基本介紹
![圈多項式](/img/b/ce7/wZwpmL3QjMwMzNzQTMxMzM1UTM1QDN5MjM5ADMwAjMwUzL0EzLxMzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![圈多項式](/img/e/2ad/wZwpmLyUjM4cDN3YjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2IzLygzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![圈多項式](/img/b/d7c/wZwpmL3gTM3kDMwkDO0ATN0UTMyITNykTO0EDMwAjMwUzL5gzLzYzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
考察有向(無向)簡單圖,其中,必要時也可以考慮有自環(loop)的有向圖,今後分別以V(S)及E(S)表示子圖S的頂點集及邊集;記號表示子圖的並運算。
相關定義
![圈多項式](/img/f/9dd/wZwpmLxEDN0QzMyMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzL1MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![圈多項式](/img/b/fbe/wZwpmLxEDN0AzN5EjM3QTN1UTM1QDN5MjM5ADMwAjMwUzLxIzLzMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![圈多項式](/img/2/9b3/wZwpmLzgDN3gTM3UjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL1IzLxIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![圈多項式](/img/f/9dd/wZwpmLxEDN0QzMyMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzL1MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
定義1取定圖G的一個子圖族;對其中每一個子圖,賦以某一整環R中之元,作為它的度量,這樣一個具有度量的子圖族便稱為“ 覆蓋單元集”;其中每一個子圖α稱為“ (覆蓋)單元”。
![圈多項式](/img/f/9dd/wZwpmLxEDN0QzMyMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzL1MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![圈多項式](/img/6/752/wZwpmLzETO5kTO0cjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3IzLyYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
定義2對於不含全不連通子圖的覆蓋單元集而言,圖G的一個“ 邊覆蓋”,是指由若干個無公共邊的單元所組成的集合,使得
![圈多項式](/img/5/7be/wZwpmLyIjN3MTO4gDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4AzLxIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![圈多項式](/img/9/e45/wZwpmLzIzNzgjNzYzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2MzLzUzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
換言之,構成邊集E(G)的一個劃分。
![圈多項式](/img/f/9dd/wZwpmLxEDN0QzMyMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzL1MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![圈多項式](/img/6/752/wZwpmLzETO5kTO0cjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3IzLyYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
定義2’對給定的覆蓋單元集而言,圖G的一個“ 點覆蓋”,是指由若干個無公共頂點的單元所組成的集合,使得
![圈多項式](/img/9/876/wZwpmLxITN3UTO2gzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4MzL3IzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![圈多項式](/img/e/c20/wZwpmL3EzN4UzM0UzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL1MzL3YzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
換言之,構成頂點集V(G)的一個劃分。
邊(點)覆蓋多項式
![圈多項式](/img/f/9dd/wZwpmLxEDN0QzMyMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzL1MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![圈多項式](/img/c/15c/wZwpmL2cDNyIDOxQTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL0EzLyczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![圈多項式](/img/b/083/wZwpmLyMDO3UjMygDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4AzL0czLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
定義3對給定的覆蓋單元集,設圖G所有邊(點)覆蓋所構成的集族為,對每一覆蓋,賦以一個單項式
![圈多項式](/img/2/416/wZwpmL3IjM2YTM0kzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5MzL3czLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
進而對圖G賦以一個多項式
![圈多項式](/img/d/bcd/wZwpmL3gDMyITNxYDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2AzL1IzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![圈多項式](/img/f/9dd/wZwpmLxEDN0QzMyMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzL1MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
這個P(G)便稱為圖G在之下的“ 邊(點)覆蓋多項式”。
![圈多項式](/img/8/d8e/wZwpmL0UjM4YjMwADN3QTN1UTM1QDN5MjM5ADMwAjMwUzLwQzLxgzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![圈多項式](/img/0/a4f/wZwpmL2QjMzIDM4kTNwMDN0UTMyITNykTO0EDMwAjMwUzL5UzLxUzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![圈多項式](/img/b/5f5/wZwpmLxITO2cjM5EjM3QTN1UTM1QDN5MjM5ADMwAjMwUzLxIzLxEzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![圈多項式](/img/5/a76/wZwpmL3ADN5UDO1YjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2IzLyYzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![圈多項式](/img/c/e25/wZwpmL0EjN1QjM3cjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3IzLxczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
