The Decycling Number on Graphs and Digraphs

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The Decycling Number on Graphs and Digraphs

Min-Yun Lien

Advisor: Hung-Lin Fu

Department of Applied Mathematics, National Chiao Tung University, Taiwan

2014-08-03

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Outline

1

Decycling Problem and Applications

2

Decycling Number and Cycle Packing Number

3

Decycling Number on Graphs

Path Product

4

Decycling Number on Digraphs

Generalized Kautz Digraph Generalized de Bruijn Digraphs

5

Objective Work

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Definition and Applications

Definition (Decycling Problem)

Given a directed/undirected graphG= (V, E), find a minimum set D⊂ V such thatG\ Dis acyclic.

H as applications in

Deadlock prevention in operating systems (Wang et al.

1985; Silberschatz et al. 2003)

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Definition and Applications

Definition (Decycling Problem)

Given a directed/undirected graphG= (V, E), find a minimum set D⊂ V such thatG\ Dis acyclic.

H as applications in

Deadlock prevention in operating systems (Wang et al.

1985; Silberschatz et al. 2003)

Example: An operating system schedules different processes.

A B

C

Process A is waiting for the resource on Process B so it can’t release its own resource.

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Definition and Applications

Definition (Decycling Problem)

Given a directed/undirected graphG= (V, E), find a minimum set D⊂ V such thatG\ Dis acyclic.

H as applications in

Deadlock prevention in operating systems (Wang et al.

1985; Silberschatz et al. 2003)

Example: An operating system schedules different processes.

B A

C Process A is waiting for no resource so it can release its resource.

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Definition and Applications

Definition (Decycling Problem)

Given a directed/undirected graphG= (V, E), find a minimum set D⊂ V such thatG\ Dis acyclic.

H as applications in

Deadlock prevention in operating systems (Wang et al.

1985; Silberschatz et al. 2003)

Example: An operating system schedules different processes.

B A

C

Deadlock:

Competing actions are each waiting for the other to finish, and thus neither ever does.

Solution:

Remove some processes to break such cycles and put them in a waiting queue.

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Applications

Monopolies in synchronous distributed systems (Peleg 1998; Peleg 2002)

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Applications

Monopolies in synchronous distributed systems (Peleg 1998; Peleg 2002)

⊲ Vertices are coloredYESorNO.

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Applications

Monopolies in synchronous distributed systems (Peleg 1998; Peleg 2002)

⊲ Vertices are coloredYESorNO.

⊲ Monotonesynchronous system: at each step aNOvertex changes toYESif more than half of its neighbors areYES.

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Applications

Monopolies in synchronous distributed systems (Peleg 1998; Peleg 2002)

⊲ Vertices are coloredYESorNO.

⊲ Monotonesynchronous system: at each step aNOvertex changes toYESif more than half of its neighbors areYES.

⊲ Problem: Set the minimum number of vertices YES at beginning such that all vertices become YES eventually.

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Applications

Monopolies in synchronous distributed systems (Peleg 1998; Peleg 2002)

⊲ Vertices are coloredYESorNO.

⊲ Monotonesynchronous system: at each step aNOvertex changes toYESif more than half of its neighbors areYES.

⊲ Problem: Set the minimum number of vertices YES at beginning such that all vertices become YES eventually.

⊲ A decycling set could be the only choice!

Example: A4-regular graph (such as toroidal mesh network).

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Applications

Monopolies in synchronous distributed systems (Peleg 1998; Peleg 2002)

⊲ Vertices are coloredYESorNO.

⊲ Monotonesynchronous system: at each step aNOvertex changes toYESif more than half of its neighbors areYES.

⊲ Problem: Set the minimum number of vertices YES at beginning such that all vertices become YES eventually.

⊲ A decycling set could be the only choice!

Example: A4-regular graph (such as toroidal mesh network).

