A Most General Edge Elimination Polynomial - Thickening of Edges

A Most General Edge Elim ination P olynomial — Thic k ening of Edges Christ i an Hoffmann No vem ber 6, 201 8 Abstract W e consider a graph p olynomial ξ ( G ; x, y , z ) introd uced by Av erb ouch, Go dlin, and Mak o wsky (2007). T his graph p olynomial sim ultaneously gener- alizes the T utte p olynomial as w ell as a biv ariate c h romatic p olynomia l defin ed b y Dohmen, P¨ onitz and Tittmann (2003). W e d er ive an iden tit y wh ic h relates the graph p olynomial of a thic k ed graph (i.e. a graph with eac h edge replaced b y k copies of it) to the graph p olynomial of the original graph. As a conse- quence, we ob s erv e that at ev ery p oint ( x, y , z ), except for p oin ts lying within some set of d imension 2, ev aluating ξ is # P -hard. 1 In tr o duction W e consider the follo wing three-v ariable graph p olynomial whic h has b een introduced b y I. Av erb ouc h, B. Go dlin, and J. A. Mak o wsky [A G M07 ]: ξ ( G ; x, y , z ) = X ( A ⊔ B ) ⊆ E x k ( A ∪ B ) − k cov ( B ) · y | A | + | B |− k cov ( B ) · z k cov ( B ) , (1) where G = ( V , E ) is a graph with multiple edges and self lo ops allo w ed, A ⊔ B denotes a vertex-disjoint union of edge sets A and B , k ( A ∪ B ) is the n um b er of comp onen ts of ( V , A ∪ B ), and k cov ( B ) is the n um b er of comp onen ts of ( V ( B ) , B ). The p olynomial ξ sim ultaneously generalizes t w o interes ting graph p olynomials: the T utte p olynomial and a biv ariate c hromatic p olynomial P ( G ; x, y ) defined by K. D ohmen, A. P¨ onitz, and P . Tittmann [D PT03]. It is known tha t the T utte p olynomial of a graph with “ thic ked” edges ev aluated at some p oin t equals the T utte p olynomial of the original graph ev aluated at a nother 1 p oint (parallel edge reduction). This prop erty can b e used to prov e that at almost ev ery p o in t ev aluating the T utte p o lynomial is hard [JVW90, Sok05, BM06, BDM07]. In Section 2 of t his note w e observ e that edge thic k ening has a similar effect on ξ as on t he T utte p olynomial (Theorem 3). In Section 3 w e conclude that for ev ery p oint ( x, y , z ) ∈ Q 3 , except o n a set o f dimension at most 2, it is # P -hard to compute ξ ( G ; x, y , z ) from G (Theorem 5). This supp orts a difficult p oint conjecture for g raph p olynomials [Mak07, Conjecture 1], [AGM07, Question 1]. 2 A p oin t-to -p oin t reductio n from t hic k eni ng In this section we apply Sok al’s approach to ξ and obta in Lemma 2 (cf. [Sok05 , Section 4 .4]), the ma in tec hnical con tribution of this not e. W e define the following auxiliary p olynomial, whic h has a differen t y -v ariable for eac h edge of the g r a ph, ¯ y = ( y e ) e ∈ E ( G ) . ψ ( G ; x, ¯ y , z ) = X ( A ⊔ B ) ⊆ E ( G ) w ( G ; x, ¯ y , z ; A, B ) , (2) where w ( G ; x, ¯ y , z ; A, B ) = x k ( A ∪ B )  Y e ∈ ( A ∪ B ) y e ) z k cov ( B ) . W e write ψ ( G ; x, y , z ) for ψ ( G ; x, ¯ y , z ) if for each e ∈ E ( G ) w e hav e y e = y . Lemma 1. We have the p olynomial identities ψ ( G ; x, y , z x − 1 y − 1 ) = ξ ( G ; x, y , z ) and ξ ( G ; x, y , z xy ) = ψ ( G ; x, y , z ) . Let G b e a graph and e ∈ E ( G ) an edge. Let E ′ := E \ { e } and G ee b e the graph G with e doubled, i.e. G ee = ( V ( G ) , E ′ ∪ { e 1 , e 2 } ) with e 1 , e 2 b eing new edges. Lemma 2. ψ ( G ee ; x, ¯ y , z ) = ψ ( G ; x, ¯ Y , z ) with Y e = (1 + y e 1 )(1 + y e 2 ) − 1 and Y ˜ e = y ˜ e for al l ˜ e ∈ E ′ . Pr o of. Let M ( G ) = { ( A, B ) | A ⊔ B ⊆ E ( G ) } and M ( G ee ) = { ( ˜ A, ˜ B ) | ˜ A ⊔ ˜ B ⊆ E ( G ee } . W e define a map τ : M ( G ) → 2 M ( G ee ) in the fo llo wing wa y . Consider ( A, B ) ∈ M ( G ). If e 6∈ A ∪ B , w e set τ ( A, B ) = { ( A, B ) } . If e ∈ A , w e let A ′ := A \ { e } and define τ ( A, B ) = { ( A ′ ∪ { e 1 } , B ) , ( A ′ ∪ { e 2 } , B ) , ( A ′ ∪ { e 1 , e 2 } , B ) } . (Note that in this case e 6∈ B , as A and B are v ertex-disjoin t.) If e ∈ B , w e let B ′ := B \ { e } and define τ ( A, B ) = { ( A, B ′ ∪ { e 1 } ) , ( A, B ′ ∪ { e 2 } ) , ( A, B ′ ∪ { e 1 , e 2 } ) } . Observ e that M ( G ee ) = ∪ ( A,B ) ∈ M ( G ) τ ( A, B ) , (3) 2 and that this union is a union of p airwise disjoint sets. Calculation yields w ( G ; x, ¯ Y , z ; A, B ) = X ( ˜ A, ˜ B ) ∈ τ ( A,B ) w ( G ee , x, ¯ y , z ; ˜ A, ˜ B ) (4) for every ( A, B ) ∈ M ( G ). Thus , ψ ( G ee ; x, ¯ y , z ) = X ( ˜ A, ˜ B ) ∈ M ( G ee ) w ( G ee ; x, ¯ y , z ; ˜ A, ˜ B ) = X ( A,B ) ∈ M ( G ) X ( ˜ A, ˜ B ) ∈ τ ( A,B ) w ( G ee ; x, ¯ y , z ; ˜ A, ˜ B ) b y (3) = X ( A,B ) ∈ M ( G ) w ( G ; x, ¯ Y , z ; A, B ) b y (4) = ψ ( G ; x, ¯ Y , z ) . Applying Lemma 2 rep eatedly and Lemma 1 t o con v ert b et w een ψ and ξ we obtain Theorem 3. L et G k b e the k -thic k e ning of G (i.e. the gr aph ob tain e d out of G b y r eplacing e ach e dge b y k c opies of it). Then ψ ( G k ; x, y , z ) = ψ ( G ; x, (1 + y ) k − 1 , z ) , (5) ξ ( G k ; x, y , z ) = ξ  G ; x, (1 + y ) k − 1 , z (1 + y ) k − 1 y  . (6) 3 Hardness The following theorem has b een pro v en indep enden tly by I. Ave rb ouch (J. A. Mako wsky , p ersonal comm unication, Octob er 2007). Theorem 4. L et P denote the biv a riate chr omatic p olynomia l define d by K. Do h men, A. P¨ onitz, and P. Tittmann [DPT03]. F or every ( x, y ) ∈ Q , y 6 = 0 , ( x, y ) 6∈ { (1 , 1) , (2 , 2) } , it is # P -ha r d to c ompute P ( G ; x, y ) fr om G . 