在樹圖上之限亮點西格瑪遊戲與其對偶遊戲

國

立

交

通

大

學

應用數學系

碩

士

論

文

在樹圖上之限亮點西格瑪遊戲與其對偶遊戲

Lit-only Sigma Game and its Dual Game on Tree

研 究 生：林育生

指導教授：翁志文 教授

在樹圖上之限亮點西格瑪遊戲與其對偶遊戲

Lit-only Sigma Game and its Dual Game on Tree

研 究 生：林育生 Student：Yu-Sheng Lin

指導教授：翁志文 Advisor：Chih-Wen Weng

國 立 交 通 大 學

應 用 數 學 系

碩 士 論 文

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

在樹圖上之限亮點西格瑪遊戲與其對偶

遊戲

學生：

指導教授

國立交通大學應用數學系碩士班

摘

要

令G是一個簡單的連通圖，G的點集為{1, 2, … , n}。若將每一個頂

點皆給定黑或是白其中一個顏色，便成G的一個配置。在每一個遊戲中

的一個走法是將一個配置換成另一個配置。在此篇論文中給兩個特別

的遊戲走法。第一個遊戲便是限亮點西格瑪遊戲，此遊戲包含了對應n

個定點的n個走法，規則為：在配置u中點i若是黑色，則走法L

將點i的

鄰居的顏色黑白互換，而且不改變其他點(包括i)的顏色。第二個遊戲

則是第一個遊戲的對偶遊戲，也包含了對應n個定點的n個走法，規則

為：在配置u中點i的鄰居中若是有奇數個黑色點，則L

_{便可以將點i的}

顏色黑白互換，而且不改變其他點的顏色。這兩種遊戲的關係在這篇

論文中也會說明，另外，在這兩種遊戲之下的任何一個規則，我們可

以利用它們對應的走法將配置的集合做出分割，並求出這些軌跡。我

們稱一個包含超過一個元素的軌跡為“非簡單＂的軌跡。若給定一些

ii

前提，我們可以猜測在限亮點西格瑪遊戲的對偶遊戲之中將有兩個非

簡單的軌跡。此外，我們知道若G是一個擁有完美配對的樹圖，則在限

亮點西格瑪遊戲之中將有三種軌跡存在，最後也給出一個演算法以及

利用其對偶遊戲的結果來描述這三種軌跡。

iii

Lit-Only σ-Game and its Dual Game

on Tree

student：

Advisors：Dr.

Department of Applied Mathematics

National Chiao Tung University

ABSTRACT

Let G be a simple connected graph with n vertices {1, 2, … , n}. A

configuration of G is an assignment of one of two colors, black or white,

to each vertex of G. A move on the set of configurations of G is a

function from the set to itself. Two different games with their own sets

of moves are investigated in this thesis. The first one which is called the

lit-only σ-game, contains n moves Li corresponding to the vertices i.

When the move Li is applied to a configuration u, the color of a vertex j

in u is changed if and only if i is a black vertex and j is a neighbor of i.

The second one which is called the lit-only dual σ-game, has n moves

Li

corresponding to the vertices i. When the move Li

is applied to a

configuration u, the color of a vertex j in u is changed if and only if i has

odd number of black neighbors and j=i. The dual relation between these

two games will be clarified. In each of the two games, the set of

configurations is partitioned into orbits by the action of its moves. An

orbit with more than one configuration is called a nontrivial orbit. When

G is a tree with some minor assumptions, we conjecture that there are

two nontrivial lit-only dual σ-game orbits. We prove the conjecture

iv

under certain assumptions. It is known that the lit-only σ-game on a

tree with perfect matchings has three orbits. We give an algorithm to

describe these three orbits by applying the results in its dual game.

