## 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

**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

**Min-Yun Lien** **2/56**

**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)

**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.

**Min-Yun Lien** **3/56**

**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.

**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.

**Min-Yun Lien** **5/56**

**Applications**

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

**Applications**

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

⊲ Vertices are coloredYESorNO.

**Min-Yun Lien** **6/56**

**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.

**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).

**Applications**

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

⊲ Vertices are coloredYESorNO.

⊲ 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**

**Min-Yun Lien** **7/56**

**Applications**

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

⊲ Vertices are coloredYESorNO.

⊲ A decycling set could be the only choice!

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

**Applications**

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

⊲ Vertices are coloredYESorNO.

⊲ A decycling set could be the only choice!

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

**Decycling Set** **Decycling Set**

**Min-Yun Lien** **9/56**

**Applications**

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

⊲ Vertices are coloredYESorNO.

⊲ A decycling set could be the only choice!

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

NOvertices on the cycle remain NO.

**Failed!**

**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).

**Min-Yun Lien** **11/56**

**Complexity**

◮

## H as been extensively studied.

NP-hard

Reduction from VERTEX COVER (R. Karp 1972).

Even for planar graphs, bipartite graphs.

**Related Problems**

◮

## I s related or equivalent to

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

Hitting cycle problem

**Min-Yun Lien** **13/56**

**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).

**Variations**

◮

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

Graph Bipartization Problem

Find*D⊂ E*such that*G\ D*has no odd cycle.

**Min-Yun Lien** **14/56**

**Variations**

◮

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

Graph Bipartization Problem

Find*D⊂ E*such that*G\ D*has no odd cycle.

Weighted Decycling Problem

Seek a minimum-weight decycling set*D*where each vertex
has a weight.

**Variations**

◮

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

Graph Bipartization Problem

Find*D⊂ E*such that*G\ D*has no odd cycle.

Weighted Decycling Problem

Seek a minimum-weight decycling set*D*where each vertex
has a weight.

Loop Cut Set Problem

Given*A(C) ⊆ V(C)*for each cycle*C*of*G, find a minimum*
set*D*such that*D∩ A(C) 6= ∅.*

**Min-Yun Lien** **14/56**

**Relations with Cycle Packing Number**

◮

## I s often compared with the following graph parameter.

Cycle Packing Number*ν(G): the maximum number of*
vertex-disjoint cycles of*G.*

### 1 ) ( *G* Q

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

**Relations with Cycle Packing Number**

⊲ Dirac and Gallai had interest in the relations between*ν(G)*
and*∇(G).*

⊲ It is clear that*ν(G) ≤ ∇(G).*

**Min-Yun Lien** **16/56**

**Relations with Cycle Packing Number**

⊲ Dirac and Gallai had interest in the relations between*ν(G)*
and*∇(G).*

⊲ It is clear that*ν(G) ≤ ∇(G).*

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

**Relations with Cycle Packing Number**

⊲ Dirac and Gallai had interest in the relations between*ν(G)*
and*∇(G).*

⊲ It is clear that*ν(G) ≤ ∇(G).*

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

⊲ ∇(1) ≤ 3;*ν(K*5) = 1and*∇(K*5) = 3(Bollob´*as 1964).*

⊲ ∇(2) = 6and9≤ ∇(3) ≤ 12(Voss 1968).

**Min-Yun Lien** **16/56**

**Relations with Cycle Packing Number**

⊲ Dirac and Gallai had interest in the relations between*ν(G)*
and*∇(G).*

⊲ It is clear that*ν(G) ≤ ∇(G).*

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

⊲ ∇(1) ≤ 3;*ν(K*5) = 1and*∇(K*5) = 3(Bollob´*as 1964).*

⊲ ∇(2) = 6and9≤ ∇(3) ≤ 12(Voss 1968).

⊲ *c*1*k log k≤ ∇(k) ≤ c*2*k log k*for some constants*c*1 and*c*2

(Erd¨os and P´osa 1964).