Decycling Set Decycling Set

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Applications

Monopolies in synchronous distributed systems (Peleg 1998; Peleg 2002)

⊲ Vertices are coloredYESorNO.

⊲ Monotonesynchronous system: at each step aNOvertex changes toYESif more than half of its neighbors areYES.

⊲ Problem: Set the minimum number of vertices YES at beginning such that all vertices become YES eventually.

⊲ A decycling set could be the only choice!

Example: A4-regular graph (such as toroidal mesh network).

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Applications

Monopolies in synchronous distributed systems (Peleg 1998; Peleg 2002)

⊲ Vertices are coloredYESorNO.

⊲ Monotonesynchronous system: at each step aNOvertex changes toYESif more than half of its neighbors areYES.

⊲ Problem: Set the minimum number of vertices YES at beginning such that all vertices become YES eventually.

⊲ A decycling set could be the only choice!

Example: A4-regular graph (such as toroidal mesh network).

Decycling Set Decycling Set

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Applications

Monopolies in synchronous distributed systems (Peleg 1998; Peleg 2002)

⊲ Vertices are coloredYESorNO.

⊲ Monotonesynchronous system: at each step aNOvertex changes toYESif more than half of its neighbors areYES.

⊲ Problem: Set the minimum number of vertices YES at beginning such that all vertices become YES eventually.

⊲ A decycling set could be the only choice!

Example: A4-regular graph (such as toroidal mesh network).

NOvertices on the cycle remain NO.

Failed!

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Applications

Constraint satisfaction problem (Dechter 1990).

Bayesian inference in artificial intelligence (Bar-Yehuda et al. 1998).

Converters’ placement problem in optical networks (Kleinberg and Kumar 1999).

VLSI chip design (Festa et al. 2000).

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Complexity

H as been extensively studied.

NP-hard

Reduction from VERTEX COVER (R. Karp 1972).

Even for planar graphs, bipartite graphs.

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Related Problems

I s related or equivalent to

Feedback vertex set problem(Wang et al. 1985).

Hitting cycle problem

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Related Problems

I s related or equivalent to

Feedback vertex set problem(Wang et al. 1985).

Hitting cycle problem

Maximum induced forest problem

(Erd ¨os, Saks and S ´os 1986 Maximum induced trees in graphs).

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Variations

H as the following variations (survey Festa et al. 2000).

Graph Bipartization Problem

FindD⊂ Esuch thatG\ Dhas no odd cycle.

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Variations

H as the following variations (survey Festa et al. 2000).

Graph Bipartization Problem

FindD⊂ Esuch thatG\ Dhas no odd cycle.

Weighted Decycling Problem

Seek a minimum-weight decycling setDwhere each vertex has a weight.

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Variations

H as the following variations (survey Festa et al. 2000).

Graph Bipartization Problem

FindD⊂ Esuch thatG\ Dhas no odd cycle.

Weighted Decycling Problem

Seek a minimum-weight decycling setDwhere each vertex has a weight.

Loop Cut Set Problem

GivenA(C) ⊆ V(C)for each cycleCofG, find a minimum setDsuch thatD∩ A(C) 6= ∅.

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Relations with Cycle Packing Number

I s often compared with the following graph parameter.

Cycle Packing Numberν(G): the maximum number of vertex-disjoint cycles ofG.

1 ) ( G Q

Definition:∇(G) :=the decycling number (minimum size of decycling set) ofG.

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Relations with Cycle Packing Number

⊲ Dirac and Gallai had interest in the relations betweenν(G) and∇(G).

⊲ It is clear thatν(G) ≤ ∇(G).

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Relations with Cycle Packing Number

⊲ Dirac and Gallai had interest in the relations betweenν(G) and∇(G).

⊲ It is clear thatν(G) ≤ ∇(G).

Definition:∇(k) := max{∇(G) : ν(G) = k}.

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Relations with Cycle Packing Number

⊲ Dirac and Gallai had interest in the relations betweenν(G) and∇(G).