v

誌

謝

本篇論文的完成，首先要感謝我的指導教授 ─ 翁志文教授。老師

不僅僅在整體內容上的編排或是文字上的改正甚至是思考的方向，都

給予我很多的建議以及修改，而且不僅僅只在論文寫作的部份，在我

碩士班兩年的課業中，老師也給予我許多的教導，對於平常待人處事

的方法以及看法，老師也是我心中一個很好的模範，讓我在往後的生

活態度有更明確的想法。而老師所著重的許多事情，學生一定謹記在

心，謝謝老師

再來要感謝的是黃皜文學長。學長所給予我的方向以及想法，才使

這篇文章得以完成，而且學長在知識上的執著以及認真的態度也是我

現階段很缺乏的一部份，在這兩年的其他科目上的學習，幸有學長仔

細的教導，我才能夠對這些科目有比較整體的了解。而在以後對於學

識的學習和研究的方式，我將以黃皜文學長為目標好好的努力。

接著要感謝的人還有游森棚教授，如果我沒遇見游教授我的人生應

該會是完全不一樣的方向，因為游教授的支持，我才能朝研究所的這

條路前進，而且游老師也給我很多的訓練與教誨，讓我一點點的進步，

我也會繼續的往前進，謝謝老師。

最後在交大學習的兩年期間，我遇到了許多的師長、學長姐與同學

們，感謝你們，沒有你們就不會有現在的我，你們的存在是不可或缺

的，謝謝你們。

最後感謝我的家人還有冠榕，謝謝你們的支持，我才有更多的動力

不迷惘的繼續往前，謝謝。

目

錄

中文摘要 ………

i

英文摘要 ………

iii

誌謝

………

v

目錄

………

vi

一、

Introduction ………

1

二、

Lit-only σ-game ………

2

三、

Lit-only dual σ-game on tree ………

4

四、

Tree with perfect matching ………

8

五、

Lit-only dual σ-game on a tree with

perfect matching ………

9

六、

Combinatorial interpretation of A

………

10

七、

Lit-only σ-game on a tree with

perfect matching ………

12

八、

Algorithm ………

13

九、

Conclusion ……… 16

Reference ………

17

1

Introduction

Let G be a simple connected graph with vertex set V (G) = {1, 2 . . . , n} and edge set E(G). A configuration of G is an assignment of one of two colors, black or white, to each vertex of G. And we call a configuration u trivial if all the vertices are white. In each game on G we have a rule on configurations to apply with and we call those steps moves. For convenience, we use the set F₂n of column vectors over F2 := {0, 1} to denote the set of configurations.

The i-th entry ui of a configuration u is 1 if and only if the vertex i is black

on this configuration. An orbit O in a game is a subset of configurations such that for any two configurations u, v ∈ O, there exists a sequence of moves that u can reach v by applying these moves in order. And we call a orbit trivial if and only if it has only one element. Our goal is to decrease the number of black vertices by applying several moves.

Here we consider in two different games, lit-only σ-game and it’s dual game which is called Reeder’s game. The lit-only game is a variation of σ-game which was investigated from 1989 [4]. The Reeder’s σ-game was appeared in the 2005 paper [3] of M. Reeder. Although the two games seem different ostensibly, there are many connections between them.

2

Lit-only

σ-game

A move Li in the lit-only σ-game is defined as follows: If a vertex i is in

black color in the configuration u, then when Li applies to u, the colors of all

neighbors of i will be changed but keep the colors of other vertices including

i unchanged. On the other hand, if i is in white color in u then Li does

nothing about the configuration. And it is the reason we called the game lit-only σ-game. Let the n × n matrix A be the adjacency matrix of the given

graph G. Note that eT

i u is the parity of ui. where {ei} is the standard basis

of Fn

2, that is, for 1 ≤ i ≤ n, the i-th entry of ei is 1 and the other entries

are 0. We have that

Li(u) = u + (eTiu)Aei = u + AeieiTu= (In+ AeieTi )u, (2.1)

where In is the n × n identity matrix. Note that for any vertex i and any

configuration u, Li(Li(u)) = u. Here we have an example.

j 1 z 2 j 3 j 4 z7 z8 j 5 z 6 u=             0 1 0 0 0 1 1 1             , L7 =             1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1             .

Figure 1: A configuration u and a move L7 in the lit-only σ-game.

