
Wan : E e y cubic g aph o pa h-wid h a mos khas a subg aph in U.
⇒The conjec u e holds o all cubic g aphs o pa h-wid h a mos k.
Ques ion: Does e e y cubic g aph o pa h-wid h a mos kcon ain a
subg aph in U?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 1 / 14
Mo i a ion
Conjec u e
E e y g aph in G
cubic g aph
sa is ies p ope y π.
Now: Find such “ ame” s uc u es and collec hem in a se U.
Wan : E e y cubic g aph o pa h-wid h a mos khas a subg aph in U.
⇒The conjec u e holds o all cubic g aphs o pa h-wid h a mos k.
Ques ion: Does e e y cubic g aph o pa h-wid h a mos kcon ain a
subg aph in U?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 1 / 14
Mo i a ion
Conjec u e
E e y g aph in G
cubic g aph
sa is ies p ope y π.
Now: Find such “ ame” s uc u es and collec hem in a se U.
Wan : E e y cubic g aph o pa h-wid h a mos khas a subg aph in U.
⇒The conjec u e holds o all cubic g aphs o pa h-wid h a mos k.
Ques ion: Does e e y cubic g aph o pa h-wid h a mos kcon ain a
subg aph in U?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 1 / 14
Mo i a ion
Conjec u e
E e y g aph in G
cubic g aph
sa is ies p ope y π.
Now: Find such “ ame” s uc u es and collec hem in a se U.
Wan : E e y cubic g aph o pa h-wid h a mos khas a subg aph in U.
⇒The conjec u e holds o all cubic g aphs o pa h-wid h a mos k.
Ques ion: Does e e y cubic g aph o pa h-wid h a mos kcon ain a
subg aph in U?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 1 / 14
Mo i a ion
Conjec u e
E e y g aph in G
cubic g aph
sa is ies p ope y π.
Now: Find such “ ame” s uc u es and collec hem in a se U.
Wan : E e y cubic g aph o pa h-wid h a mos khas a subg aph in U.
⇒The conjec u e holds o all cubic g aphs o pa h-wid h a mos k.
Ques ion: Does e e y cubic g aph o pa h-wid h a mos kcon ain a
subg aph in U?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 1 / 14
Answe ing he Ques ion
Q: Does e e y cubic g aph o pa h-wid h a mos 3con ain a K2,3?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 3 / 14
Answe ing he Ques ion
Q: Does e e y cubic g aph o pa h-wid h a mos 3con ain a K2,3?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 3 / 14
Answe ing he Ques ion
Q: Does e e y cubic g aph o pa h-wid h a mos 3con ain a K2,3?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 3 / 14
Answe ing he Ques ion
Q: Does e e y cubic g aph o pa h-wid h a mos 3con ain a K2,3?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 3 / 14
Answe ing he Ques ion
Q: Does e e y cubic g aph o pa h-wid h a mos 3con ain a K2,3?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 3 / 14
Answe ing he Ques ion
Q: Does e e y cubic g aph o pa h-wid h a mos 3con ain a K2,3?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 3 / 14
Answe ing he Ques ion
Q: Does e e y cubic g aph o pa h-wid h a mos 3con ain a K2,3?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 3 / 14
Answe ing he Ques ion
Q: Does e e y cubic g aph o pa h-wid h a mos 3con ain a K2,3?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 3 / 14
Answe ing he Ques ion
Q: Does e e y cubic g aph o pa h-wid h a mos 3con ain a K2,3?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 3 / 14
Answe ing he Ques ion
Q: Does e e y cubic g aph o pa h-wid h a mos 3con ain a K2,3?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 3 / 14
Answe ing he Ques ion
Q: Does e e y cubic g aph o pa h-wid h a mos 3con ain a K2,3?
A: No!
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 3 / 14
Answe ing he Ques ion
Q: Does e e y cubic g aph o pa h-wid h a mos 3con ain a K2,3o a ∆?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 3 / 14
Answe ing he Ques ion
Q: Does e e y cubic g aph o pa h-wid h a mos 3con ain a K2,3o a ∆?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 3 / 14
Answe ing he Ques ion
Q: Does e e y cubic g aph o pa h-wid h a mos 3con ain a K2,3o a ∆?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 3 / 14
Answe ing he Ques ion
Q: Does e e y cubic g aph o pa h-wid h a mos 3con ain a K2,3o a ∆?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 3 / 14
Answe ing he Ques ion
Q: Does e e y cubic g aph o pa h-wid h a mos 3con ain a K2,3o a ∆?
A: Yes!
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 3 / 14
Ou line
Ob aining an Algo i hm
Speeding I Up
Does I E en Te mina e?
