半電力控制集的研究

國立交通大學

應用數學系

碩士論文

半電力控制集的研究

A Study of Semi-power Dominating Sets

研究生：吳政軒

指導教授：傅恆霖 教授

半電力控制集的研究

A Study of Semi-power Dominating Sets

研 究 生：吳政軒 Student: Cheng-Hsun Wu 指導教授：傅恆霖 Advisor: Hung-Lin Fu

國立交通大學

應用數學系

碩士論文

A Thesis

Submitted to Department of Applied Mathematics College of Science

National Chiao Tung University In Partial Fulfillment of the Requirements

For the Degree of Master

In

Applied Mathematics

June 2010

Hsinchu, Taiwan, Republic of China

I

半電力控制集的研究

研究生：吳政軒 指導教授：傅恆霖 國立交通大學 應用數學系

摘要

用圖的模型來研究一個半電力控制集問題是在圖G上一些點放著測試器，依 據我們訂下的規則，若能讓圖G的所有邊都被觀察到，則我們稱這些點所成的集 合為圖G的半電力控制集合。在這篇論文中，我們先說明了電力控制集問題與半 電力控制集問題的關聯性。接著，我們證明了一些特殊圖的半電力控制集的最少 點數，也證明當圖G為連通圖，且G中每個點所連到的邊數至少為兩邊時，半 電力控制集的最少點數與回饋點集的最少點數相等。我們也提出一遞迴方法，來 建構P P_n 、_n C_nC_n的半電力控制集；於是，提供了一個半電力控制集最少點數 的上界。最後我們證出P P_n 的半電力控制集最少點數為_n F 或_n F_n ，其中1 2 2 2 3 n n n F _{ }   _  ，這結果改善了原來文獻中的最佳結論。

II

A Study of Semi-power Dominating Sets

Student: Cheng-Hsun Wu Advisor: Hung-Lin Fu

Department of Applied Mathematics Department of Applied Mathematics National Chiao Tung University National Chiao Tung University

Hsinchu, Taiwan 30050 Hsinchu, Taiwan 30050

Abstract

In a semi-power domination set (SPDS) system, we place measurement units on some vertices of a graph G, and according to the rule we defined, if all the edges of G can be observed, then we say that the vertex set is a semi-power domination set. In this thesis, we first find the relationship between PDS and SPDS, and then we prove that the minimum size of SPDS of a graph G, denoted by _sp( ),G is equal to the

minimum size of the feedback vertex set of ,G provided G is connected and

( )G 2.

  In addition, we bring up a recursive idea to produce the SPDS of a graph G.

Finally, with the recursive idea, we prove that _sp(P P_n _n) is equal to either F or _n 1, n F  where 2 2 2 3 n n n F _{ }   _

III

謝 誌

研究所經歷了四年，首先，我要感謝我的指導老師傅恆霖教授。在這四年當 中給我許多的包容及鼓勵，讓我一方面能就讀研究所，一方面能在教師之路(實 習、教檢、教甄)上走的順遂。在撰寫論文的期間，傅老師給我很多的精闢意見 及論文寫作的方法，讓我能在忙碌的教職中也能完成我的論文，順利畢業，誠摯 地感謝傅老師的悉心指導。 接著，我要感謝我的口試委員陳伯亮老師與史青林老師，撥空來為我的碩士 學位做口試，也細心、耐心地給我許多論文寫作上的建議，讓我的論文能更加完 善。感謝陳秋媛老師、黃大原老師、以及翁志文老師三位老師在課業上的指導， 讓我在唸研究所期間學到更多關於組合數學的知識。 另外，我要感謝我的任職學校雲林縣立斗南高中的同事們，陳嘉辰校長、黃 義政主任、龍文娟主任、曹富發組長、霍仲美組長、素貞老師、怡惠老師等，在 生活上及工作上的幫忙，讓我有機會能一邊任教一邊進修，且在焦頭爛耳的忙碌 生活中，還能開心的過生活，衷心的感謝妳們。 我要感謝黃明輝、羅元勳、陳宏賓、詹棨丰、張惠蘭、連敏筠等學長姐、呂 老師及施智懷，在我遇到問題的時候，能提供我一些意見，讓我能更順利的完成 論文研究，也給予我許多論文口試上的建議。感謝研究所的各位同學，奇聰、偉 帆、志文、佩純、皜文、鈺傑、若宇、雅榕等。感謝你們，讓我的研究所生活過 的這麼愉快，謝謝。我還要感謝我的好友們小不點、香菇、正隆、將兔、慧玲、 月琪、詠嘉等，以及我實習時的好夥伴元順、詩亭、巧靜、明儒、玫伶、玟靜， 在我煩惱不已不知所措時，聆聽我的苦水並給我一些實質上的幫忙，還好有你 們，我才能度過壓力最大的日子，謝謝你們。 最後，我要感謝我的家人，一直支持著我的理想及夢想，無怨無悔的在生活 上對我的協助，包容、體諒著我，也終於可以跟媽媽說我畢業了。在此謝謝媽媽 常帶我求神保佑我順利畢業；謝謝姊姊從四川成都遠方的關心；謝謝哥哥陪我玩 wii抒發壓力以及謝謝已過世的奶奶，常以我為榮。謝謝你們的支持讓我順利完 成學業，僅以此論文獻給我最愛的家人。