Now we consider in the Reeder’s game. A move L∗

i is defined as follows:

If the vertex i in the configuration u0 _{has odd number of black neighbors then}

the move L∗

i changes the color of i and keeps other vertices unchanged. And

if vertex i has even number of neighbors in black color, the move L∗

i does

nothing. Note that eT

i Au0 is the parity of the number of black neighbors of

i in u0_{. Like in lit-only σ game, we also use matrices to represent the moves}

and then L∗_i(u0) = u0+ eTi Au0
ei = u0+ eieTiAu0 = (In+ eieTi A)u0 = (In+ AeieTi ) T_u0 = LTi (u0), (2.2) 2

and L∗ i(L∗i(u0)) = u0. Here is an example. j 1 z 2 j 3 j 4 z7 z8 j 5 z 6 u0 ₌             0 1 0 0 0 1 1 1             , L∗ 7 =             1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1             .

Figure 2: A configuration u0 _{and a move L}∗

7 in the lit-only dual σ-game.

By (2.1) and (2.2) we notice that each of the two matrices representations of the moves in different games, respectively, is the transport of the other one.

We have one more relation between the two moves on G.

Proposition 2.1. For a given graph G, let A be the adjacency matrix of G.

Then LiA= AL∗i, i.e. the following diagram commutes:

u0 L∗i −→ L∗ i(u0) A↓  _{↓ A} u= Au0 Li −→ Li(u) .

Proof. Note that

AL∗_i

= A(In+ eieTi A)

= (In+ AeieTi )A

= LiA

and the proposition follows.

3

Lit-only dual

σ-game on a tree

Throughout this section let G be a tree with the vertex set {1, 2, . . . , n}.

We study the lit-only dual σ-game on G in this section. Let u ∈ Fn

2 be a

configuration. Then u is moveable if there exists a vertex with odd black neighbors, i.e. Au 6= 0. Let Bu denote the subset of vertices consisting of

black vertices in u, i.e. Bu := {i | ui = 1}, and let c(Bu) denote the number

of components in the subgraph induced by Bu. Recall that an independent

set of G is a subset of vertices in which each pair of vertices are not adjacent. For a subset S of vertices we denote N[S] as the set of closed neighbors of S, i.e. N[S] := S ∪ {a | {a, s} ∈ E(G), for some s ∈ S}.

Lemma 3.1. For any configuration u, each component of Bu can be reduced

to a black vertex by a sequence of lit-only dual σ-game moves on G. Formally, for any u ∈ Fn

2 there exists v ∈ F2n such that u, v are in the same orbit, Bv

is an independent set and c(Bv) = c(Bu).

Proof. Since G is a tree, we know that each connected component of Bu is

also a tree. Start from a component which has vertices more than 2 and select a leaf i of the component. Since i has only one neighbor in black

color, we can use the move L∗

i to get a new configuration w = u + ei. We

change the color of the vertex i by L∗

i without change the number c(Bu) then

we know that u, w are in the same orbit and c(Bu) = c(Bw). Repeat this

process we finally have a configuration v which is in the same orbit with u and c(Bu) = c(Bv) and each connected component of Bv has only one black

vertex, i.e. Bv is an independent set.

Lemma 3.2. Let u, v be two nontrivial configurations such that c(Bu) and

c(Bv) have different parities. Then u and v are in different lit-only dual

σ-game orbits.

Proof. Suppose there are two moveable configurations u, v such that u, v are in the same orbit. That is, u can reach v by applying several moves. If there is a move L∗

i changes c(Bu), i.e. c(Bu) 6= c(BL∗

i(u)), we know that L

∗

i separates

a connected component of Bu or combines several connected components

into one. By the definition of moves of Reeder’s game, we know i has odd

number of black neighbors. Then the move L∗i separates one component

into odd number of components or combines odd number of components into one. So each one of these moves can not change the parity of c(Bu) for any

configuration u to another one. In other words, if two configurations u, v in the same orbit, then c(Bu), c(Bv) have the same parity.

The special case when G is a path is easy to settle.

Proposition 3.3. Let G be a path and let u and v be two moveable

configu-rations. Then u and v are in the same lit-only dual σ-game orbit if and only if c(Bu) = c(Bv).

