Non-separable detachments of graphs

Journal of Combinatorial Theory, Series B 87 (2003) 17–37

http://www.elsevier.com/locate/jctb

Non-separable detachments of graphs Bill Jacksona and Tibor Jorda´nb,1 a

School of Mathematical Sciences, Queen Mary College, University of London, Mile End Road, London E1 4NS, UK b Department of Operations Research, Eo¨tvo¨s University, Pa´zma´ny Pe´ter se´ta´ny 1/C, 1117 Budapest, Hungary Received 13 June 2001 Dedicated to the memory of Crispin Nash-Williams

Abstract Let G ¼ ðV ; EÞ be a graph and r : V-Zþ : An r-detachment of G is a graph H obtained by ‘splitting’ each vertex vAV into rðvÞ vertices, called the pieces of v in H: Every edge uvAE corresponds to an edge of H connecting some piece of u to some piece of v: An r-degree specification for G is a function f on V ; such that, for each vertex vAV ; f ðvÞ is a partition of dðvÞ into rðvÞ positive integers. An f -detachment of G is an r-detachment H in which the degrees in H of the pieces of each vAV are given by f ðvÞ: Crispin Nash-Williams [3] obtained necessary and sufficient conditions for a graph to have a k-edge-connected r-detachment or f -detachment. We solve a problem posed by Nash-Williams in [2] by obtaining analogous results for non-separable detachments of graphs. r 2002 Elsevier Science (USA). All rights reserved.

1. Introduction All graphs considered are finite, undirected, and may contain loops and multiple edges. We shall use the term simple graph for graphs without loops or multiple edges. Let G ¼ ðV ; EÞ be a graph and r : V -Zþ : An r-detachment of G is a graph H obtained by ‘splitting’ each vertex vAV into rðvÞ vertices. The vertices v1 ; y; vrðvÞ obtained by splitting v are called the pieces of v in H: Every edge uvAE corresponds to an edge of H connecting some piece of u to some piece of v: An r-degree E-mail addresses: [email protected] (B. Jackson), [email protected] (T. Jorda´n). Supported by the Hungarian Scientific Research Fund Grant No. T029772, F034930, and FKFP Grant No. 0143/2001. 1

0095-8956/02/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved. doi:10.1016/S0095-8956(02)00026-6

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specification is a function f on V ; such that, for each vertex vAV ; f ðvÞ is a partition v of dðvÞ into rðvÞ positive integers, f1v ; f2v ; y; frðvÞ : An f -detachment of G is an r-detachment in which the degrees of the pieces of each vAV are given by f ðvÞ: Crispin Nash-Williams [3] obtained the following necessary and sufficient conditions for a graph to have a k-edge-connected r-detachment or f -detachment. For X ; Y disjoint subsets of V ðGÞ; let eðX ; Y Þ be the number of edges of G from X to Y ; eðX Þ the number of edges between the vertices of X ; bðX Þ the number of P components of G  X and rðX Þ ¼ xAX rðxÞ: When X or Y has only one element, we shall simplify the preceding notation by replacing the set by its element. Thus, for example, we shall write eðx; Y Þ instead of eðfxg; Y Þ: For vAV ; we use dðvÞ to denote the degree of v: Thus dðvÞ ¼ eðv; V  vÞ þ 2eðvÞ: Theorem 1.1 (Nash-Williams). Let G ¼ ðV ; EÞ be a graph and r : V -Zþ : Then G has a connected r-detachment if and only if eðX Þ þ eðX ; V  X ÞXrðX Þ þ bðX Þ  1 for every X DV : Furthermore, if G has a connected r-detachment then G has a connected f-detachment for every r-degree specification f. Theorem 1.2 (Nash-Williams). Let G ¼ ðV ; EÞ be a graph, r : V -Zþ ; and kX2 be an integer. Then G has a k-edge-connected r-detachment if and only if (a) (b) (c) (d)

G is k-edge-connected, dðvÞXkrðvÞ for each vAV ; and neither of the following statements is true: k is odd and G has a cut-vertex v such that dðvÞ ¼ 2k and rðvÞ ¼ 2; k is odd, jV j ¼ 2; jEj ¼ 2k; G is loopless, and rðvÞ ¼ 2 for each vertex vAV :

Furthermore, if G has a k-edge-connected r-detachment then G has a k-edge-connected f-detachment for any r-degree specification f for which fiv Xk for every vAV and every 1piprðvÞ: Let G be a graph. A vertex v is a cut-vertex of G if jEðGÞjX2 and either v is incident with a loop or G  v has more components than G: A graph is non-separable if it is connected and has no cut-vertices. Nash-Williams proposed the following problem in [2, p. 145]: It might also be worth looking at the question whether one can give necessary and sufficient conditions on a graph G and function r : V ðGÞ-Zþ for the existence of a non-separable r-detachment of G; i.e. an r-detachment of G which has no cutvertices—but of course it is not self-evident that a reasonable set of necessary and sufficient conditions for this must even exist. In this paper we answer this question by showing necessary and sufficient conditions for the existence of a non-separable r-detachment of a graph. We also solve the degree specified version. We shall need the following slight strengthening of Theorem 1.1.

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Theorem 1.3. Let G ¼ ðV ; EÞ be a graph, r : V -Zþ and V2 ¼ fvAV : rðvÞX2g: Then G has a connected r-detachment if and only if eðX Þ þ eðX ; V  X ÞXrðX Þ þ bðX Þ  1 for every X DV2 : Proof. Necessity follows from Theorem 1.1. To see sufficiency suppose that G does not have an r-detachment. By Theorem 1.1, eðX Þ þ eðX ; V  X ÞprðX Þ þ bðX Þ  2 for some X DV : If xAX  V2 ; then rðxÞ ¼ 1; and putting X 0 ¼ X  x we have eðX 0 Þ þ eðX 0 ; V  X 0 ÞprðX 0 Þ þ bðX 0 Þ  2: Hence we can construct X 00 DV2 with eðX 00 Þ þ eðX 00 ; V  X 00 ÞprðX 00 Þ þ bðX 00 Þ  2: &

2. Main results Let G be a graph and NðGÞ ¼ fvAV : dðvÞX4g: Given r : V -Zþ ; let N1 ðG; rÞ ¼ fvANðGÞ: rðvÞ ¼ 1g; and N2 ðG; rÞ ¼ fvANðGÞ: rðvÞX2g: Theorem 2.1. Let G ¼ ðV ; EÞ be a graph with at least two edges and r : V -Zþ : Then G has a non-separable r-detachment if and only if (a) (b) (c) (d)