IV

Contents

摘要... I

Abstract ... II

1. Introduction and Preliminaries ... 1

1.1. Graph Notions ... 1

1.2. Power-dominating Sets ... 4

1.3. Semi-power Dominating Sets ... 5

1.4. Feedback Vertex Sets ... 7

2. Known Results ... 8

2.1. On Power-dominating Sets... 8

2.2. On Feedback Vertex Sets ... 9

3. Main Results ... 12

4. Concluding Remark ... 19

1

1. Introduction and Preliminaries

Domination problems are the most useful ones in real world which are related to graph models. Find an efficient way to build hospitals, power plants and control centers is a typical example. One to the applications with specific needs, the problem has many variations, e.g. k-domination, independent domination, total domination, power domination, etc. This thesis provides a different version, called semi-power domination which has a suitable graph model to fit. We shall explain it in section 1.3. First, some graph notions are necessary.

1.1. Graph Notions

In this section, we first introduce the terminologies and definitions of graphs. For details, the readers may refer to the book "Introduction to Graph Theory" by D. B. West. [18]

A graph G is a triple consisting of a vertex set V G an edge set ( ),( ), E G and a relation that associates with each edge. Two vertices of an edge are called its

endpoints. A loop is an edge whose endpoints are equal. Multi-edges are edges having

the same pair of endpoints. A simple graph is a graph without loops or multi-edges. In this thesis, all the graphs we consider are simple. The size of the vertex set V G ( ),

( )

V G , is called the order of G, and the size of the edge set E G ( ), E G( ) , is called the size of G. The neighborhood of v written N v is the set of vertices adjacent to _G( ),

v. The degree of vertex v in a graph G, deg(v) is the number of edges incident to v, i.e.

deg( )v  N vG( ) . The maximum degree among all vertices of a graph G is denoted

2

( )G ( ).G

  A graph G is k-regular if the common degree is k. An isolated vertex

is a vertex of degree 0.

A path is a graph whose vertices can be ordered so that two vertices are adjacent if and only if they are consecutive in the list. A path with n vertices is denoted by P _n. In the case that path P starts at u and ends at v, we call P a (u, v)-path. A graph G is

connected if it has a (u, v)-path whenever ,u v V G ( )(otherwise, G is disconnected). The components of a graph G are its maximal connected subgraphs. A component (or graph) is trivial if it has no edges; otherwise it is nontrivial.

A subgraph of a graph G is a graph H such that (V H)V G( ) and

( ) ( )

E H E G and the assignment of endpoints to edges in H is the same as in G. A

spanning subgraph of G is a subgraph H with ( )V H V G( ).

A cycle is a graph with an equal number of vertices and edges whose vertices can be placed around a circle so that two vertices are adjacent if and only if they appear consecutively along the circle. A cycle with n vertices is denoted by C A graph _n. with no cycle is acyclic. A tree is a connected acyclic graph. A spanning tree is a spanning subgraph that is a tree.

A complete graph is a simple graph whose vertices are pair-wise adjacent; the complete graph with n vertices is denoted by K A graph G is bipartite if ( )_n. V G is the union of two disjoint independent sets called partite sets of G. A graph G is

m-partite if ( )V G can be expressed as the union of m independent sets. A complete bipartite graph is a bipartite graph such that two vertices are adjacent if and only if

they are in different partite sets. When the sets have the sizes s and t, the complete bipartite graph is denoted by K If the sets have the same size n, the complete _{s t,}.