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 4 / 14
Ob aining an Algo i hm
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 5 / 14
De eloping an algo i hm
de FindS uc u es(G,U,k):
Ini ialise a queue Qwi h Ek+1
while Q=∅do
G←Q.dequeue()
o each ib an e ex u∈Gdo
o each choice o edges Fa udo
G′←G+F+ ,uis dulled
i G′con ains a subg aph in U hen con inue
i G′con ains a coun e example H hen e u n H
Q.append(G′)
G
′
=
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 6 / 14
De eloping an algo i hm
de FindS uc u es(G,U,k):
Ini ialise a queue Qwi h Ek+1
while Q=∅do
G←Q.dequeue()
o each ib an e ex u∈Gdo
o each choice o edges Fa udo
G′←G+F+ ,uis dulled
i G′con ains a subg aph in U hen con inue
i G′con ains a coun e example H hen e u n H
Q.append(G′)
Q=
G
′
=
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 6 / 14
De eloping an algo i hm
de FindS uc u es(G,U,k):
Ini ialise a queue Qwi h Ek+1
while Q=∅do
G←Q.dequeue()
o each ib an e ex u∈Gdo
o each choice o edges Fa udo
G′←G+F+ ,uis dulled
i G′con ains a subg aph in U hen con inue
i G′con ains a coun e example H hen e u n H
Q.append(G′)
Q=G′=
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 6 / 14
De eloping an algo i hm
de FindS uc u es(G,U,k):
Ini ialise a queue Qwi h Ek+1
while Q=∅do
G←Q.dequeue()
o each ib an e ex u∈Gdo
o each choice o edges Fa udo
G′←G+F+ ,uis dulled
i G′con ains a subg aph in U hen con inue
i G′con ains a coun e example H hen e u n H
Q.append(G′)
Q=G′=
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 6 / 14
De eloping an algo i hm
de FindS uc u es(G,U,k):
Ini ialise a queue Qwi h Ek+1
while Q=∅do
G←Q.dequeue()
o each ib an e ex u∈Gdo
o each choice o edges Fa udo
G′←G+F+ ,uis dulled
i G′con ains a subg aph in U hen con inue
i G′con ains a coun e example H hen e u n H
Q.append(G′)
Q=G′=
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 6 / 14
De eloping an algo i hm
de FindS uc u es(G,U,k):
Ini ialise a queue Qwi h Ek+1
while Q=∅do
G←Q.dequeue()
o each ib an e ex u∈Gdo
o each choice o edges Fa udo
G′←G+F+ ,uis dulled
i G′con ains a subg aph in U hen con inue
i G′con ains a coun e example H hen e u n H
Q.append(G′)
Q=
G
′
=
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 6 / 14
De eloping an algo i hm
de FindS uc u es(G,U,k):
Ini ialise a queue Qwi h Ek+1
while Q=∅do
G←Q.dequeue()
o each ib an e ex u∈Gdo
o each choice o edges Fa udo
G′←G+F+ ,uis dulled
i G′con ains a subg aph in U hen con inue
i G′con ains a coun e example H hen e u n H
Q.append(G′)
Q=G
′
=
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 6 / 14
De eloping an algo i hm
de FindS uc u es(G,U,k):
Ini ialise a queue Qwi h Ek+1
while Q=∅do
G←Q.dequeue()
o each ib an e ex u∈Gdo
o each choice o edges Fa udo
G′←G+F+ ,uis dulled
i G′con ains a subg aph in U hen con inue
i G′con ains a coun e example H hen e u n H
Q.append(G′)
Q=G
′
=
u
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 6 / 14
De eloping an algo i hm
de FindS uc u es(G,U,k):
Ini ialise a queue Qwi h Ek+1
while Q=∅do
G←Q.dequeue()
o each ib an e ex u∈Gdo
o each choice o edges Fa udo
G′←G+F+ ,uis dulled
i G′con ains a subg aph in U hen con inue
i G′con ains a coun e example H hen e u n H
Q.append(G′)
Q=G
′
=
u
F
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 6 / 14
Speeding I Up
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 7 / 14
How slow is his ac ually?
Example:
5 g aphs.
Algo i hm: 45 g aphs.
x4
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 8 / 14
How slow is his ac ually?
Example:
5 g aphs.