**Relations with Cycle Packing Number**

Consider planar graphs:

**Min-Yun Lien** **17/56**

**Relations with Cycle Packing Number**

Consider planar graphs:

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

**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).*

**Min-Yun Lien** **17/56**

**Decycling number of outerplanar graphs**

Consider outerplanar graphs:

Theorem (Kloks, Lee and Liu 2002)

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

◮ An outerplanar graph*G*is called*lower-extremal*if*∇(G) = ν(G)*
and*upper-extremal*if*∇(G) = 2ν(G).*

**Decycling number of outerplanar graphs**

Consider outerplanar graphs:

Theorem (Kloks, Lee and Liu 2002)

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

◮ An outerplanar graph*G*is called*lower-extremal*if*∇(G) = ν(G)*
and*upper-extremal*if*∇(G) = 2ν(G).*

◮ Upper-Extremal Results:

We define a*sun graphS*3as follows.

*∇(S*3*) = 2 = 2ν(S*3). *S*

**Min-Yun Lien** **18/56**

**Decycling number of outerplanar graphs**

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

*An outerplanar graphGis upper-extremal if and only ifGis anS*3*-tree.*

A graph is an*S*3*-tree of ordert*if it has exactly*t*

vertex-disjoint*S*3-subdivisions and every edge not on these
*S*3-subdivisions belongs to no cycle.

**Decycling number of outerplanar graphs**

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

*An outerplanar graphGis upper-extremal if and only ifGis anS*3*-tree.*

A graph is an*S*3*-tree of ordert*if it has exactly*t*

vertex-disjoint*S*3-subdivisions and every edge not on these
*S*3-subdivisions belongs to no cycle.

Example:

*An S*3*-tree G of order 3, *
where(*G*) 6 2Q(*G*).

**Min-Yun Lien** **19/56**

**Decycling number of outerplanar graphs**

◮ Lower-Extremal Results:

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

Sun graphs*S**k*with odd number*k:*

*S*_{3} *S*_{5}

*∇(S**k*) = ⌈^{k}_{2}⌉and*ν(S**k*) = ⌊^{k}_{2}⌋.

**Decycling number of outerplanar graphs**

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

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

**Min-Yun Lien** **21/56**

**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} .

**Cartesian Product**

Definition (G✷H)

*V(G✷H) = {(u, v)|u ∈ V(G)*and*v∈ V(H)}*

*E(G✷H) = {(u, v)(u*^{′}*, v*^{′}*)|u = u*^{′}and*(v, v*^{′}*) ∈ E(H)*or*(u, u*^{′}) ∈
*E(G)*and*v= v*^{′}}

**Min-Yun Lien** **23/56**

**Cartesian Product**

Definition (G✷H)

*V(G✷H) = {(u, v)|u ∈ V(G)*and*v∈ V(H)}*

*E(G✷H) = {(u, v)(u*^{′}*, v*^{′}*)|u = u*^{′}and*(v, v*^{′}*) ∈ E(H)*or*(u, u*^{′}) ∈
*E(G)*and*v= v*^{′}}

Figure :*P*_{9}*✷P*11

**Decycling number of** *C*

_{m}*✷C*

*n*

Theorem (Pike, Zou, 2005, SIDMA)

*∇(C**m**✷C**n*) =

⌈^{3n}_{2}⌉ *ifm*= 4,

⌈^{3m}_{2} ⌉ *ifn*= 4,

⌈^{mn+2}_{3} ⌉ *otherwise*.

**Min-Yun Lien** **24/56**

**Decycling number of Path Product** *P*

_{m}*✷P*

*n*

Lower bound

Theorem (Luccio, 1998, IPL)

*Ifm, n ≥ 2, then∇(P**m**✷P** ^{n}*) ≥l

*(m−1)(n−1)+1*3

m
*.*

**Decycling number of Path Product** *P*

_{m}*✷P*

*n*

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

*In Table, A:∇(P**m**✷P*^{n}*) = F**m,n**, B:∇(P**m**✷P*^{n}*) ≤ F**m,n*+ 1, C:

*∇(P**m**✷P**n**) ≤ F**m,n*+ 2, where*F**m,n*=l*(m−1)(n−1)+1*
3

m*.*

**Min-Yun Lien** **26/56**

**Decycling number of Path Product** *P*

_{m}*✷P*

*n*

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

3 *is an integer, then each decycling set*
*Sof sizef**m,n**satisfies the following two properties:*

*(1)Scontains exactly one vertex of degree*3*and contains no vertex*
*of degree 2; and*

*(2)Sinduces a subgraph ofP**m**✷P*^{n}*with no edges.*

**Decycling number of Path Product** *P*

_{m}*✷P*

*n*

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

3 *is an integer, then each decycling set*
*Sof sizef**m,n**satisfies the following two properties:*