3

bipartite graph is called balanced, which is denoted by K_{n n,} . Similarly, the complete

m-partite graph is denoted by

1,2,...,m s s s

K .

A star S is the complete bipartite graph _k K , i.e., a tree with one internal _1,k node and k leaves. A star with 3 edges is called a claw. Let T be the tree formed from a star by subdividing any number of its edges any number of times; that is, T has at most one vertex of degree 3 or more. We call such a tree T a spider. A path, for example, is a special case of a spider.

The corona of two graphs G and H, denoted G ◦ H, is the graph formed from one copy of G and V G( ) copies of H where the ith vertex of G is adjacent to every

vertex in the ith copy of H.

The diamond is the graph D obtained from the complete graph K by deleting ₄

one edge. For each positive integer k, let D be the connected claw-free cubic graph _k formed from k disjoint copies of D by joining pair-wise 2k vertices of degree two. Note that D is just ₁ K . ₄

Let G be a graph of order m with V G( ) { : 0g_i    }, and let H be a i m 1 graph of order n with (V H) { : 0h_i    }. The Cartesian product i n 1 G H is defined to be the graph with vertex set { ( ,g h_{i j}) : 0  i m 1 and 0   } and j n 1 ( ,g h g h_{i j})( _s, )_t E G H(  ) if either g_i g_s and h h_{j t}E H( ) or h_j h_t and g g_{i s}E G( ).

4

1.2. Power-dominating Sets

Electric power companies monitor the state of their electric power system by placing phasor measurement units (PMUs) in the system. Because of the high cost of a PMU, we want to minimize the number of PMUs to monitor (observe) the entire system. A system is said to be observed if all of the state variables of the system can be determined from a set of measurements.

Let ( , G V E) be a graph representing an electric power system, where a vertex represents an electrical node and an edge represents a transmission line joining two electrical nodes. The problem of locating a smallest set of PMUs to monitor the entire system is a graph model problem closely related to the well-known vertex covering and domination problems. For a thorough study of domination, related subset problems and terminology, the readers may refer to two books [11, 12].

A PMU measures the state variable for the vertex at which it is placed and its incident edges and their endvertices. (These vertices and edges are said to be observed.) The other observation rules are as follows:

1. Any vertex that is incident to an observed edge is observed. 2. Any edge joining two observed vertices is observed.

3. If a vertex is incident to a total of k 1 edges and if k1 of these edges are observed, then all k of these edges are observed.

For a given vertex set P of representing the nodes where the PMUs are placed, to solve the power system monitoring problem we want to minimize |P|. This monitoring problem was introduced and studied in [1, 2, 3 and 17]. We define a set SV(G) to be a power dominating set (PDS) in a graph G( , )V E if every vertex and every edge

5

in G is observed by S. The cardinality of a minimum power dominating set of G is the

power domination number _p( ).G A power dominating set of G with minimum

cardinality is called a _p( )G -set. In [4, 13], it was proved that to obtain power domination set is NP-complete for planar bipartite graphs, bipartite graphs and chordal graphs, respectively.

In the following section, we will introduce the semi-power domination set problem and some observations.

1.3. Semi-power Dominating Sets

In this thesis, we try to examine an electric power system including edges and vertices in graph model. Then we place some weak measurement units (WMUs) on vertices, and we suppose that all the edges connected to the vertices that has place the WMU can be tested. Furthermore, if there are n1 edges to be tested in n edges connected to a vertex, then all of them must be tested. For economic reason, we minimize the number of WMUs.

Weak measurement units (WMUs) measure the state variable for the vertex at which it is placed and its incident edges and their endpoints. (These vertices and edges are said to be observed.) The other observation rules are as follows:

1. Any vertex that is incident to an observed edge is observed.

2. If a vertex is incident to a total of k 1 edges and if k1 of these edges are observed, then all k of these edges are observed.

6

representing the nodes where the WMUs are placed, to solve the semi-power system monitoring problem we would try to minimize |P|.

A set SV (G) is a semi-power dominating set (SPDS) in a graph G( , )V E if every vertex and every edge in G is observed by S following the rules defined above. The cardinality of a minimum semi-power dominating set of G is the semi-power

domination number _sp( ).G A semi-power dominating set of G with minimum

cardinality is called a _sp( )G -set.

We have the following two observations.