3 Pr o of (Sketch). Giv en a graph G = ( V , E ) let ˜ G denote the graph obtained o ut o f G b y inserting a new v ertex ˜ v and connecting ˜ v to a ll v ertices in V . Let P ( G ; y ) denote the c hromatic p olynomial [Rea68]. It is w ell kno wn that P ( ˜ G ; y ) = y P ( G ; y − 1 ) . (7) F rom this and [DPT03, Theorem 1] w e can derive P ( ˜ G ; x, y ) = y P ( G ; x − 1 , y − 1) + ( x − y ) P ( G ; x, y ) . (8) The pro of of the theorem now w or ks in the same fashion as a pro of that P ( G ; y ) is # P -hard to ev aluate almost ev erywhere using (7) w ould w ork: using (8) w e reduce along the lines x = y + d , whic h ev en tually enables us to ev aluate P a t (1 + d, 1) (if y is a p ositiv e in teger, w e reach (1 + d, 1 ) directly; otherwise w e obtain arbitrary man y p oints on the line x = y + d , whic h enables us to interpolate the p olynomial on this line). On the line y = 1 the p o lynomial P equ als the indep enden t set p olynomial [DPT03, Corollary 2], whic h is # P -hard to ev aluate almost ev erywhere [AM07, BH07]. Theorem 5. F or every ( x, y , z ) ∈ Q , x 6 = 0 , z 6 = − xy , ( x, z ) 6∈ { (1 , 0) , (2 , 0) } , y 6∈ {− 2 , − 1 , 0 } , the fol lowing statement holds true: It is # P -ha r d to c om pute ξ ( G ; x, y , z ) fr om G . Pr o of (Sketch). F or x, y ∈ Q , x, y 6 = 0 and ( x, y ) 6∈ { (1 , 1) , (2 , 2) } the follo wing problem is # P - hard by Theorem 4: Give n G , compute P ( G ; x, y ) = ξ ( G ; x, − 1 , x − y ) = ψ  G ; x, − 1 , y − x x  , where the first equality is by [AGM07, Pro p osition 18] and the second b y Lemma 1. W e will argue that, for any fixed ˜ y ∈ Q \ {− 2 , − 1 , 0 } , this reduce s to compute ψ ( G ; x, ˜ y , y − x x ) from G . W e hav e ψ  G ; x, ˜ y , y − x x  = ξ ( G ; x, ˜ y , ( y − x ) ˜ y ) b y Lemma 1. An easy calculation con verts the conditions on x, ˜ y , y into conditions on x, y , z and yields the statemen t of the theorem. No w assume that w e are able to ev aluate ψ a t some fixed ( x, y , z ) ∈ Q 3 , i.e. giv en G we can compute ψ ( G ; x, y , z ). Then Theorem 3 allows us to ev aluate ψ at ( x, y ′ , z ) for infinitely man y differen t y ′ = (1 + y ) k − 1 pro vided that | 1 + y | 6 = 0 and | 1 + y | 6 = 1. As ψ is a p olynomial, this enables in terp olation in y and ev entually giv es us the ability to ev aluate ψ at ( x, y ′ , z ) for any y ′ ∈ Q . In particular, b eing a ble to ev aluate ψ at ( x, ˜ y , y − x x ), ˜ y ∈ Q \ {− 2 , − 1 , 0 } , implies the abilit y to ev aluate it a t ( x, − 1 , y − x x ). 4 References [A GM07] Ilia Av erb ouc h, Benn y Go dlin, and J. A. Mak o wsky . A most general edge elimination p olynomial, 2007. Preprin t , arXiv:0712.311 2v1 (math.CO). [AM07] Ilia Av erb ouc h and J. A. Mak ows ky . The complexit y of m ultiv ariate matc hing po lynomials, F ebruary 2007. Preprint. [BDM07] M. Bl¨ aser, H. Dell, and J.A. Mako wsky . Comple xit y of the Bollob´ a s- Riordan p olynomial: Dic hotom y results and uniform reductions, 2007. Preprin t. [BH07] Markus Bl¨ aser a nd Christian Hoff ma nn. On the complexit y of the interlace p olynomial, 2007. Preprin t, arXiv:cs.CC/0707.4565. [BM06] Markus Bl¨ aser and Johann Mako wsky . Hip hip ho ora y for Sok al, 2 006. 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