Proof. Each vertex in G has at most two neighbors since G is a path. For a

configuration w, we know that any move L∗

i can not change the Bw by the

definition of moves of Reeder’s game. If two moveable configurations u, v are in the same orbit then c(Bu), c(Bv) must be equal since c(Bu) will hold by

any move L∗

i.

On the opposite side, we assume G = {1, 2, . . . , n} and two moveable configurations u, v with c(Bu) = c(Bv) = k. By Lemma 3.1 there exists two

configurations u0_{, v}0 _{such that u}0_{, v}0 _{are in the same orbit with u, v,}

respec-tively. And Bu0, B_v0 are independent sets with B_u0 = {i₁, i₂, . . . , i_k}, B_v0 =

{j1, j2, . . . , jk}.

We use these moves L∗

i1−1, L ∗ i1−2, . . . , L ∗ 1, L∗i1, L ∗ i1−1, . . . , L ∗ 2 in turn to shift

the black vertex i1 of u to the vertex 1. And we shift those black vertices

i2, i3, . . . , ikto the vertices 3, 5, . . . , 2k−1 similarly. Then we get a new

config-uration w such that u0_{, ware in the same orbit and B}

w = {1, 3, 5, . . . , 2k −1}.

If we use the same method to shift these black vertices of v0 _{then we can}

get the same configuration w. So that we know that v0_{, w are in the same}

orbit and u0_{, v}0 _{are in the same orbit, that is, u, v are in the same orbit. Then}

the proposition follows.

Definition 3.4. We call a graph G a binary star, and defined by D(n; r, s),

if all the leaves of G are adjacent to one of the endpoints of path Pn.

c 9 c 8 c 7 c 1 2c 3c 4c 5c 6c _c 12 c11 c₁₀ c₁₃ """ b b "" b b %%% e ee Figure 3: D(6; 3, 4). 5

Conjecture 3.5. Assume G is a tree but not a binary star. Let u and v be

two moveable configurations with c(Bu) = c(Bv). Then u and v are in the

same lit-only dual σ-game orbit.

The following example, D(5; 2, 0), is first found not have the conclusion of Conjecture 3.5 by Hau-wen Huang.

s s c s c c c """ b b 6 7 5 4 3 2 1 c s c s c s c """ b b 6 7 5 4 3 2 1

Figure 4: Two configurations which are not in the same lit-only

dual σ-game orbit.

Lemma 3.6. For any vertices i, j, the configurations ei and ej are in the

same lit-only dual σ-game orbit. Moreover if S consists of the vertices in the path from i to j and u is a configuration with N[Bu] ∩ S = ∅ then u + ei and

u+ ej are in the same lit-only dual σ-game orbit.

Proof. Let the unique path from i to j be i0i1i2· · · ik where i0 = i, ik = j

and S = {i0, i1, i2, . . . , ik}. Since N[Bu] ∩ S = ∅ these moves L∗i1, L

∗ i2, . . . , L ∗ ik, L∗ i0, L ∗ i1, . . . , L ∗

i_k−1 are doing nothing about u. And if we apply these moves in

turn then we have

u+ ej = L∗ik−1L ∗ ik−2· · · L ∗ i0L ∗ ikL ∗ ik−1· · · L ∗ i2L ∗ i1(u + ei),

that is, u + ei, u+ ej are in the same orbit. Let u be the configuration with no

black vertices and for any i, j, ei and ej are in the same lit-only dual σ-game

orbit.

Conjecture 3.7. Let G be a tree but not a binary star. Then the set Fn

2 of

configurations is partitioned into the following lit-only dual σ-game orbits: (i) the orbit {u} of a single non-moveable configuration;

(ii) {u ∈ Fn

2 |c(Bu) 6= 0 is even.};

(iii) {u ∈ Fn

2 | c(Bu) is odd.}.

The following proves Conjecture 3.7 under the assumption that Conjec-ture 3.5 holds.

Proof. For any non-moveable configuration u we know that each move L∗

i

does nothing on u. Then there are orbits of a single non-moveable configura-tion u.