⊲ It is clear thatν(G) ≤ ∇(G).

Definition:∇(k) := max{∇(G) : ν(G) = k}.

⊲ ∇(1) ≤ 3;ν(K5) = 1and∇(K5) = 3(Bollob´as 1964).

⊲ ∇(2) = 6and9≤ ∇(3) ≤ 12(Voss 1968).

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Relations with Cycle Packing Number

⊲ Dirac and Gallai had interest in the relations betweenν(G) and∇(G).

⊲ It is clear thatν(G) ≤ ∇(G).

Definition:∇(k) := max{∇(G) : ν(G) = k}.

⊲ ∇(1) ≤ 3;ν(K5) = 1and∇(K5) = 3(Bollob´as 1964).

⊲ ∇(2) = 6and9≤ ∇(3) ≤ 12(Voss 1968).

c1k log k≤ ∇(k) ≤ c2k log kfor some constantsc1 andc2

(Erd¨os and P´osa 1964).

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Relations with Cycle Packing Number

Consider planar graphs:

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Relations with Cycle Packing Number

Consider planar graphs:

Jones’ Conjecture (Kloks, Lee and Liu 2002) For every planar graphG,∇(G) ≤ 2ν(G).

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Relations with Cycle Packing Number

Consider planar graphs:

Jones’ Conjecture (Kloks, Lee and Liu 2002) For every planar graphG,∇(G) ≤ 2ν(G).

Theorem (Chen, Fu and Shih 2010, TJM) For every planar graphG,∇(G) ≤ 3ν(G).

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Decycling number of outerplanar graphs

Consider outerplanar graphs:

Theorem (Kloks, Lee and Liu 2002)

For every outerplanar graphG,∇(G) ≤ 2ν(G).

◮ An outerplanar graphGis calledlower-extremalif∇(G) = ν(G) andupper-extremalif∇(G) = 2ν(G).

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Decycling number of outerplanar graphs

Consider outerplanar graphs:

Theorem (Kloks, Lee and Liu 2002)

For every outerplanar graphG,∇(G) ≤ 2ν(G).

◮ An outerplanar graphGis calledlower-extremalif∇(G) = ν(G) andupper-extremalif∇(G) = 2ν(G).

◮ Upper-Extremal Results:

We define asun graphS3as follows.

∇(S3) = 2 = 2ν(S3). S

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Decycling number of outerplanar graphs

Theorem (Chang, Fu, Lien, 2011, JCO)

An outerplanar graphGis upper-extremal if and only ifGis anS3-tree.

A graph is anS3-tree of ordertif it has exactlyt

vertex-disjointS3-subdivisions and every edge not on these S3-subdivisions belongs to no cycle.

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Decycling number of outerplanar graphs

Theorem (Chang, Fu, Lien, 2011, JCO)

An outerplanar graphGis upper-extremal if and only ifGis anS3-tree.

A graph is anS3-tree of ordertif it has exactlyt

vertex-disjointS3-subdivisions and every edge not on these S3-subdivisions belongs to no cycle.

Example:

An S3-tree G of order 3, where’(G) 6 2Q(G).

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Decycling number of outerplanar graphs

◮ Lower-Extremal Results:

The following graphs are NOT lower-extremal (∇(G) 6= ν(G)):

Sun graphsSkwith odd numberk:

S3 S5

∇(Sk) = ⌈k2⌉andν(Sk) = ⌊k2⌋.

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Decycling number of outerplanar graphs

Theorem (Chang, Fu, Lien, 2011, JCO)

For an outerplanar graphG, ifGhas noSk-subdivision for all odd numberk, thenGis lower-extremal.

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Decycling number of Graphs

Lower Bound of Undirected Graphs

Lemma (Beineke, 1997, JGT)

IfGis a connected graph withpvertices (p> 2),qedges, and maximum degree∆, then∇(G) ≥q−p+1∆−1 .