Algo i hm: 45 g aphs.
x4
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 8 / 14
Teaching he algo i hm o ecognise symme ies
De ini ion
A ib an au omo phism φo a g aph Gis
▶an au omo phism φ
▶ ha maps 7→ and 7→
Using ib an isomo phisms
▶ o each ib an e ex u∈Gdo
▶ o each choice o edges Fa udo
x4
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 9 / 14
Teaching he algo i hm o ecognise symme ies
De ini ion
A ib an au omo phism φo a g aph Gis
▶an au omo phism φ
▶ ha maps 7→ and 7→
Using ib an isomo phisms
▶ o each ib an e ex u∈Gdo
▶ o each choice o edges Fa udo
x4
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 9 / 14
Teaching he algo i hm o ecognise symme ies
De ini ion
A ib an au omo phism φo a g aph Gis
▶an au omo phism φ
▶ ha maps 7→ and 7→
Using ib an isomo phisms
▶ o each ib an e ex u∈Gdo
▶ o each choice o edges Fa udo x4
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 9 / 14
Teaching he algo i hm o ecognise symme ies
De ini ion
A ib an au omo phism φo a g aph Gis
▶an au omo phism φ
▶ ha maps 7→ and 7→
Using ib an isomo phisms
▶ o each ib an e ex u∈Gdo
▶ o each choice o edges Fa udo x4
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 9 / 14
Teaching he algo i hm o ecognise symme ies
De ini ion
A ib an au omo phism φo a g aph Gis
▶an au omo phism φ
▶ ha maps 7→ and 7→
Using ib an isomo phisms
▶ o each ib an e ex u∈Gdo
▶ o each choice o edges Fa udo x4
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 9 / 14
Teaching he algo i hm o ecognise symme ies
De ini ion
A ib an au omo phism φo a g aph Gis
▶an au omo phism φ
▶ ha maps 7→ and 7→
Using ib an isomo phisms
▶ o each ib an e ex u∈Gdo
▶ o each choice o edges Fa udo
x4
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 9 / 14
Teaching he algo i hm o ecognise symme ies
De ini ion
A ib an au omo phism φo a g aph Gis
▶an au omo phism φ
▶ ha maps 7→ and 7→
Using ib an isomo phisms
▶ o each ib an e ex u∈Gdo
▶ o each choice o edges Fa udo
x4
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 9 / 14
Teaching he algo i hm o ecognise symme ies
De ini ion
A ib an au omo phism φo a g aph Gis
▶an au omo phism φ
▶ ha maps 7→ and 7→
Using ib an isomo phisms
▶ o each ib an e ex u∈Gdo
▶ o each choice o edges Fa udo
x4
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 9 / 14
Teaching he algo i hm o ecognise symme ies
De ini ion
A ib an au omo phism φo a g aph Gis
▶an au omo phism φ
▶ ha maps 7→ and 7→
Using ib an isomo phisms
▶ o each ib an e ex u∈Gdo
▶ o each choice o edges Fa udo
x4
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 9 / 14
Teaching he algo i hm o ecognise symme ies
De ini ion
A ib an au omo phism φo a g aph Gis
▶an au omo phism φ
▶ ha maps 7→ and 7→
Using ib an isomo phisms
▶ o each ib an e ex u∈Gdo
▶ o each choice o edges Fa udo
x4
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 9 / 14
Wha ’s he wo s ha could happen?
▶I a coun e example exis s, hen he algo i hm inds (a smalles ) one.
▶O he wise, we migh be in ouble.
Fo example, le Gcon ain he ollowing g aphs:
We can cons uc hese as ollows:
The iangle appea s a bi a ily la e!
Lemma
De e mining whe he e e y g aph in Ghas a subg aph in Uis undecidable.
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 11 / 14
Wha ’s he wo s ha could happen?
▶I a coun e example exis s, hen he algo i hm inds (a smalles ) one.
▶O he wise, we migh be in ouble.
Fo example, le Gcon ain he ollowing g aphs:
We can cons uc hese as ollows:
The iangle appea s a bi a ily la e!
Lemma
De e mining whe he e e y g aph in Ghas a subg aph in Uis undecidable.
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 11 / 14
Wha ’s he wo s ha could happen?
▶I a coun e example exis s, hen he algo i hm inds (a smalles ) one.
▶O he wise, we migh be in ouble.
Fo example, le Gcon ain he ollowing g aphs:
We can cons uc hese as ollows:
The iangle appea s a bi a ily la e!
Lemma
De e mining whe he e e y g aph in Ghas a subg aph in Uis undecidable.
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 11 / 14
Wha ’s he wo s ha could happen?
▶I a coun e example exis s, hen he algo i hm inds (a smalles ) one.
▶O he wise, we migh be in ouble.
Fo example, le Gcon ain he ollowing g aphs:
We can cons uc hese as ollows:
The iangle appea s a bi a ily la e!