There is a vertex i with at least three neighbors i1, i2, i3 since G is not

a path. We apply these moves L∗

i, L∗i3, L

∗ i2, L

∗

i1 on ei and get a moveable

configuration v = L∗ iL∗i3L

∗ i2L

∗

i1(ei) which c(v) = 3. Then we know that v, ei

are in the same orbit. By this method, in each process we have a moveable configuration u and find a vertex j with degree greater or equal to 3. First we shift black vertices out of N[N[{j}]] and then shift one black vertex to j and then apply these moves L∗

j1, L ∗ j2, L ∗ j3, L ∗

j and get a new moveable configuration

v with c(Bv) = c(Bu) + 2.

By Conjecture 3.5 and the previously method we know that for two move-able configurations u, v if c(Bu), c(Bv) have the same parity then u, v are in

the same orbit. And by lemma 3.2 we know that the set Fn

2 of configurations

is partitioned into the following lit-only dual σ-game orbits: (i) Trivial orbits {u} which u is a non-moveable configuration; (ii) {u ∈ Fn

2 |c(Bu) 6= 0 is even.};

(iii) {u ∈ Fn

2 | c(Bu) is odd.}.

4

Tree with perfect matching

Lemma 4.1. Let G be a tree with perfect matching. Then the perfect

match-ing is unique.

Proof. Since G is a tree with perfect matching, we know that G has even vertices. So that we prove this lemma by induction on |V (G)| = 2k. For k = 1, we have that the perfect matching is unique. We assume the lemma holds for 1 ≤ k ≤ d − 1. Let G be a tree with perfect matching with |V (G)| = 2d. Since G is a tree, we can find a leaf i with neighbor j. If G has a perfect matching π then we know that {i, j} belongs to π. Then we consider the

graph G0 _{= G − {i, j} which is an union of connected components. Since G}

is a tree with perfect matching π then each component of G0 _{is also a tree}

with even vertices. Moreover, we know that G has perfect matching π then

each component of G0 _{must have a perfect matching π}0 _{such that π}0 _{⊂ π.}

Since the number of vertices of each component is less or equal to 2(d − 1)

we have that the perfect matching π0 _{is unique by the assumption. Then G}

has an unique perfect matching π and the lemma follows.

5

Lit-only dual

σ-game on a tree with perfect

matching

In this section we collect a known result to support Conjecture 3.7 and then Conjecture 3.5. M. Reeder uses the property of quadratic form to prove the following theorem [3, page 33].

Theorem 5.1. Let G be a tree with perfect matching but not a path. Then

there are three lit-only dual σ-game orbits.

Hau-wen Huang quotes the above theorem to describe the three orbits combinatorially [2].

Proposition 5.2. Assume Conjecture 3.5 hold. Then the set Fn

2 of

con-figurations is partitioned into the following three orbits: {0}, {u |c(Bu) 6=

0 is even.}, {u | c(Bu) is odd.}.

Proof. G is a tree with perfect matching so that the adjacency matrix of G is invertible then there is only one non-moveable configuration {0}. And since a non-moveable configuration is an orbit, we have an orbit {0}. By Lemma 3.2 we know if two configurations u, v such that c(Bu), c(Bv) have different

parities then u, v are not in the same orbit. And by Theorem 5.1 we know that there are only three lit-only dual σ-game orbits. So that if two moveable configurations u, v such that c(Bu), c(Bv) have the same parity then u, v must

be in the same orbit otherwise the number of orbits is greater than 3. Then we have the three orbits: {0}, {u |c(Bu) 6= 0 is even.}, {u | c(Bu) is odd.}.

6

Combinatorial interpretation of

A

For the completeness, we shall provide a combinatorial proof of the following well-known theorem, See for example [1, page 21].

Theorem 6.1. If G is a tree with perfect matching, then the adjacency matrix A of G is invertible.

Proof. A graph with perfect matching must has even number of vertices, then we prove this by induction on the number of vertex set |V (G)| = 2k.

1. For k = 1, the adjacency matrix of G is A(G) = 0 1

1 0 

then we have det(A(G)) = −1. Since the determinant of A(G) is not 0, we know that A(G) is invertible.