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Cartesian Product

Definition (G✷H)

V(G✷H) = {(u, v)|u ∈ V(G)andv∈ V(H)}

E(G✷H) = {(u, v)(u, v)|u = uand(v, v) ∈ E(H)or(u, u) ∈ E(G)andv= v}

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Cartesian Product

Definition (G✷H)

V(G✷H) = {(u, v)|u ∈ V(G)andv∈ V(H)}

E(G✷H) = {(u, v)(u, v)|u = uand(v, v) ∈ E(H)or(u, u) ∈ E(G)andv= v}

Figure :P9✷P11

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Decycling number of C

m

✷C

n

Theorem (Pike, Zou, 2005, SIDMA)

∇(Cm✷Cn) =

3n2ifm= 4,

3m2ifn= 4,

mn+23otherwise.

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Decycling number of Path Product P

m

✷P

n

Lower bound

Theorem (Luccio, 1998, IPL)

Ifm, n ≥ 2, then∇(Pm✷Pn) ≥l(m−1)(n−1)+1 3

m .

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Decycling number of Path Product P

m

✷P

n

Theorem (Madelaine and Stewart, 2008, DISC) Table :

In Table, A:∇(Pm✷Pn) = Fm,n, B:∇(Pm✷Pn) ≤ Fm,n+ 1, C:

∇(Pm✷Pn) ≤ Fm,n+ 2, whereFm,n=l(m−1)(n−1)+1 3

m.

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Decycling number of Path Product P

m

✷P

n

New Lower bound Proposition (Observation) Ifm≥ 5andfm,n=(m−1)(n−1)+1

3 is an integer, then each decycling set Sof sizefm,nsatisfies the following two properties:

(1)Scontains exactly one vertex of degree3and contains no vertex of degree 2; and

(2)Sinduces a subgraph ofPm✷Pnwith no edges.

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Decycling number of Path Product P

m

✷P

n

New Lower bound Proposition (Observation) Ifm≥ 5andfm,n=(m−1)(n−1)+1

3 is an integer, then each decycling set Sof sizefm,nsatisfies the following two properties:

(1)Scontains exactly one vertex of degree3and contains no vertex of degree 2; and

(2)Sinduces a subgraph ofPm✷Pnwith no edges.

Theorem (Lien, Fu, Shih, 2014, DMAA) Ifm≥ 5,mnis even andfm,nis an integer, then

∇(Pm✷Pn) ≥ fm,n+ 1 = Fm,n+ 1,wherefm,n= (m−1)(n−1)+1

3 .

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Decycling number of Path Product P

m

✷P

n

Proof.

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Decycling number of Path Product P

m

✷P

n

Proof.

Suppose not.

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Decycling number of Path Product P

m

✷P

n

Proof.

Suppose not.

Assume that∇(Pm✷Pn) = fm,n = Fm,nandSis a decycling set with sizefm,n.

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Decycling number of Path Product P

m

✷P

n

Proof.

Suppose not.

Assume that∇(Pm✷Pn) = fm,n = Fm,nandSis a decycling set with sizefm,n.

By above Proposition, we may letvi,1be the vertex ofSwith degree3where2≤ i ≤m

2

.

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Decycling number of Path Product P

m

✷P

n

Proof.

Suppose not.

Assume that∇(Pm✷Pn) = fm,n = Fm,nandSis a decycling set with sizefm,n.

By above Proposition, we may letvi,1be the vertex ofSwith degree3where2≤ i ≤m

2

.

Letvi,1∈ S, 2 ≤ i ≤ ⌊m2⌋.

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Decycling number of Path Product P

m

✷P

n

Proof.

Suppose not.

Assume that∇(Pm✷Pn) = fm,n = Fm,nandSis a decycling set with sizefm,n.

By above Proposition, we may letvi,1be the vertex ofSwith degree3where2≤ i ≤m

2

.