Observation 1. For each graph G ( ), _sp G _p( ) 1G  .

Observation 2. G is a graph and H is a subgraph of G then , _sp(H) may be larger than _sp( ).G

Example：In Figure 1, H is a subgraph of G and, _sp( )G 3_sp(H) . 4

:

G H:

Figure 1:

While studying SPDS problem, in some conditions, we found semi-power domination and feedback vertex sets are quite the same. In the following section, we will introduce feedback vertex sets.

7

1.4. Feedback Vertex Sets

A feedback vertex set (FVS) of a connected graph G( , )V E is a subset V'

of V(G) such that the graph G' induced by V \V' is a forest. The cardinality of a minimum feedback vertex set (MFVS) in G is the feedback vertex number ( ). G A feedback vertex set of G with minimum cardinality is called a ( ) G -set.

The problem of finding a minimum feedback vertex set in a graph is one of the classic NP-complete problems [14] and is NP-hard for general graphs [7]. We refer to [10] for a rather complete and recent survey on the feedback vertex set problem.

8

2. Known Results

Some known results on PDS will be introduced as following.

2.1. On Power-dominating Sets

T. W. Haynes, S. M. Hedetniemi, S. T. Hedetniemi and M. A. Henning had mentioned the following result in [13].

Theorem 2.1.1. [13] For any tree T , _p( )T = 1 if and only if T is a spider.

Theorem 2.1.2. [13] If T is a tree having k vertices of degree at least 3, then 2 ( ) 3 p k T

   and this bound is sharp.

Theorem 2.1.3. [13] For any tree T of order n ≥ 3, ( ) 3

p

n T

  with equality if and

only if T is the corona T ◦K , where T is any tree. ₂

M. Dorfling and M. A. Henning had mentioned the following results in [8] for the graphP P_n . _m

Theorem 2.1.4. [8] If G is an n × m grid graph P P_n _m where m n 1, then

1 4 (mod 8); 4 ( ) . 4 p n if n and G n otherwise     _               

M. Zhao, L. Kang and G. J. Chang had mentioned the following results in [19]. Let F be the family of graphs obtained from connected graphs H by adding two new vertices v and v to each vertex v of H and new edges vv and vv, while v v  may be added or not.

9

Theorem 2.1.5. [19] If G( , )V E is a connected graph of order n 3, then ( )

3

p

n G

  with equality if and only if G F {K_3,3}.

Corollary 2.1.6. [19] If each component G_i of a graph G of order n contains at least

three vertices, then ( ) 3

p

n G

  with equality if and only if each component

3,3

{ }.

i

G  F K

Theorem 2.1.7. [19]If G( , )V E is a connected claw-free cubic graph of order n,

then ( ) 4

p

n G

  with equality if and only if G A where , A{D k_k| 1}.

C.C. Chuang had mentioned the following results in [6].

Theorem 2.1.8. [6] G K _nP_m, _p( )G 1. Theorem 2.1.9. [6] , ( ) 1, 1 ( , ) (2,3). 2, . n m p n or n m G K C G otherwise       _{ }  Theorem 2.1.10. [6] G K _nK_m, where 2 n m, _p( )G  n 1. Theorem 2.1.11. [6] G C _nC_m, 3 n m then, 1, 2 (mod 4). 2 ( ) , . 2 p n if n G n otherwise                       

While studying SPDS problem, we will use some results on feedback vertex sets and it will be introduced in the following section.

2.2. On Feedback Vertex Sets

In [15], Luccio proved upper and lower bounds on the sizes of minimal feedback vertex sets in grids. Subsequently, both, Caragiannis, Kaklamanis, Kanellopoulos in [5] and Madelaine, Stewart in [16] improved the upper bounds, respectively.

10 Theorem 2.2.1. [15] For all ,n m N ,

( 1)( 1) 1 ( ) ( , ) . 3 n m 3 6 m n mn m n P P o m n       _{  } _{ }         

Theorem 2.2.2. [5] For all ,n m N , ( ) 5 .