2. Suppose for k = n − 1, it is true.

3. Let G is a tree with matching and |V (G)| = 2(n − 1). And another

graph G with V (G0_{) = V (G) ∪ {2n − 1, 2n}, and E(G}0_{) = E(G) ∪}

{{i, 2n − 1}, {2n − 1, 2n}}, for some 1 ≤ i ≤ 2n − 2. G0 _{is also a tree}

with perfect matching. Then the (2n) × (2n) adjacency matrix A(G0₎

of G0 _is               0 0 ... 0 0 A(G) 1 ... 0 0 ... 0 0 · · · 0 1 0 · · · 0 1 0 · · · 0 1 0              

And we get det(A(G0_{)) = − det(A(G)) 6= 0}

That is, a tree with perfect matching has an invertible adjacency matrix. Definition 6.2. Let G be a tree with perfect matching π. A path i0i1. . . it

of length t is alternating if t is odd and for 0 ≤ j ≤ t − 1, ijij+1 ∈ π, if j is even;_{6∈ π, if j is odd.}

The following proposition gives a combinatorial interpretation of A−1_.

Proposition 6.3. (A−1₎

ij =

 1, if the path from i to j is alternating; 0, else.

Proof. The n×n matrix B is defined as : If the path from i to j is alternating

then Bij = 1, otherwise, Bij = 0. And we want to show that AB = In. We

have that (AB)ij = n X k=i AikBkj = X k∈N [{i}]−{i} Bkj.

In other words, (AB)ij stands for the number of neighbors k of i such that

the paths from these neighbors k to j are alternating with odd length. If i= j: Since G is a tree with perfect matching, for each vertex i there is

only one neighbor k of i such that the path ik is an alternating path, and then (AB)ii = 1.

If i6= j: We assume ik ∈ π. If k is in the unique path from i to j, then

there is not an alternating path from the neighbor of i to j and we have (AB)ij = 0. If k is not in the unique path from i to j, and there

is at most one neighbor l 6= k of i such that the path from l to j is an alternating path, then the path from k to j is also an alternating path and we have (AB)ij = Bkj+ Blj = 0 in F2.If no such l exists, we know

that (AB)ij = 0.

Finally we have AB = In and then A−1 = B.

7

Lit-only

σ-game on a tree with perfect

match-ing

Here we have a relation between lit-only σ-game orbits and lit-only dual σ-game orbits on tree with perfect matching.

Proposition 7.1. Let G be a tree with perfect matching, and O are O0 _are

the sets of orbits in lit-only σ-game and lit-only dual σ-game respectively. Then O = {AO0 _{| O}0 _{∈ O}0_{} and O}0 _{= {A}−1_{O | O ∈ O}.}

Proof. Let u0_{, v}0_{be in the same orbit O}0_{and O}0 _{∈ O}0_{.If v}0 _{= L}∗ i1(L ∗ i2· · · L ∗ ik(u 0_)), by proposition 2.1 we have Av0 = A(L∗i1(L ∗ i2· · · L ∗ ik(u 0₎₎₎ = Li1(A(L ∗ i2· · · L ∗ ik(u 0₎₎₎ = ... = Li1(Li2· · · Li_k(Au 0_)).

So that Au0_{, Av}0 _{are in the same orbit of lit-only σ game. And we know that}

if O0 _{is an orbit in lit-only dual σ-game then AO}0 _{is an orbit in lit-only}

σ-game. We have that O = {AO0 _{| O}0 _{∈ O}0_{}. Moreover, since G is a tree with}

perfect matching then A−1 _{exists so that we prove O}0 _{= {A}−1_{O | O ∈ O}}

similarly.

By using Proposition 7.1, for any configuration u we can know u is in which lit-only σ-game orbit by checking the lit-only dual σ-game orbit of A−1u. And the following propositions are Hau-wen Huang’s result [2].

Proposition 7.2. Let G be a tree with perfect matching but not a path.

Then there are three lit-only σ-game orbits. Moreover, the three orbits are {0}, {Au |c(Bu) 6= 0 is even.}, {Au | c(Bu) is odd.}.

Proposition 7.3. There exist distinct vertices i, j such that ei and ej are in

different lit-only σ-game orbits.