Letvi,1∈ S, 2 ≤ i ≤ ⌊m2⌋.

vm−1,2∈ Sandvm−1,3∈ S./

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Decycling number of Path Product P

m

✷P

n

Proof.

Suppose not.

Assume that∇(Pm✷Pn) = fm,n = Fm,nandSis a decycling set with sizefm,n.

By above Proposition, we may letvi,1be the vertex ofSwith degree3where2≤ i ≤m

2

.

Letvi,1∈ S, 2 ≤ i ≤ ⌊m2⌋.

vm−1,2∈ Sandvm−1,3∈ S./

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Decycling number of Path Product P

m

✷P

n

Proof.

Suppose not.

Assume that∇(Pm✷Pn) = fm,n = Fm,nandSis a decycling set with sizefm,n.

By above Proposition, we may letvi,1be the vertex ofSwith degree3where2≤ i ≤m

2

.

Letvi,1∈ S, 2 ≤ i ≤ ⌊m2⌋.

vm−1,2∈ Sandvm−1,3∈ S./ vm−1,2, vm−1,4, · · · , vm−1,n−1∈ S.

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Decycling number of Path Product P

m

✷P

n

Proof.

Suppose not.

Assume that∇(Pm✷Pn) = fm,n = Fm,nandSis a decycling set with sizefm,n.

By above Proposition, we may letvi,1be the vertex ofSwith degree3where2≤ i ≤m

2

.

Letvi,1∈ S, 2 ≤ i ≤ ⌊m2⌋.

vm−1,2∈ Sandvm−1,3∈ S./ vm−1,2, vm−1,4, · · · , vm−1,n−1∈ S.

Hence,n− 1is even andnis odd.

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Decycling number of Path Product P

m

✷P

n

Proof.

Suppose not.

Assume that∇(Pm✷Pn) = fm,n = Fm,nandSis a decycling set with sizefm,n.

By above Proposition, we may letvi,1be the vertex ofSwith degree3where2≤ i ≤m

2

.

Similarly,

vm−3,n−1, vm−5,n−1, · · · , v2,n−1∈ S.

Thus,mis odd, a contradiction.

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Decycling number of Path Product P

m

✷P

n

New Lower bound Corollary

Form≥ 5, ifm≡ 0(mod6) andn≡ 2 (mod3)or (m, n) ≡ (3, 2) (mod6), then∇(Pm✷Pn) ≥ Fm,n+ 1.

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Decycling number of Path Product P

m

✷P

n

New Lower bound Corollary

Form≥ 5, ifm≡ 0(mod6) andn≡ 2 (mod3)or (m, n) ≡ (3, 2) (mod6), then∇(Pm✷Pn) ≥ Fm,n+ 1.

Theorem

Form≥ 5, if(m, n) ≡ (0, 2), (0, 5), (3, 2), (2, 0), (5, 0), (2, 3) (mod6), then∇(Pm✷Pn) = Fm,n+ 1.

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Decycling number of Path Product P

m

✷P

n

New Lower bound Corollary

Form≥ 5, ifm≡ 0(mod6) andn≡ 2 (mod3)or (m, n) ≡ (3, 2) (mod6), then∇(Pm✷Pn) ≥ Fm,n+ 1.

Theorem

Form≥ 5, if(m, n) ≡ (0, 2), (0, 5), (3, 2), (2, 0), (5, 0), (2, 3) (mod6), then∇(Pm✷Pn) = Fm,n+ 1.

Theorem (Lien, Fu, Shih, 2014, DMAA) Form, n ≥ 6,∇(Pm✷Pn) ≤ Fm,n+ 1.

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Decycling number of Digraphs

5

3 1 0

4 2

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Decycling number of Digraphs

Notation:

Letvbe a vertex in a digraph. The out-neighborhood or successor setN+(v)is{x ∈ V(G) : v → x}.