G is 2-edge connected, dðvÞX2rðvÞ for all vAV ; eðvÞ ¼ 0 for all vAN1 ðG; rÞ; and eðX ; V  X  yÞ þ eðX ÞXrðX Þ þ bðX þ yÞ  1 X DN2 ðG; rÞ:

for

all

yAN1 ðG; rÞ

and

The degree specified version is as follows. Theorem 2.2. Let G ¼ ðV ; EÞ be a graph with at least two edges, r : V -Zþ ; and let f v v be an r-degree specification, where f ðvÞ ¼ ðf1v ; f2v ; y; frðvÞ Þ and f1v Xf2v X?XfrðvÞ ; for each vAV : Then G has a non-separable f-detachment if and only if (a) G is 2-edge connected, (b) fiv X2 for all vAV and all 1piprðvÞ; (c) eðX þ v; V  X  vÞ þ eðX þ vÞ  f1v XrðX þ vÞ þ bðX þ vÞ  2 for all vANðGÞ and X DN2 ðG; rÞ  v: Note that condition (c) of Theorem 2.2 implies that eðvÞ ¼ 0 for all vAN1 ðGÞ by taking X ¼ |: We shall need the following lemmas. Since loops create complications in notation, and since we only need the lemmas for loopless graphs, we add the hypothesis to the lemmas that the graphs are loopless. Note however that they may be applied to graphs with loops by subdividing their loops. Lemma 2.3. Let G be a 2-edge-connected loopless graph and vANðGÞ: Define r : V -Zþ by rðvÞ ¼ 2 and rðuÞ ¼ 1 for all uAV  v: Then there exists a

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2-edge-connected r-detachment H of G such that, at least one of the pieces of v in H, has degree two. Furthermore, if v is a cut-vertex of G, then there exists a 2-edgeconnected r-detachment H 0 of G such that, for the pieces v1 ; v2 of v in H 0 ; we have dH 0 ðv2 Þ ¼ bðvÞ; and neither v1 nor v2 is a cut-vertex of H 0 : Proof. The first part of the lemma is easy and well known (it follows for example from Theorem 1.2). To prove the second part, let C1 ; C2 ; y; Cb be the components of G  v; where b ¼ bðvÞ: Let H 0 be an r-detachment of G such that H 0 has exactly one edge from v2 to each component Ci : Since G is 2-edge-connected, there is also at least one edge from v1 to each Ci : To see that H 0 is 2-edge-connected, it suffices to show that H 0 has a cycle containing v1 ; v2 : (Since the 2-edge-connectivity of G implies that every cut-edge of H 0 must separate v1 and v2 :) We can construct such a cycle by choosing edges v1 xi ; v2 yi in H 0 with xi ; yi ACi and an xi yi -path in Ci for each iAf1; 2g: The fact that neither v1 nor v2 is a cut-vertex of H 0 follows easily from the construction of H 0 : & Let G be a graph. A block B of G is a non-separable subgraph of G which is maximal with respect to subgraph inclusion. We say that G is a block if G is a block of itself (or equivalently, if G is non-separable). A vertex vAV ðBÞ is an internal vertex of B (in G) if v is not a cut-vertex of G: An end-block of G is a block which contains at most one cut-vertex of G: Note that if G is separable then G has at least two end-blocks. We say that G is a uv-block-path if G is connected with exactly two end-blocks, B1 and B2 say, and u; v are internal vertices of B1 and B2 ; respectively, in G: For edges ux; vz in a graph G; we define the graph Gðux; vzÞ obtained by switching ux and vz by putting Gðux; vzÞ ¼ G  fux; vzg,fuz; vxg: Note that switching preserves the degree sequence of G: We shall use the following lemmas to determine when switching can be used to reduce the number of blocks in a detachment of G: Lemma 2.4. Let G be a loopless graph and ux; vzAEðGÞ such that ux; vz belong to vertex disjoint cycles in G. Suppose that G is either a block or a uv-block-path. Then Gðux; vzÞ is a block. Proof. Choose disjoint cycles C1 ; C2 containing ux; vz; respectively. Then ðC1  uxÞ,ðC2  vzÞ,fuz; vxg induces a cycle in Gðux; vzÞ containing u; x; v; z: Since every end-block of G  fux; vzg contains either u; x; v or z as an internal vertex, it follows that Gðux; vzÞ is a block. & We shall use the following rather technical lemmas to show that, if a graph G has an f -detachment with a unique cut-vertex yAN1 ðGÞ; then either G has a nonseparable f -detachment, or we can find a set X DN2 ðGÞ such that rðX Þ þ bðX þ yÞ is large. Let G be a graph, yAV ðGÞ; r : V -Zþ with rðyÞ ¼ 1; and f be an r-degree specification for G: Let H be an f -detachment of G; W DV ðHÞ; and u; vAV ðHÞ  W : We say that u and v are W -separated in H if u and v belong to different