3 6 n m mn m n P P   _    _  

Lemma 2.2.3. [16] For n m N,  , let a be the upper bound of (_{n m,}  P P_n _m),

(i) if n=3k+1, m=2r, k1, 2r , then a_{n m,} F_{n m,} ;

(ii) if n=3k+1, m=2r+1, k1, 3,r then a_{n m,} F_{n m,} ;

(iii) if n=3k, m=3r or 3r+2, 3, 2k r and r is even, then an m, Fn m,  ; 1

(vi) if n=3k, m=3r, k 3, 3r and r is odd, then a_{n m,} F_{n m,}  ; 1

(v) if n=3k, m=3r+2, k3, 3r and r is odd, then a_{n m,} F_{n m,}  ; 2

(vi) if n=3k+2, m=3r or 3r+2, 3, 2k r and r is even, then a_{n m,} F_{n m,}  ; 1

(vii) if n=3k+2, m=3r, k3, 3r and r is odd, then a_{n m,} F_{n m,}  ; 2

(viii) if n=3k+2, m=3r+2, 3, 2k r and r is odd, then an m, Fn m,  . 2

Theorem 2.2.4. [16] If ( , ) {( , ) | {2, 3,5} n m  i j i or j or i j{ , }{6,8}}, then ( P P_n _m)=F , _{n m,} F_{n m,}  or 1 F_{n m,}  , where 2 _, ( 1)( 1) 1 3 n m m n F _{ }    _  . Lemma 2.2.5. [16] If n N , n then 2, 2 1 ( ) 2 n n P P   _{ }  _  .

Lemma 2.2.6. [16] For each rN, r 3,

(i) 3 2 1 3( 1) ( ) . 2 r r P P   _{  }  _   (ii) 3( 1) ( _{3 2} ) 3( 1) 1. 2 r 2 r r P P     _{  } _        

Lemma 2.2.7. [16] For all p and the grid 0 P P₅ _m with m we have 2,

5 8 5 8 1 5 8 2 5 8 3 5 8 4 5 8 5 5 8 6 5 8 7 ( ) 11 1; ( ) 11 ; ( ) 11 2; ( ) 11 3; ( ) 11 5; ( ) 11 6; ( ) 11 8; ( ) 11 9. p p p p p p p p P P p P P p P P p P P p P P p P P p P P p P P p                                      

11

Consequently, by Theorem 2.2.4, Lemma 2.2.5, Lemma 2.2.6 and Lemma 2.2.7 we have the following theorem.

Theorem 2.2.8. [16] There exists a computable function ( , )f n m such that (P P_{n m})

  is equal to one of f n m( , ), f n m( , ) 1 or ( , ) 2,f n m  where



n m,









12

3. Main Results

First, we prove the relationship between SPDS and FVS.

Lemma 3.1. If P is a _sp( )G set, then G P\ has no cycles, i.e., P is a

feedback vertex set of G. Thus, _sp( )G ( ).G

Proof: Suppose not. Then G P\ has a cycle and each edge on the cycle is not

observed. Hence, P is not a _sp( )G set, a contradiction. ■

Lemma 3.2. Let G be a connected graph with ( ) 2 G  . Then S is a semi-power

dominating set provided that S is a ( ) G set. Thus, _sp( )G ( ).G

Proof: Let S be a ( )G set. Then, G S\ has no cycles and thus, every component in G S\ is a tree. Let them be T T₁, , ₂ ..., .T Moreover, let the _k maximum height of the above k trees be h. We claim that all vertices and edges can be observed after h rounds. In the first round, let V₁{ |v v V G S ( \ ) and deg_{G S\} ( )v

1}.

 Then   , v is adjacent to a vertex of v V₁ S. Clearly, v is observed thus uv is also observed. Now, consider v'V G S V( \ (  ₁)) with degree 1 in G S V\ (  ₁). Since v' is observed, u v' ' is also observed, where u' is a parent of v'. We continue this step for h times. All of the vertices and edges of G are observed, hence

S is also a semi-power dominating set.

■

Theorem 3.3. If G is connected with ( ) 2 G  , then ( ) G _sp( )G .

Proof: By Lemma 3.1 and Lemma 3.2 we have the proof. ■

Now, we prove the number _sp( )G for some special graphs G.