下文始終採取“空和為零,空積為1“的習慣約定,因此,當時,P(G)=0,當G=(空圖)時,只有一個覆蓋;而,所以。其次,在邊覆蓋多項式的情況,增減一些孤立頂點時邊覆蓋不變,因而多項式亦不變;故此時不妨假定圖G並無孤立頂點 。
圈多項式的定義
![圈多項式](/img/f/9dd/wZwpmLxEDN0QzMyMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzL1MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![圈多項式](/img/3/2f8/wZwpmLwIDNxAjMxMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzLyIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
在無向圖的情況,設由G中所有的K(一點生成子圖)、K(一邊生成子圖)及初級圈(點數≥3)所構成,R為實數域上的多項式環,由此產生出的點覆蓋多項式通常稱為“ 圈多項式”,特別當單元K的度量為(未定元),K的度量為-1,其它初級圈的度量為-2時,點覆蓋多項式
![圈多項式](/img/3/0ec/wZwpmL3QzN0ADMwIzM3QTN1UTM1QDN5MjM5ADMwAjMwUzLyMzL2UzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
就是熟知的特徵多項式,其中p(S)為S中有邊的單元數,q(S)為S中的圈數。
相關概念
子圖多項式
![圈多項式](/img/f/9dd/wZwpmLxEDN0QzMyMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzL1MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![圈多項式](/img/f/9dd/wZwpmLxEDN0QzMyMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzL1MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![圈多項式](/img/b/fbe/wZwpmLxEDN0AzN5EjM3QTN1UTM1QDN5MjM5ADMwAjMwUzLxIzLzMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![圈多項式](/img/b/569/wZwpmLwYzNwcDM5YDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2AzL0IzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
當G為無向圖,由G的一切連通子圖所構成時,圖G在之下的點覆蓋多項式稱為“ 子圖多項式”,而當單元的度量不同時,可得一些特殊的多項式,例如:
![圈多項式](/img/b/fbe/wZwpmLxEDN0AzN5EjM3QTN1UTM1QDN5MjM5ADMwAjMwUzLxIzLzMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![圈多項式](/img/1/712/wZwpmLzIDMyIDMykjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5IzL1EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![圈多項式](/img/a/813/wZwpmL3MDNzkTNxYDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2AzLyYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
(1°) 對每一單元,賦以度量,其中為連通子圖α的圈秩,那么圖G在這種覆蓋意義下的多項式
![圈多項式](/img/2/769/wZwpmL0cDO5AzN1YDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2AzLzAzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![圈多項式](/img/3/9f5/wZwpmLwcjN5gzMxIzM3QTN1UTM1QDN5MjM5ADMwAjMwUzLyMzL2EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
就是 雙色多項式,其中p(S)及q(S)分別表示由S的單元的並所構成的支撐子圖的分圖數及圈秩。
![圈多項式](/img/3/e35/wZwpmL1ADOygDO5QTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL0EzLyYzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![圈多項式](/img/9/caa/wZwpmLxYjNwEjMxgTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4EzLxUzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
(2°) 若令,其中為連通子圖α的秩,則
![圈多項式](/img/0/00a/wZwpmLyAjMyEjNwAjM3QTN1UTM1QDN5MjM5ADMwAjMwUzLwIzL1gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![圈多項式](/img/6/9f5/wZwpmL4QTOyIDMxkTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5EzL0czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
就是 秩多項式,其中,表示由S所成的支撐子圖的秩。
![圈多項式](/img/3/a57/wZwpmLyMDMzUDN4YDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2AzL1czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![圈多項式](/img/6/49d/wZwpmLyIDN1QTOzADN3QTN1UTM1QDN5MjM5ADMwAjMwUzLwQzL3czLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
(3°) 若令,其中為連通子圖α的邊數,則
![圈多項式](/img/8/4db/wZwpmL3YjN3gDMwkDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5AzL2MzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![圈多項式](/img/8/3b3/wZwpmLwQjM2YTOwkzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5MzLzUzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![圈多項式](/img/c/9ee/wZwpmLzETOzkjN4cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzLyMzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
就是 色多項式,事實上,記,並以表示H的生成子圖的秩,則上式可改寫為
![圈多項式](/img/d/d48/wZwpmL4UTN1YDNzIjM3QTN1UTM1QDN5MjM5ADMwAjMwUzLyIzLwAzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