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components of H  W : Define sequences of sets R1 ; R2 ; yDV ðGÞ; S1 ; S2 ; yDV ðHÞ; and W0 DW1 D?DV ðHÞ; recursively, as follows. Let W0 ¼ fyg; and, for iX1; let Ri ¼ fvAV ðGÞ: at least two pieces of v are Wi1 -separated in Hg; Si ¼ fvj AV ðHÞ: vj is a piece of some vARi g; and Wi ¼ Si ,Wi1 : It follows from these definitions that Si -Sj ¼ | ¼ Ri -Rj for iaj and Wi ¼ fyg,S1 ,S2 ,?,Si : Also note that Si ¼ | for all iX1 if y is not a cut-vertex of H: Lemma 2.5. Let H be a connected f-detachment of G. Let Z be a component of H  Wi1 for some iX1 and uv; wxAEðZÞ: Suppose Zðuv; wxÞ is connected. Then H 0 ¼ Hðuv; wxÞ is a connected f-detachment of G and Sm ðH 0 Þ ¼ Sm ðHÞ for all 1pmpi: Proof. Since H and H 0 have the same degree sequence, H 0 is an f -detachment of G: Furthermore H 0 is connected since H and Zðuv; wxÞ are connected. We shall show that Sm ðH 0 Þ ¼ Sm ðHÞ for all 1pmpi by induction on i: If i ¼ 1 then W0 ðH 0 Þ ¼ fyg ¼ W0 ðHÞ: Since Zðuv; wxÞ is connected, we have R1 ðHÞ ¼ R1 ðH 0 Þ and hence S1 ðHÞ ¼ S1 ðH 0 Þ: Suppose iX2: By induction, Sm ðH 0 Þ ¼ Sm ðHÞ for all 1pmpi  1: Note that Z is contained in a component Z 0 of H  Wm and Z0 ðuv; wxÞ is connected since Zðuv; wxÞ is connected. Thus Wi1 ðH 0 Þ ¼ Wi1 ðHÞ: Since Zðuv; wxÞ is connected, we have Ri ðHÞ ¼ Ri ðH 0 Þ and hence Si ðHÞ ¼ Si ðH 0 Þ: & Let H be a graph and y be a vertex of H and B be an end-block of H: We say that H is a block-star centered on y if every block of H contains y: We say that H is an extended block-star centered on y with distinguished end-block B if every end-block of H; with the possible exception of B; contains y: Note that every block-star is an extended block-star and every block is a block-star. An edge e of H is a cut-edge of H if H  e has more components than H: Lemma 2.6. Let G be a loopless graph, yAV ðGÞ; r : V -Zþ with rðyÞ ¼ 1; and f be an r-degree specification for G such that fiv X2 for all vAV ðGÞ and all 1piprðvÞ: Suppose that G has an f-detachment which is a block-star centered on y, and that H has been chosen amongst all such f-detachments so that bH ðyÞ is as small as possible. Then each edge of H  y incident to a vertex in Si is a cut-edge of H  y for all iX1: Proof. We proceed by contradiction. Suppose the lemma is false. Since the lemma is vacuously true if y is not a cut-vertex of H we have bH ðyÞX2: Choose an f -detachment K of G such that: (i) bK ðyÞ ¼ bH ðyÞ; (ii) K is an extended block-star centered on y with distinguished end-block B:

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(iii) for some edge vj xAEðB  yÞ and iX1 we have vj ASi and vj x is not a cut-edge of K  y; (iv) each edge of K  y which is incident to a vertex of Sm is a cut-edge of K  y for all 1pmpi  1; (v) subject to (i)–(iv), i is as small as possible. Note that K exists since if H is a counterexample to the lemma and we choose an edge which is not a cut-edge of H  y and is incident to a vertex of Si such that i is as small as possible, then H will satisfy (i)–(iv). Our proof technique forces us to work with extended block-stars rather than block-stars because the switching operations we use preserve the property of being an extended block-star, but may not preserve the property of being a block-star. Since vj x is not a cut-edge of K  y; vj x is contained in a cycle C of K  y: Let B1 ; B2 ; y; Bt be blocks of K such that yAV ðB1 Þ; V ðBs Þ-V ðBs0 Þa| if and only if js  s0 jp1; Bt ¼ B; and yeV ðB2 Þ if tX2: Then vj xAEðBt Þ and CDBt : Since vj ASi ; vj is a piece of some vertex vARi : Thus we may choose another piece vk of v such that vj and vk are Wi1 -separated in K: Claim 2.7. iX2: Proof. Suppose i ¼ 1: Then vj and vk are y-separated in K: Hence vj and vk belong to different end-blocks Bt and B0 of K: Choose an edge vk zAEðB0 Þ: Since K has minimum degree at least two, vk z is contained in a cycle C 0 in B0 : Since yeV ðCÞ; C and C 0 are vertex disjoint. Applying Lemma 2.4 to the block-path F ¼ B0 ,B1 ,?,Bt ; we deduce that F ðvj x; vk zÞ is a block. Thus H 0 ¼ Kðvj x; vk zÞ is an f -detachment of G which is a block-star centered on y; and bH 0 ðyÞ ¼ bH ðyÞ  1: This contradicts the hypothesis that bH ðyÞ is as small as possible. & Since vj ; vk eSi1 they are not Wi2 -separated in K: Thus they both belong to the same component Z of K  Wi2 : In particular vj ; vk are not y-separated in K so vk AV ðBs Þ for some s; 1pspt: By (iv), V ðCÞ-Wi2 ¼ |; and hence CDZ: Let P0 be a path from vk to C in Z: We may extend P0 around C if necessary to obtain a vk vj -path P in Z which avoids the edge vj x: Let vk z be the edge of P incident with vk : Since vj ; vk are Wi1 separated but not Wi2 -separated, we can choose uAV ðPÞ-Si1 : Let K 0 ¼ Kðvj x; vk zÞ: We shall show that K 0 contradicts the above choice of K: Claim 2.8. (a) K 0 is a connected f-detachment of G and Sj ðK 0 Þ ¼ Sj ðKÞ for all 1pmpi  1: (b) u and vj z are contained in a common cycle of K 0  y:

Proof. Part (a) follows from Lemma 2.5 since P½z; vj ,ðC  vj xÞ,fxvk g is a connected subgraph of Zðvj x; vk zÞ:

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Part (b) follows since P½z; vj ,fvj zg is a cycle in K 0  y containing u and vj z: & Let F1 ¼ B1 ,B2 ,?,Bt : Since uASi1 ; (iv) implies that each edge of K  y incident with u is a cut-edge of K  y: Thus there is exactly one edge from u to each component of F1  fy; ug: Let X be the component of F1  fy; ug which contains vj : Since u has a unique neighbour in X and u lies on the vj vk -path P in F1  y; vk eV ðX Þ: Since C is a cycle of K  y containing vj ; CDX : Let F2 be the graph obtained from F1 by adding a new edge yu and put F3 ¼ F2  X : Claim 2.9. F3 is a block. Proof. Suppose F3 is not a block. Since F1  y is connected and X is a component of F1  fy; ug; F3  y ¼ ðF1  X Þ  y is connected. Thus we can choose an end-block Bn of F3 such that yeV ðBn Þ: Then Bn is an end-block of K which does not contain y: Since CDX -Bt and Bn -X ¼ |; Bn aBt : This gives us a contradiction since (ii) implies that every end-block of K other than Bt must contain y: & Since F3 is a block, we can choose a cycle C 0 in F3 which contains vk z: Then C 0 is vertex disjoint from C since CDX : Furthermore, F2 is either a block or a block-path with distinct end-blocks F3 and Bt : Since vk zAEðF3 Þ-EðC 0 Þ and vj xAEðBt Þ-EðCÞ; Lemma 2.4 implies that F2 ðvj x; vk zÞ is a block. Thus F1 ðvj x; vk zÞ ¼ F2 ðvj x; vk zÞ  yu is either a block or a yu-block-path. Combining this with Claim 2.8, we deduce: Claim 2.10. K 0 ¼ Kðvj x; vk zÞ is an extended block-star, centered on y, with distinguished end-block B0 ; where B0 is the block of F1 ðvj x; vk zÞ which contains u. Furthermore bK 0 ðyÞ ¼ bK ðyÞ; u is an internal vertex of K 0 ; uASi1 ðK 0 Þ and u and vj z are contained in a common cycle of K 0  y which is contained in B0 : Choose u0 x0 AEðB0  yÞ such that u0 x0 is not a cut-edge of K 0  y; u0 ASp ðK 0 Þ for some pX1 and p is as small as possible. Then ppi  1 since uASi1 ðK 0 Þ-V ðB0 Þ: To show that K 0 contradicts our choice of K to minimise i; it only remains to show that (iv) holds for K 0 : Claim 2.11. Each edge of K 0  y which is incident to a vertex of Sm ðK 0 Þ is a cut-edge of K 0  y for all 1pmpp  1: Proof. Suppose the claim is false and let C n be a cycle of K 0  y which contains a vertex of Sm ðK 0 Þ for some m; 1pmpp  1: The minimality of p implies that C n contains no edge of B0 : Since vj zAEðB0 Þ by Claim 2.10, vj zeEðC n Þ: Since vk and x belong to different components of K  fy; ug; every path from vk to x in K  y contains u: Thus every cycle of K 0  y which contains vk x also contains either u or vj z and hence is contained in B0 : Thus vk xeEðC n Þ: Since vk x; vj zeEðC n Þ we have C n DK  y: Since Sm ðK 0 Þ ¼ Sm ðKÞ by Claim 2.8(a), the existence of C n now contradicts condition (iv) in the choice of K: &

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This completes the proof of the lemma. & To simplify notation, we shall first prove Theorem 2.1 for the special case when G is loopless and NðGÞ is an independent set of vertices in G: The general case follows easily from this special case by the simple procedure of subdividing every edge of G and extending r by putting rðvÞ ¼ 1 for each subdivision vertex v: Thus we shall prove: Theorem 2.12. Let G ¼ ðV ; EÞ be a loopless graph with at least two edges and r : V -Zþ : Suppose that NðGÞ is an independent set of vertices in G. Then G has a nonseparable r-detachment if and only if (a) G is 2-edge connected, (b) dðvÞX2rðvÞ for all vAV ; and (c) eðX ; V  X ÞXrðX Þ þ bðX þ yÞ  1 for all yAN1 ðG; rÞ and X DN2 ðG; rÞ:

Proof. We first prove necessity. Suppose H is a non-separable r-detachment of G: Then H is 2-edge-connected and since ‘detaching’ vertices cannot increase edgeconnectivity, (a) holds. Since H has minimum degree at least two we also have (b). Condition (c) follows from the easy part of Theorem 1.1, since H  y is a connected rjV y -detachment of G  y: We next prove sufficiency. We proceed by contradiction. Suppose that the theorem is false and choose a counterexample ðG; rÞ such that X gðG; rÞ :¼ jN1 ðG; rÞj þ ðdðvÞ  3Þ vANðGÞ

is as small as possible, and, subject to this condition, jV ðGÞj is as small as possible. If NðGÞ ¼ | then G has maximum degree at most three and, by (b), rðvÞ ¼ 1 for all vAV ðGÞ: Using (a) we deduce that G is a non-separable r-detachment of itself. Hence we may suppose that NðGÞa| and hence gðG; rÞX1: Claim 2.13. Suppose that UCV such that eðU; V  UÞ ¼ 2: Then either jUj ¼ 1 or jV  Uj ¼ 1: Proof. Suppose jUjX2 and jV  UjX2: Let U1 ¼ U and U2 ¼ V  U: For iAf1; 2g; let Gi be the graph obtained from G by contracting Ui to a single vertex ui of degree two. Define ri : V ðGi Þ-Zþ by putting ri ðui Þ ¼ 1 and ri ðvÞ ¼ rðvÞ for vAV ðGi Þ  ui : Then gðGi ; ri ÞpgðG; rÞ and jV ðGi ÞjojV ðGÞj: Since contraction preserves edge-connectivity, Gi satisfies (a). Clearly ðGi ; ri Þ also satisfies (b). Suppose ðGi ; ri Þ does not satisfy (c). Then eGi ðX ; V ðGi Þ  X Þpri ðX Þ þ bGi ðX þ yÞ  2 for some yAN1 ðGi ; ri Þ and X DN2 ðGi ; ri Þ: Since ui eNðGi Þ; ui belongs to some component of Gi  ðX þ yÞ; and X þ yDV ðGÞ: Thus X ; y contradict the fact that (c) holds for G: Hence (c) holds for ðGi ; ri Þ and, by induction, Gi has a non-separable

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ri -detachment for i ¼ 1; 2: Since eðU; V  UÞ ¼ 2; this implies that G has a nonseparable r-detachment. & Claim 2.14. N1 ðG; rÞa|: Proof. Suppose N1 ðG; rÞ ¼ |: Choose vAN2 ðG; rÞ such that rðvÞ is as large as possible and dðvÞ is as small as possible. Define rv : V ðGÞ-Zþ by rv ðvÞ ¼ 2 and rv ðuÞ ¼ 1 for all uAV  v: By Lemma 2.3, we can construct a 2-edge-connected rv detachment H of G such that, for the pieces v1 ; v2 of v in H; we have dH ðv2 Þ ¼ 2: Define r0 : V ðHÞ-Zþ by r0 ðv1 Þ ¼ rðvÞ  1; r0 ðv2 Þ ¼ 1; and r0 ðuÞ ¼ rðuÞ for all uAV ðHÞ  fv1 ; v2 g: Then gðH; r0 ÞogðG; rÞ: By construction ðH; r0 Þ satisfies (a) and (b). If ðH; r0 Þ also satisfies (c), then, by induction H has a non-separable r0 detachment H 0 : Clearly, H 0 is the required r-detachment of G: Hence eH ðX ; V ðHÞ  X Þpr0 ðX Þ þ bH ðX þ yÞ  2

ð1Þ

for some yAN1 ðH; r0 Þ and X DN2 ðH; r0 Þ: Since N1 ðG; rÞ ¼ | and dH ðv2 Þ ¼ 2; we must have y ¼ v1 ; and r0 ðv1 Þ ¼ 1: Thus rðvÞ ¼ 2: The choice of v now implies rðuÞ ¼ 2

for all uANðGÞ:

ð2Þ

In particular r0 ðX Þ ¼ rðX Þ ¼ 2jX j: The choice of v also implies that dH ðuÞ ¼ dG ðuÞXdG ðvÞ ¼ dH ðv1 Þ þ 2 for all uANðGÞ  v: Thus eH ðX ; V ðHÞ  X ÞXjX jðdH ðv1 Þ þ 2Þ: Since H is 2-edge-connected, each component of H  ðX þ v1 Þ has at least two edges to X þ v1 : Thus 2bH ðX þ v1 ÞpeH ðX ; V ðHÞ  X Þ þ dH ðv1 Þ: Substituting these inequalities into (1), we deduce that dH ðv1 ÞðjX j  1Þp2ðjX j  2Þ: Hence X ¼ |: Now (1) implies that bH ðv1 ÞX2: Thus v1 is a cut-vertex of H and hence v is a cut-vertex of G: By Lemma 2.3, we can construct a 2-edge-connected rv -detachment H 0 of G such that, for the pieces v01 ; v02 of v in H 0 ; we have dH 0 ðv02 Þ ¼ bG ðvÞ; and neither v01 nor v02 is a cut-vertex of H 0 : Defining r0 as above (i.e. r0 ðv01 Þ ¼ rðvÞ  1 ¼ 1; r0 ðv02 Þ ¼ 1; and r0 ðuÞ ¼ rðuÞ for all uAV ðH 0 Þ  fv01 ; v02 g) we have gðH 0 ; r0 ÞogðG; rÞ; and ðH 0 ; r0 Þ satisfies (a) and (b). Thus we may again deduce that ðH 0 ; r0 Þ fails to satisfy (c). Hence eH 0 ðX ; V ðH 0 Þ  X Þpr0 ðX Þ þ bH 0 ðX þ yÞ  2 for some yAN1 ðH 0 ; r0 Þ and X DN2 ðH 0 ; r0 Þ: Since N1 ðG; rÞ ¼ | we must have y ¼ v0i for some iAf1; 2g: Using (2), we may now deduce as above that X ¼ | and hence v0i is a cut-vertex of H 0 : This contradicts the choice of H 0 : & Choose yAN1 ðG; rÞ such that dðyÞ is as large as possible. Define ry : V ðGÞ-Zþ by ry ðyÞ ¼ 2 and ry ðuÞ ¼ rðuÞ for all uAV  y: Clearly gðG; ry ÞogðG; rÞ; and ðG; ry Þ satisfies (a) and (b). Claim 2.15. ðG; ry Þ satisfies ðcÞ: Proof. Suppose we have eG ðX ; V  X Þpry ðX Þ þ bG ðX þ y0 Þ  2 for some y0 AN1 ðG; ry Þ and X DN2 ðG; ry Þ: Since (c) holds for ðG; rÞ we must have

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yAX and eG ðX ; V  X Þ ¼ rðX Þ þ bG ðX þ y0 Þ  1:

ð3Þ

Let C1 ; C2 ; y; Cb be the components of G  ðX þ y0 Þ: Since (c) holds for ðG; rÞ; we may apply Theorem 1.3 to deduce that G  y0 has a connected r0 -detachment H; where r0 ¼ rjV y0 : Let X n be the set of all pieces of vertices of X in H: Since (b) holds for G; Theorem 1.1 implies that H may be constructed to have the additional property that dH ðxi ÞX2 for all xi AX n : Let H 0 be the detachment of G  y0 obtained from H by ‘re-attaching’ all the pieces of v; for each vAV  X  y0 : Thus H 0 is a connected r00 -detachment of G  y0 ; where r00 ðvÞ ¼ rðvÞ for vAX and r00 ðvÞ ¼ 1 for vAV  X  y0 : Using the fact that equality holds in (3), we have exactly rðX Þ þ b  1 edges in H 0 joining the vertices in X n and the components C1 ; C2 ; y; Cb : Since H 0 is connected and jX n j ¼ rðX Þ; the graph T obtained from H 0 by contracting each component Ci to a single vertex ci ; is a tree. Since dT ðxi ÞX2 for all xi AX n ; no vertex of X n is an end-vertex of T: Since r00 ðyÞ ¼ 1; we can label the unique piece of y in H as y: We then have yAX n and dT ðyÞ ¼ dG ðyÞ: Thus T has at least dG ðyÞ end-vertices, all of which belong to fc1 ; c2 ; y; cb g: Furthermore, if T has exactly dG ðyÞ endvertices, then all vertices of T other than y have degree one or two. Let S ¼ fCi : dT ðci Þ ¼ 1; 1pipbg: Then eH 0 ðCi ; V ðH 0 Þ  Ci Þ ¼ 1 for all Ci AS: Since G is 2-edgeconnected and r00 ðvÞ ¼ 1 for all vACi ; there is at least one edge in G from Ci to y0 for each Ci AS: Since jSjXdG ðyÞ; it follows that dG ðy0 ÞXdG ðyÞ: It now follows from the initial choice of y that we must have dG ðy0 Þ ¼ jSj ¼ dG ðyÞ; that there is exactly one edge in G from y0 to each Ci AS and to no other vertices of G; and that all vertices of T other than y have degree one or two. Again, since r00 ðvÞ ¼ 1 for all vACi ; we have eG ðCi ; V ðGÞ  Ci Þ ¼ 2 for all 1pipb: Using Claim 2.13, we deduce that jV ðCi Þj ¼ 1 for all 1pipb: Thus V ðCi Þ ¼ fci g; H 0 ¼ H and since G is loopless, dG ðci Þ ¼ 2: By (b), rðci Þ ¼ 1 ¼ r00 ðci Þ for all 1pipb and H ¼ T: Let G0 be the graph obtained from H by adding y0 and the edge y0 ci for each Ci AS: (Thus G0 is obtained by adding an edge from y0 to each end-vertex of T:) Then G0 is the required non-separable r-detachment of G: & Since (c) holds for ðG; ry Þ; we may apply induction to deduce that G has a nonseparable ry -detachment. It follows that G has an r-detachment H such that H is a block-star centered on y: We may suppose that H has been chosen such that the number of blocks of H is as small as possible. Let f be the r-degree specification for G given by H: Since G has no non-separable f -detachment, bH ðyÞX2: For iX0; let Siþ1 and Wi be the subsets of V ðHÞ defined as for Lemma 2.6. Since the sets Si are pairwise disjoint and H is finite, we may choose i such that Siþ1 ¼ |: Let X 0 ¼ Wi  y and X ¼ fxAV ðGÞ: some piece of x in H belongs to X 0 g: By Lemma 2.6, every edge x1 vAEðH  yÞ with x1 AX 0 is a cut-edge in H  y: Thus the graph F we get from H  y by contracting each component of H  X 0  y to a single vertex is a forest with bH ðyÞ components and bH ðX 0 þ yÞ þ jX 0 j vertices. Since X þ yDNðGÞ by construction, and NðGÞ is an independent set of vertices in G; we deduce that