ForS⊆ V(G),N+(S) = ∪v∈SN+(v).

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Decycling number of Digraphs

Notation:

Letvbe a vertex in a digraph. The out-neighborhood or successor setN+(v)is{x ∈ V(G) : v → x}.

ForS⊆ V(G),N+(S) = ∪v∈SN+(v).

Construction Method Lemma

LetSbe a set of vertices in a digraphG. ThenSis a decycling set of Gif and only if we can find a sequence of subsets ofV(G),

S= S0, S1, · · · , St= V(G)such that (1)Si⊆ Si+1; and

(2)N+(Si+1\ Si) ⊆ Sifori= 0, 1, · · · , t − 1.

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Decycling number of Digraphs

Example 1.

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Decycling number of Digraphs

Example 1.

5

3 1 0

4 2

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Decycling number of Digraphs

Example 1.

5

3 1 0

4

2 0

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Decycling number of Digraphs

Example 1.

5

3 1 0

4

2 0 2

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Decycling number of Digraphs

Example 1.

5

3 1

4 2

There is a directed cycle(1, 3, 5).

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Decycling number of Digraphs

Example 2.

5

3 1 0

4 2

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Decycling number of Digraphs

Example 2.

5

3 1 0

4

2 0

5

S

1

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Decycling number of Digraphs

Example 2.

5

3 1 0

4

2 0 2

5 3

S

1

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Decycling number of Digraphs

Example 2.

5

3 1 0

4

2 0 2

5 3

S

1

S

2

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Decycling number of Digraphs

Example 2.

5

3 1 0

4

2 0 2

5 3

1 4

S

1

S

2

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Decycling number of Digraphs

Example 2.

5

3 1 0

4

2 0 2

5 3

1 4

S

1

S

2 3

S

We haveS3= V(G).

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Decycling number of Digraphs

Example 2.

3 1

4 2

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de Bruijn Digraphs and Kautz Digraphs

Definition (de Bruijn digraphB(d, n))

V(B(d, n)) = {x1x2· · · xn: xi∈ {0, 1, · · · , d − 1}, 1 ≤ i ≤ n}.

Edge: X = x1x2· · · xn−→ Y = x2x3· · · xnαwhereα ∈ {0, 1, · · · , d − 1}.

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de Bruijn Digraphs and Kautz Digraphs

Definition (de Bruijn digraphB(d, n))

V(B(d, n)) = {x1x2· · · xn: xi∈ {0, 1, · · · , d − 1}, 1 ≤ i ≤ n}.

Edge: X = x1x2· · · xn−→ Y = x2x3· · · xnαwhereα ∈ {0, 1, · · · , d − 1}.

Definition (Kautz digraphK(d, n))

V(K(d, n)) = {x1x2· · · xn : xi∈ {0, 1, · · · , d}, 1 ≤ i ≤ nandxi6= xi+1, 1 ≤ i≤ n − 1}.

Edge: X = x1x2· · · xn−→ Y = x2x3· · · xnαwhereα ∈ {0, 1, · · · , d}.

Remark:K(d, n) ⊆ B(d + 1, n).

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Generalized de Bruijn Digraphs Generalized Kautz Digraphs

Definition (Generalized de Bruijn digraphGB(d, n)) V(GB(d, n)) = {0, 1, · · · , n − 1}.

E(GB(d, n)) = {(x, y)|y ≡ dx + i (mod n), 0 ≤ i ≤ d − 1}.

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Generalized de Bruijn Digraphs Generalized Kautz Digraphs

Definition (Generalized de Bruijn digraphGB(d, n)) V(GB(d, n)) = {0, 1, · · · , n − 1}.

E(GB(d, n)) = {(x, y)|y ≡ dx + i (mod n), 0 ≤ i ≤ d − 1}.

Definition (Generalized Kautz digraphGK(d, n)) V(GK(d, n)) = {0, 1, · · · , n − 1}.