*(1)Scontains exactly one vertex of degree*3*and contains no vertex*
*of degree 2; and*

*(2)Sinduces a subgraph ofP**m**✷P*^{n}*with no edges.*

Theorem (Lien, Fu, Shih, 2014, DMAA)
*Ifm*≥ 5,*mnis even andf**m,n**is an integer, then*

*∇(P**m**✷P**n**) ≥ f**m,n**+ 1 = F**m,n*+ 1,where*f** _{m,n}*=

*(m−1)(n−1)+1*

3 *.*

**Min-Yun Lien** **27/56**

**Decycling number of Path Product** *P*

_{m}*✷P*

*n*

**Proof.**

**Decycling number of Path Product** *P*

_{m}*✷P*

*n*

**Proof.**

Suppose not.

**Min-Yun Lien** **28/56**

**Decycling number of Path Product** *P*

_{m}*✷P*

*n*

**Proof.**

Suppose not.

Assume that*∇(P**m**✷P*^{n}*) = f**m,n* *= F**m,n*and*S*is a decycling set
with size*f**m,n*.

**Decycling number of Path Product** *P*

_{m}*✷P*

*n*

**Proof.**

Suppose not.

Assume that*∇(P**m**✷P*^{n}*) = f**m,n* *= F**m,n*and*S*is a decycling set
with size*f**m,n*.

By above Proposition, we may let*v**i,1*be the vertex of*S*with
degree3where2*≤ i ≤*_{m}

2

.

**Min-Yun Lien** **28/56**

**Decycling number of Path Product** *P*

_{m}*✷P*

*n*

**Proof.**

Suppose not.

Assume that*∇(P**m**✷P*^{n}*) = f**m,n* *= F**m,n*and*S*is a decycling set
with size*f**m,n*.

By above Proposition, we may let*v**i,1*be the vertex of*S*with
degree3where2*≤ i ≤*_{m}

2

.

Let*v**i,1**∈ S, 2 ≤ i ≤ ⌊*^{m}_{2}⌋.

**Decycling number of Path Product** *P*

_{m}*✷P*

*n*

**Proof.**

Suppose not.

Assume that*∇(P**m**✷P*^{n}*) = f**m,n* *= F**m,n*and*S*is a decycling set
with size*f**m,n*.

By above Proposition, we may let*v**i,1*be the vertex of*S*with
degree3where2*≤ i ≤*_{m}

2

.

Let*v**i,1**∈ S, 2 ≤ i ≤ ⌊*^{m}_{2}⌋.

*v**m−1,2**∈ S*and*v**m−1,3**∈ S.*/

**Min-Yun Lien** **30/56**

**Decycling number of Path Product** *P*

_{m}*✷P*

*n*

**Proof.**

Suppose not.

Assume that*∇(P**m**✷P*^{n}*) = f**m,n* *= F**m,n*and*S*is a decycling set
with size*f**m,n*.

By above Proposition, we may let*v**i,1*be the vertex of*S*with
degree3where2*≤ i ≤*_{m}

2

.

Let*v**i,1**∈ S, 2 ≤ i ≤ ⌊*^{m}_{2}⌋.

*v**m−1,2**∈ S*and*v**m−1,3**∈ S.*/

**Decycling number of Path Product** *P*

_{m}*✷P*

*n*

**Proof.**

Suppose not.

Assume that*∇(P**m**✷P*^{n}*) = f**m,n* *= F**m,n*and*S*is a decycling set
with size*f**m,n*.

By above Proposition, we may let*v**i,1*be the vertex of*S*with
degree3where2*≤ i ≤*_{m}

2

.

Let*v**i,1**∈ S, 2 ≤ i ≤ ⌊*^{m}_{2}⌋.

*v**m−1,2**∈ S*and*v**m−1,3**∈ S.*/
*v**m−1,2**, v**m−1,4**, · · · , v**m−1,n−1**∈ S.*

**Min-Yun Lien** **32/56**

**Decycling number of Path Product** *P*

_{m}*✷P*

*n*

**Proof.**

Suppose not.

Assume that*∇(P**m**✷P*^{n}*) = f**m,n* *= F**m,n*and*S*is a decycling set
with size*f**m,n*.