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27

jEðF Þj ¼ eH ðX 0 ; V ðHÞ  X 0 Þ: Thus eH ðX 0 ; V ðHÞ  X 0 Þ ¼ bH ðX 0 þ yÞ þ jX 0 j  bH ðyÞ:

ð4Þ

We have eH ðX 0 ; V ðHÞ  X 0 Þ ¼ eG ðX ; V ðGÞ  X Þ; jX 0 j ¼ rðX Þ and bH ðyÞX2: Furthermore, for each vAV ðGÞ  X  y; all pieces of v in H belong to the same component of H  X 0  y; since Siþ1 ¼ |: Thus bG ðX þ yÞ ¼ bH ðX 0 þ yÞ: Substituting into (4) we obtain eG ðX ; V ðGÞ  X ÞprðX Þ þ bG ðX þ yÞ  2: This contradicts hypothesis (c) of the theorem and completes our proof. & Proof of Theorem 2.1. Let G 0 be obtained from G by subdividing every edge of G: Then G 0 is loopless, NðG 0 Þ ¼ NðGÞ and NðG0 Þ is independent in G 0 : Extend r to r0 by putting r0 ðvÞ ¼ rðvÞ for all vAV ðGÞ and r0 ðvÞ ¼ 1 for all vAV ðG 0 Þ  V ðGÞ: Then N1 ðG 0 ; r0 Þ ¼ N1 ðG; rÞ and N2 ðG 0 ; r0 Þ ¼ N2 ðG; rÞ: We shall show that conditions (a)– (d) of Theorem 2.1 hold for ðG; rÞ if and only if conditions (a)–(c) of Theorem 2.12 hold for ðG 0 ; r0 Þ: Clearly Theorem 2.1(a) and (b) hold for ðG; rÞ if and only if Theorem 2.12(a) and (b) hold for ðG 0 ; r0 Þ: Furthermore for yAN1 ðG; rÞ ¼ N1 ðG 0 ; r0 Þ and X DN2 ðG; rÞ ¼ N2 ðG 0 ; r0 Þ; we have rðX Þ ¼ r0 ðX Þ; and eG ðX ; V  X  yÞ þ eG ðX Þ  bG ðX þ yÞ  eG ðyÞ ¼ eG0 ðX ; V  X Þ  bG0 ðX þ yÞ: If Theorem 2.1(c) and (d) hold for ðG; rÞ; then eG ðyÞ ¼ 0 and the above equalities imply that Theorem 2.1(c) holds for ðG 0 ; r0 Þ: Suppose, on the other hand, that Theorem 2.12(c) holds for ðG 0 ; r0 Þ: Taking X ¼ | we have bG0 ðyÞp1 for all yAN1 ðG 0 ; r0 Þ and hence eG ðyÞ ¼ 0 for all yAN1 ðG; rÞ: Thus Theorem 2.1(c) holds for ðG; rÞ: The above equalities now imply that Theorem 2.1(d) also holds for ðG; rÞ: & We shall next prove Theorem 2.2. Given a graph G ¼ ðV ; EÞ and X DV ; let GðX Þ be the set of vertices of V  X which are adjacent to vertices in X and put gðX Þ ¼ jGðX Þj: We shall use the following operation to adjust the degree sequence in a detachment of a graph. For vertices x; y; z of G with xzAEðGÞ; we define the graph Gðxz-yzÞ obtained by flipping xz to yz by putting Gðxz-yzÞ ¼ G  xz þ yz: The following lemma characterises when we may flip edges in a non-separable graph and preserve non-separability. Lemma 2.16. Let G ¼ ðV ; EÞ be a non-separable graph and let x; yAV be distinct vertices of G. Let xz1 ; xz2 ; y; xzt be distinct edges of G  y with tX3: Then Gðxzi -yzi Þ is separable for all 1pipt if and only if there exist distinct components C1 ; C2 ; y; Ct of G  fx; yg with zi AV ðCi Þ and eðx; Ci Þ ¼ 1 for all 1pipt: Proof. Sufficiency is easy to see. To prove necessity first note that Gðxzi -yzi Þ has no loops for all 1pipt since G is non-separable and zi ay: Suppose that Gðxzi -yzi Þ has a cut-vertex for all 1pipt: It is easy to see that flipping an edge xzi to yzi creates a cut-vertex if and only if there is a set W CV  x in G with eðW ; xÞ ¼ 1; gðW Þ ¼ 2; and zi AW ; yAW ,GðW Þ: We call W a certificate for zi :