Then we know that in each orbit of lit-only σ-game there is a configuration with at most one black vertex.

8

Algorithm

Let G be a tree with perfect matching but not a path. By the above result 5.2, 7.1 we know that two configurations ei and ej are in the same lit-only

σ-game orbit if and only if c(BA−1_ei) and c(B_A−1_ej) have the same parity.

We shall give the algorithm to determine which orbit the configuration ei is

belonging to.

Algorithm. For a configuration ei is given, we want to find the

corre-sponding configuration u0 _{such that e}

i = Au0.

Input Set u0 _{= 0}

Step 1 Start from the subset X = {i} of V (G).

Step 2 If the vertex j is in the same matching with i, set u0 _{:= u}0_{+ e} j, and

X := N[X].

Step 3 If vertex k ∈ N[X] − X is adjacent to a black vertex in u0_{, and the}

vertex l is in the same matching with k, then set u0 _{:= u}0_{+ e} l.

Step 4 Set X := X ∪ N[X].

Step 5 Repeat Step 3 and Step 4 until X = V (G).

Output We get a configuration u0_.

Here is an example.

Example 8.1. G is shown and u1 = e12. And these thick edges are the

matching of G. j11 j10 j 9 j8 j7 z12 j1 j2 j3 j4 j5 j6 j13 j14 j15 j16 j17 j18 13

Start form G with all vertices in white. First we change {11} to black, then {9}, and then {1, 7, 4}, and finally {6, 16}. We have e0

12 shown as z11 j10 z 9 j8 z7 j12 z1 j2 j3 z4 j5 z6 j13 j14 j15 z16 j17 j18 u2 = e3, j11 j10 j 9 j8 j7 j12 j1 j2 z3 j4 j5 j6 j13 j14 j15 j16 j17 j18 then we have e03 j11 j10 j 9 j8 j7 j12 j1 j2 j3 z4 j5 z6 j13 j14 j15 z16 j17 j18 u₃ = e16, 14

j11 j10 j 9 j8 j7 j12 j1 j2 j3 j4 j5 j6 j13 j14 j15 z16 j17 j18 e0 16 is shown j11 z10 j 9 j8 j7 z12 j1 j2 z3 j4 j5 j6 j13 z14 z15 j16 j17 z18

Since c(e12) = 7, c(e3) = 3, c(e16) = 6, we know that e12, e3 are in the

same orbit and e16 is not.

9

Conclusion

Let G be a tree with perfect matching but not a path. For any moveable configurations u, v ∈ Fn

2, there are two configurations u0, v0 ∈ F2n such that

u0 = A−1_{u, v}0 _{= A}−1_{v,and we know that u, v are in the same orbit in lit-only}

σ-game if and only if u0_{, v}0 _{are in the same orbit in lit-only dual σ-game if}

and only if c(Bu0), c(B_v0) have the same parity. So we can know that whether

two configurations u, v are in the same orbit or not by checking the parities of c(BA−1_u), c(B_A−1_v).

Moreover, for a moveable configuration u in lit-only σ-game there is a moveable configuration u0 _{= A}−1_{uin lit-only dual σ-game and by Proposition}

7.3 we know that: Whether Bu0 is odd or even, there is a configuration e_i

which has only one vertex in black color in the same orbit with u. And by applying the algorithm, we can find these ei’s which are in the same orbit

with u.

That is: Given a tree G with perfect matching but not a path, and any initial configuration u with at least one vertex in black color, then we can

reach a configuration ei with a single vertex i in black color by applying

several moves in lit-only σ-game. Moreover, we know the single black vertex appearing at which vertex of G.

References

[1] C. D. Godsil, Algebraic Combinatorics, Benjamin/Cummings, Menlo Park, 1984.

[2] Hau-wen Huang, Lit-only sigma-game and its dual game

arXiv:0902.3550, preprint

[3] M. Reeder, Level-two structure of simply-laced Coxeter groups, Journal of Algebra 285 (2005) 29-57.

[4] K. Sutner, Linear cellular automata and the Garden-of-Eden, Math. Intelligencer 11 (1989), no.2, 49-53.