By above Proposition, we may let*v**i,1*be the vertex of*S*with
degree3where2*≤ i ≤*_{m}

2

.

Let*v**i,1**∈ S, 2 ≤ i ≤ ⌊*^{m}_{2}⌋.

*v**m−1,2**∈ S*and*v**m−1,3**∈ S.*/
*v**m−1,2**, v**m−1,4**, · · · , v**m−1,n−1**∈ S.*

Hence,*n*− 1is even and*n*is odd.

**Decycling number of Path Product** *P*

_{m}*✷P*

*n*

**Proof.**

Suppose not.

Assume that*∇(P**m**✷P*^{n}*) = f**m,n* *= F**m,n*and*S*is a decycling set
with size*f**m,n*.

By above Proposition, we may let*v**i,1*be the vertex of*S*with
degree3where2*≤ i ≤*_{m}

2

.

Similarly,

*v**m−3,n−1**, v**m−5,n−1**, · · · , v*2,n−1*∈ S.*

Thus,*m*is odd, a contradiction.

**Min-Yun Lien** **33/56**

**Decycling number of Path Product** *P*

_{m}*✷P*

*n*

New Lower bound Corollary

*Form*≥ 5, if*m*≡ 0*(mod*6) and*n*≡ 2 (mod3)*or*
*(m, n) ≡ (3, 2) (mod*6), then*∇(P**m**✷P*^{n}*) ≥ F**m,n*+ 1.

**Decycling number of Path Product** *P*

_{m}*✷P*

*n*

New Lower bound Corollary

*Form*≥ 5, if*m*≡ 0*(mod*6) and*n*≡ 2 (mod3)*or*
*(m, n) ≡ (3, 2) (mod*6), then*∇(P**m**✷P*^{n}*) ≥ F**m,n*+ 1.

Theorem

*Form*≥ 5, if*(m, n) ≡ (0, 2), (0, 5), (3, 2), (2, 0), (5, 0), (2, 3) (mod*6),
*then∇(P**m**✷P*^{n}*) = F**m,n*+ 1.

**Min-Yun Lien** **34/56**

**Decycling number of Path Product** *P*

_{m}*✷P*

*n*

New Lower bound Corollary

*Form*≥ 5, if*m*≡ 0*(mod*6) and*n*≡ 2 (mod3)*or*
*(m, n) ≡ (3, 2) (mod*6), then*∇(P**m**✷P*^{n}*) ≥ F**m,n*+ 1.

Theorem

*Form*≥ 5, if*(m, n) ≡ (0, 2), (0, 5), (3, 2), (2, 0), (5, 0), (2, 3) (mod*6),
*then∇(P**m**✷P*^{n}*) = F**m,n*+ 1.

Theorem (Lien, Fu, Shih, 2014, DMAA)
*Form, n ≥ 6,∇(P**m**✷P*^{n}*) ≤ F**m,n*+ 1.

**Decycling number of Digraphs**

5

3 1 0

4 2

**Min-Yun Lien** **35/56**

**Decycling number of Digraphs**

Notation:

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

For*S⊆ V(G),N*^{+}*(S) = ∪**v∈S**N*^{+}*(v).*

**Decycling number of Digraphs**

Notation:

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

For*S⊆ V(G),N*^{+}*(S) = ∪**v∈S**N*^{+}*(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= S*0*, S*1*, · · · , S**t**= V(G)such that*
*(1)S**i**⊆ S**i+1**; and*

*(2)N*^{+}*(S**i+1**\ S**i**) ⊆ S**i**fori= 0, 1, · · · , t − 1.*

**Min-Yun Lien** **36/56**

**Decycling number of Digraphs**

Example 1.

**Decycling number of Digraphs**

Example 1.

5

3 1 0

4 2

**Min-Yun Lien** **37/56**

**Decycling number of Digraphs**

Example 1.

5

3 1 0

4

2 0

**Decycling number of Digraphs**

Example 1.

5

3 1 0

4

2 0 2

**Min-Yun Lien** **39/56**

**Decycling number of Digraphs**

Example 1.

5

3 1

4 2

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

**Decycling number of Digraphs**

Example 2.

5

3 1 0

4 2

**Min-Yun Lien** **41/56**

**Decycling number of Digraphs**

Example 2.

5

3 1 0

4

2 0

5

*S*

1
**Decycling number of Digraphs**

Example 2.