Lemma
De e mining whe he e e y g aph in Ghas a subg aph in Uis undecidable.
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 11 / 14
Wha ’s he wo s ha could happen?
▶I a coun e example exis s, hen he algo i hm inds (a smalles ) one.
▶O he wise, we migh be in ouble.
Fo example, le Gcon ain he ollowing g aphs:
We can cons uc hese as ollows:
The iangle appea s a bi a ily la e!
Lemma
De e mining whe he e e y g aph in Ghas a subg aph in Uis undecidable.
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 11 / 14
Wha ’s he wo s ha could happen?
▶I a coun e example exis s, hen he algo i hm inds (a smalles ) one.
▶O he wise, we migh be in ouble.
Fo example, le Gcon ain he ollowing g aphs:
We can cons uc hese as ollows:
The iangle appea s a bi a ily la e!
Lemma
De e mining whe he e e y g aph in Ghas a subg aph in Uis undecidable.
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 11 / 14
Wha ’s he wo s ha could happen?
▶I a coun e example exis s, hen he algo i hm inds (a smalles ) one.
▶O he wise, we migh be in ouble.
Fo example, le Gcon ain he ollowing g aphs:
We can cons uc hese as ollows:
The iangle appea s a bi a ily la e!
Lemma
De e mining whe he e e y g aph in Ghas a subg aph in Uis undecidable.
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 11 / 14
Wha ’s he wo s ha could happen?
▶I a coun e example exis s, hen he algo i hm inds (a smalles ) one.
▶O he wise, we migh be in ouble.
Fo example, le Gcon ain he ollowing g aphs:
We can cons uc hese as ollows:
The iangle appea s a bi a ily la e!
Lemma
De e mining whe he e e y g aph in Ghas a subg aph in Uis undecidable.
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 11 / 14
Wha ’s he wo s ha could happen?
▶I a coun e example exis s, hen he algo i hm inds (a smalles ) one.
▶O he wise, we migh be in ouble.
Fo example, le Gcon ain he ollowing g aphs:
We can cons uc hese as ollows:
The iangle appea s a bi a ily la e!
Lemma
De e mining whe he e e y g aph in Ghas a subg aph in Uis undecidable.
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 11 / 14
Wha ’s he wo s ha could happen?
▶I a coun e example exis s, hen he algo i hm inds (a smalles ) one.
▶O he wise, we migh be in ouble.
Fo example, le Gcon ain he ollowing g aphs:
We can cons uc hese as ollows:
The iangle appea s a bi a ily la e!
Lemma
De e mining whe he e e y g aph in Ghas a subg aph in Uis undecidable.
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 11 / 14
Wha ’s he wo s ha could happen?
▶I a coun e example exis s, hen he algo i hm inds (a smalles ) one.
▶O he wise, we migh be in ouble.
Fo example, le Gcon ain he ollowing g aphs:
We can cons uc hese as ollows:
The iangle appea s a bi a ily la e!
Lemma
De e mining whe he e e y g aph in Ghas a subg aph in Uis undecidable.
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 11 / 14
Wha ’s he wo s ha could happen?
▶I a coun e example exis s, hen he algo i hm inds (a smalles ) one.
▶O he wise, we migh be in ouble.
Fo example, le Gcon ain he ollowing g aphs:
We can cons uc hese as ollows:
The iangle appea s a bi a ily la e!
Lemma
De e mining whe he e e y g aph in Ghas a subg aph in Uis undecidable.
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 11 / 14
Wha ’s he wo s ha could happen?
▶I a coun e example exis s, hen he algo i hm inds (a smalles ) one.
▶O he wise, we migh be in ouble.
Fo example, le Gcon ain he ollowing g aphs:
We can cons uc hese as ollows:
The iangle appea s a bi a ily la e!
Lemma
De e mining whe he e e y g aph in Ghas a subg aph in Uis undecidable.
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 11 / 14
Wha ’s he wo s ha could happen?
▶I a coun e example exis s, hen he algo i hm inds (a smalles ) one.
▶O he wise, we migh be in ouble.
Fo example, le Gcon ain he ollowing g aphs:
We can cons uc hese as ollows:
The iangle appea s a bi a ily la e!
Lemma
De e mining whe he e e y g aph in Ghas a subg aph in Uis undecidable.
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 11 / 14
Wha ’s he wo s ha could happen?
▶I a coun e example exis s, hen he algo i hm inds (a smalles ) one.
▶O he wise, we migh be in ouble.
Fo example, le Gcon ain he ollowing g aphs:
We can cons uc hese as ollows:
The iangle appea s a bi a ily la e!