這就是 色多項式的展開式(極易由容斥原理推出)。
匹配多項式
![圈多項式](/img/f/9dd/wZwpmLxEDN0QzMyMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzL1MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![圈多項式](/img/3/f77/wZwpmL4AjMzQzN1kzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5MzLzgzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![圈多項式](/img/f/aa3/wZwpmL4MDM1cTM4kjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5IzLzUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
設由G中所有子圖K及K所構成,它們的度量分別為x及y,由這種單元組成的點覆蓋S就是圖G的“ 匹配”(matching);它對應的單項式為,其中為S中單元K的數目(邊數),於是
![圈多項式](/img/2/dfc/wZwpmLyYTO3YjMxYzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2MzL1MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![圈多項式](/img/0/8af/wZwpmL1MzM5kjN3ADO3EDN0UTMyITNykTO0EDMwAjMwUzLwgzLyAzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
就是圖G的 匹配多項式,其中為k邊匹配的數目 。
相關定理
![圈多項式](/img/f/9dd/wZwpmLxEDN0QzMyMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzL1MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
設圖G= (V,E)在單元集之下的邊(點)覆蓋多項式為P(G),下面將給出這兩類多項式的一般消去定理。
![圈多項式](/img/7/949/wZwpmL4YDNwETNyQTO2UzM1UTM1QDN5MjM5ADMwAjMwUzL0kzLxMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![圈多項式](/img/2/f7a/wZwpmLwEjN4MDMzQDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL0gzL0gzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![圈多項式](/img/4/8f7/wZwpmLzETMxITO5gzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4MzL3MzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
定義4對任意的頂點子集及邊子集,二元組稱為G的“偽子圖”。
![圈多項式](/img/f/bf6/wZwpmLwIDMwgjM0kjN0YzM1UTM1QDN5MjM5ADMwAjMwUzL5YzL4gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![圈多項式](/img/7/687/wZwpmL3MDM0YTNwADN3UzM1UTM1QDN5MjM5ADMwAjMwUzLwQzL2IzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![圈多項式](/img/4/8f7/wZwpmLzETMxITO5gzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4MzL3MzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![圈多項式](/img/f/7bb/wZwpmLwAzM1AzN4kTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5EzLwczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![圈多項式](/img/0/a4f/wZwpmL2QjMzIDM4kTNwMDN0UTMyITNykTO0EDMwAjMwUzL5UzLxUzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
當中每條邊所關聯的頂點均屬於時,就是通常意義的子圖,將簡記為。
定義5給定G的一個偽子圖H,圖G關於H的部分邊(點) 覆蓋S,是指由若干個無公共邊(點)的單元所組成的集合,這些單元的並包含了H的全部頂點和邊,且每個單元都至少含有H的一個頂點或一條邊。
![圈多項式](/img/4/667/wZwpmLyEzMzIjN2IzM3QTN1UTM1QDN5MjM5ADMwAjMwUzLyMzLyAzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
對於圖G的一個邊(點)覆蓋S 而言,如果S覆蓋著H的點和邊,則將S 中含有H的頂點或邊的單元取出來,這樣構成的子集S一定是關於H的部分覆蓋,即有,但反過來說,一個部分覆蓋卻不一定能成為某個覆蓋的子集(它未必能添上一些單元後構成G的覆蓋)。
![圈多項式](/img/2/14d/wZwpmLzITMyAjN1kjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5IzLxIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
定義6對部分邊(點)覆蓋而言,其相應的單項式定義為
![圈多項式](/img/a/afe/wZwpmL3YzM0AzMxEjM3QTN1UTM1QDN5MjM5ADMwAjMwUzLxIzLyYzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![圈多項式](/img/7/dd3/wZwpmLyIDMxUjNwYzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2MzLwAzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![圈多項式](/img/d/dc5/wZwpmL4YzNxADO1UzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL1MzLwgzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![圈多項式](/img/1/5ee/wZwpmL4gDO4cjM3cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzL4IzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
特別當,因而時,。
![圈多項式](/img/c/c72/wZwpmLxMzNygTM1cDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3AzL4czLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
此外,我們記。
現在,分兩種情況討論。
邊覆蓋多項式的消去定理
![圈多項式](/img/2/66f/wZwpmL4AjNzcDNzQTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL0EzLyUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
定義7一個關於H的部分邊覆蓋S稱為極大的,如果不存在以它為真子集的另一個關於H的部分邊覆蓋。
![圈多項式](/img/b/083/wZwpmLyMDO3UjMygDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4AzL0czLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
命題 對任一個邊覆蓋,由其中所有含有H的頂點或邊的單元所構成的部分邊覆蓋S一定是極大的。