Theorem 3.4. For n 2, _sp(P_n) . 1

Proof: Let V P( _n){ , ,...,v v_{0 1} v_n1} and E P( _n){v v_{i i1}| 0  i n 2}. It is clear that

( ) 1.

sp Pn

13 Theorem 3.5. For n ( ) 13, _sp C_n  .

Proof: Let V C( _n){ , ,...,v v_{0 1} v_n1} and (E C_n){v v_{i j}| j i 1(mod ), 0n   i n 1}. It is clear that _sp(C_n) 1. Hence, the proof follows by letting S { }.v₀ ■

Theorem 3.6. If T is a spider, then _sp( )T  . 1

Proof: Since T is a spider, T has at most one vertex v with deg( )v 3. If T has no vertex v with deg( )v  then T is a path. By Theorem 3.4, we have the 3, proof. If T has exactly one vertex v with deg( )v  then the proof follows by 3, letting { }.S  v ■

Theorem 3.7. For n3, (_sp K_n) n 2.

Proof: Suppose the size of a _sp(K_n)set is less than n2. Then \ _sp( _n)

G  K set contains a K which has a cycle. This is a contradiction. Hence ₃

the number _sp(K_n) is at least n2. On the other hand, it is clear that

( ) 2.

sp Kn n

   Hence, we have the proof. ■

Theorem 3.8. _sp(K_{n m,} ) n 1, where 2 n m.

Proof: Let V K( _{n m,} ) V₁ V₂ and E K( _{n m,} ){x y_{i j} | 0  i n 1, 0  j m 1}, where V₁ { , ,...,x x_{0 1} x_n1} and V₂ { ,y y_{0 1},...,y_m1}. Since two vertices of V and ₁ two vertices of V will induce a cycle, ₂ _sp(K_{n m,} )min m.{ 2, n2}  Hence, n 2. the proof follows by letting S V 1\ { }.x0 ■ Theorem 3.9. _sp(K_nK_n)(n1) , 2 where n2.

Proof: Let V K( _nK_n){v_{i j,} | 0i j n,  1} and E K( _nK_n){v v_{i j k l, ,} |i k or

, 0 , , , 1 }.

j l i j k l n  First, we prove the lower bound of _sp(K_nK_n). Since

( ) 2

sp Kn n

14

two vertices in each row and each column which has no PMUs. W.L.O.G., let

,

{ _{i j}| , 1, ..., 3(mod ), 0 1}

S  v j i i  m i  m   i n with |S| n n( 2). Then \

n n

K K S has a cycle, S is not an SPDS. _sp(K_nK_n)n n



 2



1. Hence, the proof follows by letting S' { v_{i j,} | j i i , 1,  ..., m i 3(mod ), m 0    i n 1}

2, 1

{v_{n n}} with |S| n n(   2) 1. ■

Theorem 3.10. _sp(K_nK_m)n m( 2), where 2 n m.

Proof: Let V K( nKm){vi j, | 0  i n 1, 0   andj m 1} E K( nKm){v vi j k l, , |

, 0 , 1 and 0 , 1}

i k or j l  i k n   j l m  . First, we find the lower bound of

( ).

sp Kn Km

  Since _sp(K_m) m 2, _sp(K_nK_m)n m



2 .



Hence, the proof follows by letting S {v_{i j,} | j i i , 1,  ..., m i 3(mod ), m 0  i n 1} with

|S| n m( 2). ■

Theorem 3.11. _sp(K_nP_m)m n( 2), where n m, 3.

Proof: Let V K( _nP_m){v_{i j,} | 0  i n 1, 0   and j m 1} E K( _nP_m){v v_{i j k j, ,} | 0i k n,  1 and 0  j m 1}{v v_{i k i k, , 1}| 0  i n 1 and 0   . First, we k n 2} find the lower bound of _sp(K_nP_m). Since _sp(K_n)  , (n 2 _sp K_nP_m)



2 .



n m Hence, the proof follows by letting S {v_{i j,} |i j j, 1,  ..., n j  3 (mod ), n 0   with | |j m 1} S m n( 2). ■

Theorem 3.12. _sp(K_nC_m)m n( 2), where 4 n m.

Proof: Let V K( _nC_m){v_{i j,} | 0  i n 1, 0   and j m 1} E K( _nC_m){v v_{i j k j, ,} |

, ,

0i k n,  1 and 0   j m 1} {v v_{i k i l}|l k 1 (mod ), 0n   i n 1 and 0 k m 1}.