E(GK(d, n)) = {(x, y)|y ≡ −dx − i (mod n), 1 ≤ i ≤ d}.

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Decycling number of Generalized Kautz Digraphs

Theorem (Lien, Kuo and Fu)

∇(G

k

(d, n)) ≤

 

 

 

 

 

 

2

9

n + 3t + 1,

where

n ≡ t (mod 36), for d = 2 ,

n

3

+

94

t + 6

where

n ≡ t (mod 36), for d = 3, (

12

d−12d2

)n +

d2

(d − t + 5) − 2 ,

where

n ≡ t (mod d + 1), for d ≥ 4.

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Decycling number of Generalized de Bruijn Digraphs

Theorem (Lien, Kuo and Fu)

∇(G

B

(d, n)) ≤

d+12d

n + 2(d − 1).

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Objective Work

For a planar graphG,∇(G)≤ 2ν(G)? (Jones’ conjecture 2002).

For a planar graphG,∇(G)≤ |V(G)|/2? (Albertson and Berman 1979).

For a bipartite planar graphG,∇(G)≤ 3|V(G)|/8? (Albertson and Berman 1979).

We have⌈(m−1)(n−1)+1

3 ⌉ ≤ ∇(Pm✷Pn) ≤ ⌈(m−1)(n−1)+1

3 ⌉ + 1.Find the exact value of∇(Pm✷Pn).

Find the lower bound of directed graphs.

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References

Albertson MO, Berman DM (1979) A conjecture on planar graphs, Bondy JA, Murty USR, Graph theory and related topics 357.

H. Chang, H. L. Fu and M. Y. Lien, The decycling number of outerplanar graphs, J. Comb. Optim. 25 (2013)

Erd ¨os P, Saks M, S ´os VT (1986) Maximum induced trees in graphs. J Combin Theory Ser B 41:61-79.

Bau S, Beineke LW, Vandell RC (1998) Decycling snakes. Congr Numer 134:79-87.

Bodlaender HL (1994) On disjoint cycles. Int J Found Comput Sci 5:59-68.

Erd ¨os P, Saks M, S ´os VT (1986) Maximum induced trees in graphs. J Combin Theory Ser B 41:61-79.

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Festa P, Pardalos PM, Resende MGC (2000) Feedback set problems, Handbook of Combinatorial Optimization, Du D-Z, Pardalos PM, Eds, Kluwer Academic Publishers, Supplement A, pp 209-259.

Kloks T, Lee C-M, Liu J (2002) New algorithms fork-face cover, k-feedback vertex set, andk-disjoint cycles on plane and planar graphs. in Proceedings of the 28th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2002) Springer-Verlag, 2573:282-295.

M. Y. Lien, H. L. Fu and C. H. Shih, The decycling number of Pm✷Pn, Discrete Math., Alg. and Appl. DOI:

10.1142/S1793830914500335, 2014.

F. R. Madelaine and I. A. Stewart, Improved upper and lower bounds on the feedback vertex numbers of grids and butterflies,

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D. A. Pike and Y. Zou, Decycling Cartesian products of two cycles, SIAM J. Discrete Math., 19 (2005) 651-663.

X. Xu, Y. Cao, J-M. Xu and Y. Wu, Feedback numbers of de Bruijn digraphs, Computers and Mathematics with Application, 59 (2010) 716-723.

F. R. Madelaine and I. A. Stewart, Improved upper and lower bounds on the feedback vertex numbers of grids and butterflies, Discrete Math., 308 (2008) 4144-4164.

D. A. Pike and Y. Zou, Decycling Cartesian products of two cycles, SIAM J. Discrete Math., 19 (2005) 651-663.

X. Xu, Y. Cao, J-M. Xu and Y. Wu, Feedback numbers of de Bruijn digraphs, Computers and Mathematics with Application, 59 (2010) 716-723.

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Thank you for your attention!

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