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Let us choose a minimal family F ¼ fW1 ; y; Wm g which contains a certificate for zi for all 1pipt: Since eðWi ; xÞ ¼ 1 for 1pipm and by the minimality of F we have t ¼ m and GðxÞ-Wi -Wj ¼ | for all 1piojpt: Furthermore, since eðWi ; xÞ ¼ 1 and G is non-separable, each Wi induces a connected subgraph of G: First we show that Wi -Wj ¼ | for 1piojpt: Suppose that Wi -Wj a| holds for two distinct Wi ; Wj AF: Since eðWi ; xÞ ¼ eðWj ; xÞ ¼ 1 and dðxÞX3; it follows that Z :¼ V  ðWi ,Wj Þ  fxg is non-empty. The subgraphs G½Wi  and G½Wj  are connected, hence we have that both GðWi Þ-Wj and GðWj Þ-Wi are non-empty. Since gðWi Þ ¼ gðWj Þ ¼ 2 and xAGðWi Þ-GðWj Þ; this implies that eðWi ,Wj ; ZÞ ¼ 0: Hence x is a cut-vertex in G; a contradiction. Now suppose that yAWi holds for some 1pipt: Since Wi -Wj ¼ |; we must have yAGðWj Þ for all Wj AF  Wi : Since xAGðWi Þ and tX3; this implies gðWi ÞX3; a contradiction. Thus yAGðWi Þ for all 1pipt: This, and the facts that gðWj Þ ¼ 2 and xAGðWj Þ for all 1pjpt; imply that Ci ¼ G½Wi ; 1pipt; are the required components of G  fx; yg: & Corollary 2.17. Let G ¼ ðV ; EÞ be a non-separable graph and let x; yAV be distinct vertices of G. Let xz1 ; xz2 ; y; xzt be distinct edges of G  y with tX3: If tXdðyÞ  eðfx; ygÞ þ 1; then Gðxzi -yzi Þ is non-separable for some 1pipt: Proof. Suppose that for all 1pipt the graph Gðxzi -yzi Þ is separable. We may apply Lemma 2.16 and deduce that bðfx; ygÞXt: Since tXdðyÞ  eðfx; ygÞ þ 1; it follows that eðC; yÞ ¼ 0 for some component C of G  fx; yg: Thus x is a cut-vertex in G; a contradiction. & Corollary 2.18. Let G ¼ ðV ; EÞ be a non-separable graph and let x; y; wAV be distinct vertices of G such that dðxÞX3 and xy; xweE: Then there exists a zAGðxÞ such that either Gðxz-yzÞ or Gðxz-wzÞ is non-separable. Proof. Suppose that for all zAGðxÞ the graph Gðxz-yzÞ is separable. By Lemma 2.16 we have bðfx; ygÞ ¼ dðxÞ: Let C1 ; C2 ; y; CdðxÞ be the components of G  fx; yg; where wAV ðC1 Þ: Then each neighbour of x other than the unique neighbour in C1 belongs to the same component of G  fx; wg: Thus Lemma 2.16 implies that Gðxz-wzÞ is non-separable for some zAGðxÞ: & Lemma 2.19. Let G ¼ ðV ; EÞ be a non-separable graph and suppose that bðfx; ygÞ ¼ dðxÞX3 for some pair x; yAV : Let w be a vertex in some component C of G  fx; yg with eðw; yÞ ¼ eðw; xÞ ¼ 0 and let zAGðxÞ  C: Then either Gðxz-wzÞðwz0 -yz0 Þ is non-separable for some z0 AGðwÞ or every edge incident to w in G is a cut-edge in C. Proof. Let the components of G  fx; yg be C1 ; C2 ; y; CdðxÞ and, without loss of generality, suppose that zAV ðC1 Þ: Then C1 aC: We first observe that H :¼ Gðxz-wzÞ is non-separable. This follows from the fact that, since dðxÞX3; there is a

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cycle containing x and z in H: Thus we need to show that either there is flip from w to y in H which creates no cut-vertices or C has the required property. Suppose that Hðwz0 -yz0 Þ has a cut-vertex for every z0 AGðwÞ-V ðCÞ: Using the facts that dH ðwÞX3; eH ðw; yÞ ¼ 0; and y is a cut-vertex in Hðwz-yzÞ; we may apply Lemma 2.16 to deduce that bH ðfw; ygÞ ¼ dH ðwÞ ¼ dG ðwÞ þ 1: This implies that every edge incident to w in H is a cut-edge in H  y: Since zeV ðCÞ and H is obtained from G by flipping xz to wz; and yeV ðCÞ; we may deduce that every edge incident to w in G is a cut-edge in C: This proves the lemma. & We next apply the above results to obtain some preliminary results on f -detachments. Lemma 2.20. Let G ¼ ðV ; EÞ be a loopless graph and r : V ðGÞ-Zþ : Suppose that G has a non-separable r-detachment H. Let f be the r-degree specification for G given by v f1v ¼ dðvÞ if rðvÞ ¼ 1; and otherwise f ðvÞ ¼ ðf1v ; f2v ; y; frðvÞ Þ where f1v ¼ JdðvÞ=2n  v v rðvÞ þ 2; f2 ¼ IdðvÞ=2m  rðvÞ þ 2; and fi ¼ 2 for all vAN2 ðG; rÞ and 3piprðvÞ: Then G has a non-separable f-detachment. Proof. For each vAV ; let v1 ; v2 ; y; vrðvÞ be the pieces of v in H; where dH ðv1 ÞXdH ðv2 ÞX?XdH ðvrðvÞ Þ: Note that the pieces of v are independent in H since G is loopless, and that dG ðvÞX2rðvÞ since H is non-separable. Suppose dH ðvi ÞX3 for some 3piprðvÞ: Using Corollary 2.18, we can construct a new non-separable r-detachment by flipping an edge vi z to either v1 z or v2 z for some zAGH ðvi Þ: Applying this operation iteratively, and relabelling v1 and v2 if necessary, we may construct a non-separable r-detachment H 0 of G on the same vertex set as H with dH 0 ðv1 ÞXdH 0 ðv2 Þ and dH 0 ðvi Þ ¼ 2 for all vAN2 ðG; rÞ and ip3prðvÞ: Suppose dH 0 ðv1 ÞXdH 0 ðv2 Þ þ 2: By Corollary 2.17, we can construct a new nonseparable r-detachment by flipping an edge v1 z to v2 z for some zAGH ðv1 Þ: Applying this operation iteratively to H 0 we may construct the required non-separable f -detachment G: & Let S be the set of all sequences of integers of length r in decreasing order of magnitude. Let Xr be the lexicographic ordering on S (hence f ¼ ðf1 ; f2 ; y; fr ÞXr g ¼ ðg1 ; g2 ; y; gr Þ if and only if either f ¼ g; or, for some 1pipr; f1 ¼ g1 ; f2 ¼ g2 ; y; fi1 ¼ gi1 ; fi 4gi ). Let G ¼ ðV ; EÞ be a graph, r : V ðGÞ-Zþ ; and f be an r-degree specification for G: We shall say that f is ordered if f ðvÞ is a decreasing sequence for each vAV : Lemma 2.21. Let G ¼ ðV ; EÞ be a loopless graph, r : V ðGÞ-Zþ ; and f and g be two ordered r-degree specifications for G. Suppose that G has a non-separable f-detachment, f ðvÞXrðvÞ gðvÞ and gvi X2 for all vAV and 1piprðvÞ: Then G has a non-separable g-detachment.