5

3 1 0

4

2 0 2

5 3

*S*

1
**Min-Yun Lien** **43/56**

**Decycling number of Digraphs**

Example 2.

5

3 1 0

4

2 0 2

5 3

*S*

1
*S*

2
**Decycling number of Digraphs**

Example 2.

5

3 1 0

4

2 0 2

5 3

1 4

*S*

1
*S*

2
**Min-Yun Lien** **45/56**

**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 have*S*3*= V(G).*

**Decycling number of Digraphs**

Example 2.

3 1

4 2

**Min-Yun Lien** **47/56**

**de Bruijn Digraphs and Kautz Digraphs**

Definition (de Bruijn digraph*B(d, n))*

*V(B(d, n)) = {x*1*x*2*· · · x**n**: x**i**∈ {0, 1, · · · , d − 1}, 1 ≤ i ≤ n}.*

*Edge: X = x*1*x*2*· · · x**n**−→ Y = x*2*x*3*· · · x**n*α*whereα ∈ {0, 1, · · · , d − 1}.*

**de Bruijn Digraphs and Kautz Digraphs**

Definition (de Bruijn digraph*B(d, n))*

*V(B(d, n)) = {x*1*x*2*· · · x**n**: x**i**∈ {0, 1, · · · , d − 1}, 1 ≤ i ≤ n}.*

*Edge: X = x*1*x*2*· · · x**n**−→ Y = x*2*x*3*· · · x**n*α*whereα ∈ {0, 1, · · · , d − 1}.*

Definition (Kautz digraph*K(d, n))*

*V(K(d, n)) = {x*1*x*2*· · · x**n* *: x**i**∈ {0, 1, · · · , d}, 1 ≤ i ≤ nandx**i**6= x**i+1*, 1 ≤
*i≤ n − 1}.*

*Edge: X = x*1*x*2*· · · x**n**−→ Y = x*2*x*3*· · · x**n*α*whereα ∈ {0, 1, · · · , d}.*

Remark:*K(d, n) ⊆ B(d + 1, n).*

**Min-Yun Lien** **48/56**

**Generalized de Bruijn Digraphs Generalized Kautz** **Digraphs**

Definition (Generalized de Bruijn digraph*G**B**(d, n))*
*V(G**B**(d, n)) = {0, 1, · · · , n − 1}.*

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

**Generalized de Bruijn Digraphs Generalized Kautz** **Digraphs**

Definition (Generalized de Bruijn digraph*G**B**(d, n))*
*V(G**B**(d, n)) = {0, 1, · · · , n − 1}.*

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

Definition (Generalized Kautz digraph*G**K**(d, n))*
*V(G**K**(d, n)) = {0, 1, · · · , n − 1}.*

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

**Min-Yun Lien** **49/56**

**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

### +

^{9}

_{4}

*t* + 6

*where*

*n* *≡ t (mod 36),* *for* *d* = 3, (

^{1}

_{2}

### −

^{d−1}*2*

_{2d}*)n +*

^{d}_{2}

*(d − t + 5) − 2* ,

*where*

*n* *≡ t (mod d + 1),* *for* *d* ≥ 4.

**Decycling number of Generalized de Bruijn Digraphs**

### Theorem (Lien, Kuo and Fu)

*∇(G*

*B*

*(d,* *n)) ≤*

^{d+1}

_{2d}*n* *+ 2(d − 1).*

**Min-Yun Lien** **51/56**

**Objective Work**

For a planar graph*G,∇(G)≤ 2ν(G)*^{?} (Jones’ conjecture 2002).

For a planar graph*G,∇(G)≤ |V(G)|/2*^{?} (Albertson and Berman
1979).

For a bipartite planar graph*G,∇(G)≤ 3|V(G)|/8*^{?} (Albertson and
Berman 1979).

We have⌈*(m−1)(n−1)+1*

3 *⌉ ≤ ∇(P**m**✷P** ^{n}*) ≤ ⌈

*(m−1)(n−1)+1*

3 ⌉ + 1.Find
the exact value of*∇(P**m**✷P**n*).

Find the lower bound of directed graphs.

**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.

**Min-Yun Lien** **53/56**

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 for*k-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
*P**m**✷P** ^{n}*, 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,

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.

**Min-Yun Lien** **55/56**