 First, we prove the lower bound of _sp(K_nC_m). Since _sp(K_n)  , n 2 (

sp Kn

 C_m) m n



2



. Now, we give an upper bound of _sp(K_nC_m). Case (a). m0 (mod )n

15

Case (b). m1, 2 (mod )n

Let S {v_{i j,} |i j j, 1,  ..., n j 3(mod n), 0   j m 2} {v_{i m, 1}| 2  i n 1}. Case (c). m0, 1, 2 (mod )n

Let S {v_{i j,} |i j j, 1, ..., n j 3(mod n), 0  j m 1}. Then \K_nC_m S has

no cycles. Thus, _sp(K_nP_m) is at most m n



2



. Consequently, _sp(K_nC_m)

( 2), where 4 .

m n  n m ■

Theorem 3.13. _sp(K_nC_m)m n( 2), where 3 m n.

Proof: Let V K( nCm){vi j, | 0  i n 1, 0   and j m 1} E K( nCm){v vi j k j, , | , ,

0i k n,  1 and 0   j m 1} {v v_{i k i l}|l k 1(mod ), 0n   i n 1 and 0 k m 1}.

 First, we find the lower bound of _sp(K_nC_m). Since _sp(K_n)  n 2, _sp(K_n





) 2 .

m

C m n

   Hence, the proof follows by letting S {v_{i j,} |i j j, 1, ...,n j

3(mod ), 0n j m 1}

    with | |S m n( 2). ■

Now, we use the relation between SPDS and FVS to improve the result of FVS on P_m . P_n

Lemma 3.14. LetG be a connected graph with ( ) 2 G  and e xy be an

arbitrary edge of G. Let G G e xz zy    where z V G ( ). Then ( )G ( ).G

Proof: Let S be a feedback vertex set of G with S ( ).G Then G S\ is a forest. So, it follows that G S \ is also a forest, i.e., S is also a feedback vertex set of G. Hence, ( )G ( ).G On the other direction, let S be a feedback vertex set of G with S ( ).G First, if S V G( ), then G S\ is a forest and thus S is also a feedback vertex set of G. This implies that ( )G  S ( ).G On the other hand, z S. Now, let S    Clearly, S is also a feedback vertex set of S z x .

16 

G of size S Since S is a feedback vertex set of , . G the proof follows by above argument. Therefore, ( )G ( ).G ■

Let n 2. P P_n is the graph with vertex set (_n V P P_n _n)defined as {v_{i j,} : 0 , 1}

i j n

   and edge set (E P P_n _n) defined as {(v_{i j,} , v_i1,j) : 0  i n 2, 0  j

, , 1

1} {( _{i j}, _{i j} ) : 0 1, 0 2}.

n  v v _   i n   j n

Lemma 3.15. _sp(P P₂ ₂) 1,  _sp(P P₃ ₃)2, _sp(P P₄ ₄) 4.

Proof: (i) Since P P₂ ₂ C₄, _sp(P P₂ ₂)_sp(C₄) 1.

(ii) Since  v V P P( ₃ ₃), P P₃ ₃\ { }v always has a cycle, and thus

3 3

( ) 2.

sp P P

   Let S {v0,0,v1,1}. Then P P S3 3\ has no cycles, S is

an SPDS. Hence, _sp(P P₃ ₃) 2. (iii) Let V P P( ₄ ₄) S₁ S₂  , S₃ S₄

1 { 0,0, , , }, {0,1 1,0 1,1 2 0,2, , , }, {0,3 1,2 1,3 3 2,0, , , },2,1 3,0 3,1

S  v v v v S  v v v v S  v v v v

4 { 2,2, , , }.2,3 3,2 3,3

S  v v v v Then each S induces a subgraph of G which _i

has a cycle. Hence, _sp(P P₃ ₃) . Let 4 S {v_1,1, , , }.v_1,3 v_2,0 v_2,2 Then

4 4 \

P P S has no cycles, S is an SPDS. Hence, _sp(P P₄ ₄) 4. ■

Lemma 3.16. For , 1,k r _sp(P_2k1P_2r1)kr_sp(P_k1P_r1).

Proof: Let X_{2k1,2r1} {v_{i j,} : , are odd, 1i j  i 2 , 1k  j 2 }.r We have the result

2 1 2 1 2 1,2 1 1 1

( \ ) ( )

sp Pk Pr X k r sp Pk Pr

 _  _{  }  _  _ by using Theorem 3.3 and Lemma 3.14.

17

Lemma 3.17.For , 1k r , _sp(P_2k2P_2r2)(k1)(r 1) _sp(P P_k _r).

