團寬

圖論中，圖G的團寬（clique-width）是描述圖的結構複雜性的參數，與樹寬密切相關，但對稠密圖來說可以很小。團寬的定義是通過以下4種操作，構造G所需的最少標號數：

創建標籤為i的新頂點v，記作 $i(v)$ ；
兩有標圖G、H的不交並，記作 $G\oplus H$ ；
用邊連接標i的每個頂點與標j的每個頂點，記作 $\eta (i,\ j),\ i\neq j$ ；
將標籤i改為標籤j，記作 $\rho (i,\ j)$

團寬有界圖包括余圖與距離遺傳圖。在無界時計算團寬是NP困難的，而有界時能否在多項式時間內計算團寬也是未知的，不過已經有一些高效的團寬近似算法。基於這些算法與古賽爾定理，很多對任意圖來說NP難的圖優化問題都可在團寬有界圖上快速求解或逼近。

Courcelle、Engelfriet、Rozenberg在1990年^[1]、Wanke (1994)提出了作為團寬概念基礎的構造序列。「團寬」一詞始見於Chlebíková (1992)，是用於另一個概念。1993年，這個詞有了現在的含義。^[2]

特殊圖類

余圖是團寬不大於2的圖。^[3]距離遺傳圖的團寬不大於3。單位區間圖的團寬無界（基於網格結構）。^[4] 相似地，二分置換圖團寬無界（基於相似的網格結構）。^[5] 余圖是沒有任何導出子圖與4頂點路徑同構的圖，根據這特徵，對許多由禁導出子圖定義的圖類的團寬進行了分類。^[6]

其他團寬有界圖如k值有界的k葉冪，它們是樹T的葉在冪 $T^{k}$ 中的導出子圖。然而，指數無界的葉冪不具有有界團寬。^[7]

界

Courcelle & Olariu (2000)、Corneil & Rotics (2005)證明，特定圖的團寬有下列邊界：

若圖的團寬不超過k，則所有導出子圖也不超過k。^[8]
k團寬圖的補圖的團寬不大於2k。^[9]
w樹寬圖的團寬不大於 $3\cdot 2^{w-1}$ 。此約束中的指數依賴是必須的：存在團寬比樹寬大指數級的圖。^[10]換一種說法，團寬有界圖可能有無界的樹寬，例如n頂點完全圖的團寬為2，而樹寬為 $n-1$ 。而沒有完全二分圖 $K_{t,\ t}$ 為子圖的k團寬圖，樹寬不大於 $3k(t-1)-1$ 。因此，所有稀疏圖族，樹寬有界等價於團寬有界。^[11]
秩寬的上下界都受團寬約束： ${\text{秩宽}}\leq {\text{团宽}}\leq 2^{{\text{秩宽}}+1}$ 。^[12]}}

另外，若圖G的團寬為k，則圖的次冪 $G^{c}$ 的團寬不大於 $2kc^{k}$ 。^[13]從樹寬得出的團寬約束與圖冪的團寬約束存在指數級差距，不過約束並不互相複合：若圖G的樹寬為w，則 $G^{c}$ 的團寬不大於 $2(c+1)^{w+1}-2$ ，只是樹寬的單指數級。^[14]

計算複雜度

未解決的數學問題：能否在多項式時間內識別團寬有界圖？

很多對一般圖來說NP困難的優化問題，限定了團寬有界、已知圖的構造序列的條件後可用動態規劃高效解決。^[15]^[16]特別是，根據古賽爾定理的一種形式，可用MSO₁一元二階邏輯（允許量化頂點集的邏輯形式）表達的圖屬性，對團寬有界圖都有線性時間算法。^[16]

已知構造序列時，也有可能在多項式時間內為團寬有界圖找到最優圖着色或哈密頓環，但多項式的指數隨團寬增加，計算複雜度理論的證據表明這種依賴可能是必要的。^[17] 團寬有界圖是χ-有界的，這是說它們的色數最多是其最大團大小的函數。^[18]