Lemma
De e mining whe he e e y g aph in Ghas a subg aph in Uis undecidable.
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 11 / 14
The good news
Theo em
The algo i hm can be modi ied such ha i e mina es o he cubic case i
Ucon ains connec ed g aphs.
Idea.
▶Disca d mo e g aphs.
▶Take ca e ha no all coun e examples a e los .
This is small!
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 12 / 14
The good news
Theo em
The algo i hm can be modi ied such ha i e mina es o he cubic case i
Ucon ains connec ed g aphs.
Idea.
▶Disca d mo e g aphs.
▶Take ca e ha no all coun e examples a e los .
This is small!
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 12 / 14
The good news
Theo em
The algo i hm can be modi ied such ha i e mina es o he cubic case i
Ucon ains connec ed g aphs.
Idea.
▶Disca d mo e g aphs.
▶Take ca e ha no all coun e examples a e los .
This is small!
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 12 / 14
The good news
Theo em
The algo i hm can be modi ied such ha i e mina es o he cubic case i
Ucon ains connec ed g aphs.
Idea.
▶Disca d mo e g aphs.
▶Take ca e ha no all coun e examples a e los .
This is small!
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 12 / 14
The good news
Theo em
The algo i hm can be modi ied such ha i e mina es o he cubic case i
Ucon ains connec ed g aphs.
Idea.
▶Disca d mo e g aphs.
▶Take ca e ha no all coun e examples a e los .
This is small!
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 12 / 14
The good news
Theo em
The algo i hm can be modi ied such ha i e mina es o he cubic case i
Ucon ains connec ed g aphs.
Idea.
▶Disca d mo e g aphs.
▶Take ca e ha no all coun e examples a e los .
This is small!
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 12 / 14
The good news
Theo em
The algo i hm can be modi ied such ha i e mina es o he cubic case i
Ucon ains connec ed g aphs.
Idea.
▶Disca d mo e g aphs.
▶Take ca e ha no all coun e examples a e los .
This is small!
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 12 / 14
The good news
Theo em
The algo i hm can be modi ied such ha i e mina es o he cubic case i
Ucon ains connec ed g aphs.
Idea.
▶Disca d mo e g aphs.
▶Take ca e ha no all coun e examples a e los .
This is small!
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 12 / 14
E en be e news
Theo em
The algo i hm can be modi ied such ha i e mina es when
▶Gis “locally ce i iable”,
▶Ghas bounded maximum deg ee, and
▶Uis a ini e se o connec ed g aphs.
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 13 / 14
E en be e news
Theo em
The algo i hm can be modi ied such ha i e mina es when
▶Gis “locally ce i iable”,
▶Ghas bounded maximum deg ee, and
▶Uis a ini e se o connec ed g aphs.
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 13 / 14
Summa y
We ha e:
▶de eloped an algo i hm ha checks whe he e e y g aph in Go
pa h-wid h a mos kcon ain a subg aph in U.
▶inco po a ed symme ies o speed up he algo i hm, and
▶showed ha we can achie e e mina ion in special cases.
Fo he scep ics: a o mal e sion is in he a Xi .
Con ac : bach le @ma hema ik.uni-kl.de
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 14 / 14
Summa y
We ha e:
▶de eloped an algo i hm ha checks whe he e e y g aph in Go
pa h-wid h a mos kcon ain a subg aph in U.
▶inco po a ed symme ies o speed up he algo i hm, and
▶showed ha we can achie e e mina ion in special cases.
Fo he scep ics: a o mal e sion is in he a Xi .
Con ac : bach le @ma hema ik.uni-kl.de
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 14 / 14
Summa y
We ha e:
▶de eloped an algo i hm ha checks whe he e e y g aph in Go
pa h-wid h a mos kcon ain a subg aph in U.
▶inco po a ed symme ies o speed up he algo i hm, and
▶showed ha we can achie e e mina ion in special cases.
Fo he scep ics: a o mal e sion is in he a Xi .
Con ac : bach le @ma hema ik.uni-kl.de
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 14 / 14
Fo une elling: someone will ask abou . . .
PCP
x1=0,x2=01,x3=110
y1=100,y2=00,y3=11
(1,2,3): 001110 1000011
(3,2,3,1): 110011100 110011100
k3 3 4 5 5 6 6 7
U U3U4U4U4U5U5U6U6
Resul K3,3None None Pe e sen None Heawood None None
Base 6 5 81 12484 3841 – – –
Cubic 3 2 3 7 5 15 9 19
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 1 / 2
Re e ences
li
O. Bach le and I. Hein ich Finding S uc u e(s) in G aphs FRICO 2022 2 / 2