![圈多項式](/img/a/196/wZwpmL4QzN5QDO4kjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5IzL2QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![圈多項式](/img/a/966/wZwpmLyUjN4czM1YDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2AzLyIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![圈多項式](/img/c/cad/wZwpmL4QDO4gzN4cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzLxczLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![圈多項式](/img/1/e2b/wZwpmL3YTO5kzN4gDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4AzLxEzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
證明:設是按上述方式導出的部分邊覆蓋,那么H中的邊()及H中的頂點所關聯的邊(的端點)均被S的單元所覆蓋;而任一單元均非孤立點,所以不存在這樣的單元,它與S的單元無公共邊而又包含H的頂點或邊,故S為極大的。
![圈多項式](/img/f/9dd/wZwpmLxEDN0QzMyMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzL1MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![圈多項式](/img/9/75a/wZwpmL4AjM5ITOyQTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL0EzL2YzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
定理1設圖G在單元集之下的邊覆蓋多項式為P(G),並設為G的一個偽子圖,則
![圈多項式](/img/8/942/wZwpmLzYTM2EjM5YjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2IzL3czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![圈多項式](/img/8/1ff/wZwpmL4gDO2gTOykDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5AzLxYzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
其中為G關於H的一切極大的部分邊覆蓋的集合。
點覆蓋多項式的消去定理
![圈多項式](/img/1/05e/wZwpmL3EzNxkTM3kTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5EzLxAzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
取定圖G的一個偽子圖,考察由它除去一些邊所成的偽子圖H的集合
![圈多項式](/img/3/b5e/wZwpmLzMzNykzN3kDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5AzLzIzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![圈多項式](/img/3/108/wZwpmLxcDOzADM0YDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2AzLzYzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
對於每一個,又考察部分點覆蓋的集合
![圈多項式](/img/8/1ff/wZwpmL4gDO2gTOykDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5AzLxYzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![圈多項式](/img/6/a1c/wZwpmLwIDN3UDM3AzNwMzM1UTM1QDN5MjM5ADMwAjMwUzLwczL1czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![圈多項式](/img/5/47e/wZwpmLzIDO3ADNxkDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5AzLyYzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
{部分點覆蓋}. (4)
![圈多項式](/img/3/43a/wZwpmLwEDN4kDO5cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzLzMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![圈多項式](/img/a/a0d/wZwpmL2MDOxYzM1gjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4IzL3gzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![圈多項式](/img/e/c57/wZwpmL3MDNzEDOxkjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5IzL1UzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![圈多項式](/img/4/be0/wZwpmL4QTM0MjN3EjM3QTN1UTM1QDN5MjM5ADMwAjMwUzLxIzLyIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![圈多項式](/img/3/43a/wZwpmLwEDN4kDO5cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzLzMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
就是G關於H的部分點覆蓋,它的單元包含J' 的邊,而不包含的邊。為更明顯起見,對於,我們令。那么,就是圖G關於H的部分點覆蓋。
![圈多項式](/img/f/9dd/wZwpmLxEDN0QzMyMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzL1MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![圈多項式](/img/1/05e/wZwpmL3EzNxkTM3kTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5EzLxAzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
定理2 設圖G在單元集之下的點覆蓋多項式為P(G),並設為G的一個偽子圖,則
![圈多項式](/img/3/070/wZwpmL1IDM1kzMwkzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5MzLxgzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![圈多項式](/img/0/786/wZwpmLyIjN3gzMwgTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4EzLxAzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![圈多項式](/img/8/1ff/wZwpmL4gDO2gTOykDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5AzLxYzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
其中集合及分別由(3)(4)式所定義 。