運用基於裂分解的算法，可在多項式時間內識別出團寬為3的圖，並找到構造序列。^[19] 對團寬無界圖，精確計算團寬是NP困難的，獲得亞線性加性誤差的近似也是NP困難的。^[20]而團寬有界時，就可能在多項式時間內^[21]（特別是在頂點數的平方時間內^[22]）獲得寬度（比實際團寬大指數級）有界的構造序列。能否在固定參數可解時間內算得確切團寬或更優的近似值、能否在多項式時間內算得團寬的每個固定邊界、能否在多項式時間內識別團寬為4的圖，目前仍是未知的。^[20]

註釋

^ Courcelle, Engelfriet & Rozenberg (1993).
^ Courcelle (1993).
^ Courcelle & Olariu (2000).
^ Golumbic & Rotics (2000).
^ Brandstädt & Lozin (2003).
^ Brandstädt et al. (2005); Brandstädt et al. (2006).
^ Brandstädt & Hundt (2008); Gurski & Wanke (2009).
^ Courcelle & Olariu (2000), Corollary 3.3.
^ Courcelle & Olariu (2000), Theorem 4.1.
^ Corneil & Rotics (2005), strengthening Courcelle & Olariu (2000), Theorem 5.5.
^ ^11.0 ^11.1 Gurski & Wanke (2000).
^ Oum & Seymour (2006).
^ Todinca (2003).
^ Gurski & Wanke (2009).
^ Cogis & Thierry (2005).
^ ^16.0 ^16.1 Courcelle, Makowsky & Rotics (2000).
^ Fomin et al. (2010).
^ Dvořák & Král' (2012).
^ Corneil et al. (2012).
^ ^20.0 ^20.1 Fellows et al. (2009).
^ Oum & Seymour (2006); Hliněný & Oum (2008); Oum (2008); Fomin & Korhonen (2022).
^ Fomin & Korhonen (2022).
^ Gurski & Wanke (2007).
^ Bonnet et al. (2022).

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[FOOTNOTECourcelleEngelfrietRozenberg1993-1] Courcelle, Engelfriet & Rozenberg (1993).

[FOOTNOTECourcelle1993-2] Courcelle (1993).

[FOOTNOTECourcelleOlariu2000-3] Courcelle & Olariu (2000).

[FOOTNOTEGolumbicRotics2000-4] Golumbic & Rotics (2000).

[FOOTNOTEBrandstädtLozin2003-5] Brandstädt & Lozin (2003).

[6] Brandstädt et al. (2005); Brandstädt et al. (2006).

[7] Brandstädt & Hundt (2008); Gurski & Wanke (2009).

[8] Courcelle & Olariu (2000), Corollary 3.3.

[9] Courcelle & Olariu (2000), Theorem 4.1.

[10] Corneil & Rotics (2005), strengthening Courcelle & Olariu (2000), Theorem 5.5.

[FOOTNOTEGurskiWanke2000-11] 11.0 ^11.1 Gurski & Wanke (2000).

[FOOTNOTEOumSeymour2006-12] Oum & Seymour (2006).

[FOOTNOTETodinca2003-13] Todinca (2003).

[FOOTNOTEGurskiWanke2009-14] Gurski & Wanke (2009).

[FOOTNOTECogisThierry2005-15] Cogis & Thierry (2005).

[FOOTNOTECourcelleMakowskyRotics2000-16] 16.0 ^16.1 Courcelle, Makowsky & Rotics (2000).

[FOOTNOTEFominGolovachLokshtanovSaurabh2010-17] Fomin et al. (2010).

[FOOTNOTEDvořákKrál'2012-18] Dvořák & Král' (2012).

[FOOTNOTECorneilHabibLanlignelReed2012-19] Corneil et al. (2012).

[FOOTNOTEFellowsRosamondRoticsSzeider2009-20] 20.0 ^20.1 Fellows et al. (2009).

[21] Oum & Seymour (2006); Hliněný & Oum (2008); Oum (2008); Fomin & Korhonen (2022).

[FOOTNOTEFominKorhonen2022-22] Fomin & Korhonen (2022).

[FOOTNOTEGurskiWanke2007-23] Gurski & Wanke (2007).

[FOOTNOTEBonnetKimThomasséWatrigant2022-24] Bonnet et al. (2022).

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特殊圖類

界

計算複雜度

相關寬參數

註釋

參考文獻