Proof: Let X_{2k2,2r2}  A_{2k2,2r2}B_{2k2,2r2}C_{2k2,2r2} where

2k 2,2r 2 { i j, : , are odd and 1 2 1, 1 2 1},

A _{ }  v i j  i k  j r

2k 2,2r 2 { i r,2 : is even , 2 2 } { 2 ,k j: is even , 2 2 } and

B _{ }  v i  i k  v j  j r

2k 2,2r 2 { 0,2k, 2 ,0r }.

C _{ }  v v

We have the result _sp(P_2k2P_2r2 \X_{2k2,2r2})_sp(P P_k _r) by using Theorem 3.3 and Lemma 3.14. Hence, we have the proof. ■

Theorem 3.18. For n 2, _sp(P P_n _n)F or F_{n n} where 1,

2 ( 1) 1 . 3 n n F _{ }   _  

Proof: By induction on n. Let a be an upper bound of (_n  P P_n _n). By Lemma 2.2.3 and Theorem 3.3. We know that

(i)n6k 4, k0, a_n F_n; (ii)n6k 1, k1, a_n F_n; (iii)n6 ,k k2, 1a_n F_n ; (vi)n6k 3, k1, 1a_n F_n ; (v)n6k 2, k2, 1a_n F_n ; (iv)n6k 5, k1, 2a_n F_n . Now it suffices to prove that a_n F_n t, t0 or 1, when n5 (mod 6).

By Lemma 3.15, Lemma 3.16 and Lemma 3.17 with direct checking, we have

2 1 2

a  F , a₃  2 F₃, a₅  6 F₅, a₆ 10F₆ , 1 a₈ 18F₈ Hence, the 1. basic cases hold.

By Lemma 3.16. 2 2 2 6 5 3 3 2 2 2 2 (3 3 1) 1 (3 2) (3 2) 1 3 3(3 2) (3 2) 1 4(3 2) 1 (6 4) 1 1 1 1 . 3 3 3 k k k a k a k k k k k             _{ }                 _{ } _{ } _{ }      

18

Theorem 3.18. If ,n m5 (mod 6) then for all ,n m11,

, , ( ) or 1, sp P Pn m Fn m Fn m     where , ( 1)( 1) 1 3 n m n m F _{ }    _  .

Proof: By Lemma 2.2.3, Theorem 3.3 and Lemma 3.16. Let a be an upper bound _{n m,} of (_sp P P_n _m). Then 6 5,6 5 (3 3)(3 3) 3 3,3 3 (3 3 1)(3 3 1) 1 (3 3)(3 3) 3 3(3 2)(3 2) (3 2)(3 2) 1 3 4(3 2)(3 2) 1 ((6 4)(6 4) 1 3 3 k r k r a k r a k r k r t k r k r t k r k r t t                   _{ }           _{ }             _{ } _{ }     6 5,6 5 Fk r t, where t0 or 1. ■

Now, we consider the product of cycles.

Theorem 3.19.For k 2, _sp(C_2k2C_2k2)(k1)2_sp(C_k1C_k1).

Proof: Let X_2k2 {v_{i j,} : , are odd, 1i j i j, 2k1}. We have the result

2 2 2 2 2 2 1 1

( \ ) ( )

sp C k C k X k sp Ck Ck

 _  _{ }  _  _ by using Theorem 3.3 and Lemma 3.14.

Hence, we have the proof. ■

Theorem 3.20. For k 3, _sp(C_2k1C_2k1)(k1)2_sp(P P_k _k).

Proof: Let X_2k1A_2k1B_2k1{v_{2 ,2k k}}, where A_2k1{v_{i j,} : , are even and 0i j

, 2 1}

i j k

   and B_2k1{v_{i k,2} : is odd, 1i  i 2k 2} {v_{2 ,k j}: is odd, 1j  j 2k 2}.

 We have the result _sp(C_2k1C_2k1\X_2k1)_sp(P P_k _k) by using Theorem 3.3 and Lemma 3.14. Hence, we have the proof. ■

19

4. Concluding Remark

In this thesis, we first introduce a new notion called semi-power dominating set to relax the well-known power dominating set as a graph model in applications. This new SPDS turns out to be exactly the same as the feedback vertex set of a connected graph G with ( ) G  Therefore, if the graphs 2. G fit the above conditions which we can find _sp( )G , then we also determine ( ). G Indeed, we have done just that by considering the product of two paths and we are very close to determine

( ).

sp P Pn m

20

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