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PrðvÞ v PrðvÞ v Proof. Note that i¼1 fi ¼ dG ðvÞ ¼ i¼1 gi for all vAV : Given an integer z let zn ¼ z if zX0; and otherwise let zn ¼ 0: We may suppose that f has been chosen to satisfy the hypotheses of the lemma and such that: P P v v n (i) y1 ðf Þ ¼ vAV 3piprðvÞ ðfi  gi Þ is as small as possible; P P (ii) subject to (i), y2 ðf Þ ¼ vAV ðgv  f v Þn is as small as possible; P3piprðvÞv i v i (iii) subject to (i) and (ii), y3 ðf Þ ¼ vAV ðf1  g1 Þ is as small as possible. Let H be a non-separable f -detachment of G: Let v1 ; v2 ; y; vrðvÞ be the pieces of v in H; where dH ðvi Þ ¼ fiv for all vAV and all 1piprðvÞ: Suppose y1 ðf ÞX1: Then we may choose vAV such that fiv Xgvi þ 1 for some 3piprðvÞ and i is as large as possible. Then fiv X3 and by Corollary 2.18 there exists a zAGH ðvi Þ such that either Hðvi z-v1 zÞ or Hðvi z-v2 zÞ is non-separable. In both cases, the resulting non-separable graph H 0 is an h-detachment of G for a unique ordered r-degree specification h: When H 0 ¼ Hðvi z-v1 zÞ or f1v ¼ f2v then the maximality of i implies that hv1 ¼ f1v þ 1; hvi ¼ fiv  1; and hvj ¼ fjv for all 1pjprðvÞ with ja1; i: Otherwise the maximality of i implies that hv2 ¼ f2v þ 1; hvi ¼ fiv  1; and hvj ¼ fjv for all 1pjprðvÞ with ja2; i: Both alternatives give hðuÞXrðuÞ gðuÞ for all uAV and y1 ðhÞ ¼ y1 ðf Þ  1: This contradicts the choice of f : Thus y1 ðf Þ ¼ 0 and fiv pgvi for all vAV and all 3piprðvÞ: Suppose y2 ðf ÞX1: Then we may choose vAV such that gvi Xfiv þ 1 for some 3piprðvÞ and i is as small as possible. Consider the following cases: Case 1: f1v Xgv1 þ 1: Then f1v 4fiv and by Corollary 2.17 there exists a zAGH ðv1 Þ such that H 0 ¼ Hðv1 z-vi zÞ is non-separable. Then H 0 is an h-detachment of G for a unique ordered r-degree specification h: v Subcase 1.1: fi1 Xfiv þ 1: If f1v 4f2v then hv1 ¼ f1v  1; hvi ¼ fiv þ 1; and hvj ¼ fjv for all 1pjprðvÞ with ja1; i: If f1v ¼ f2v then hv2 ¼ f2v  1; hvi ¼ fiv þ 1; and hvj ¼ fjv for all 1pjprðvÞ with ja2; i: In both cases we have hðuÞXrðuÞ gðuÞ for all uAV ; y1 ðhÞ ¼ 0; and y2 ðhÞ ¼ y2 ðf Þ  1: (Note that hðvÞXrðvÞ gðvÞ since y1 ðf Þ ¼ 0 and hence either hv1 4gv1 ; or else hv1 ¼ gv1 and hv2 Xgv2 with equality only if hðvÞ ¼ gðvÞ:) This contradicts the choice of f : v ¼ fiv : The choice of i implies that i ¼ 3: Since f ðvÞXrðvÞ gðvÞ and Subcase 1.2: fi1 v v g3 Xf3 þ 1 we must have f1v 4f2v : Thus hv1 ¼ f1v  1; hv2 ¼ f2v þ 1; and hvj ¼ fjv for all 3pjprðvÞ: Thus hðuÞXrðuÞ gðuÞ for all uAV ; y1 ðhÞ ¼ 0; y2 ðhÞ ¼ y2 ðf Þ; and y3 ðhÞ ¼ y3 ðf Þ  1: (Note that hðvÞXrðvÞ gðvÞ since y1 ðf Þ ¼ 0 and f2v ¼ f3v pgv3  1pgv2  1 imply that hv1 ¼ f1v  14gv1 :) This contradicts the choice of f : Case 2: f1v pgv1 : Since f ðvÞXrðvÞ gðvÞ; y1 ðvÞ ¼ 0 and y2 ðvÞX1; we must have f1v ¼ gv1 and f2v Xgv2 þ 1: We can now obtain a contradiction as in Subcase 1.1 by using Corollary 2.17 to show that there exists a zAGH ðv2 Þ such that H 0 ¼ Hðv2 z-vi zÞ is v non-separable. (Note that in this case we must have fi1 Xfiv þ 1:) Since all the above cases yield a contradiction we have y2 ðf Þ ¼ 0 ¼ y1 ðf Þ and hence fiv ¼ gvi for all vAV and all 3piprðvÞ:

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Suppose y3 ðf ÞX1: Then we may choose vAV such that f1v Xgv1 þ 1: Then and by Corollary 2.17 there exists a zAGH ðv1 Þ such that Hðv1 z-v2 zÞ is non-separable. The resulting graph H 0 is a non-separable h-detachment of G for a unique ordered r-degree specification h with hðuÞXrðuÞ gðuÞ for all uAV ; y1 ðhÞ ¼ 0 ¼ y2 ðhÞ; and y3 ðhÞ ¼ y3 ðf Þ  1: This contradicts the choice of f and hence y3 ðf Þ ¼ 0: Thus f ¼ g and the lemma is trivially true. &

f1v 4gv2 4f2v

We shall need one more lemma which is an extension of Lemma 2.3. Lemma 2.22. Let G be a 2-edge-connected loopless graph, vANðGÞ; and r : V -Zþ be such that rðvÞX2 and rðuÞ ¼ 1 for all uAV  v: Suppose that v is a cut-vertex of G and that bðvÞXrðvÞ: Then there exists a 2-edge-connected r-detachment H of G such that, for the pieces v1 ; v2 ; y; vrðvÞ of v in H, we have dH ðv2 Þ ¼ bG ðvÞ  rðvÞ þ 2; dH ðvi Þ ¼ 2 for 3piprðvÞ and neither v1 nor v2 is a cut-vertex of H. Proof. We use induction on rðvÞ: If rðvÞ ¼ 2 then the lemma follows from Lemma 2.3. Hence suppose rðvÞX3 and choose x; yAGðvÞ belonging to different components of G  v: Let G 0 be the graph obtained from G by detaching v into two vertices v0 and v00 ; where dG0 ðv00 Þ ¼ 2 and GG0 ðv00 Þ ¼ fx; yg: It can be seen that G0 is 2-edge connected and bG0 ðv0 Þ ¼ bG ðvÞ  1: The lemma now follows by applying induction to ðG 0 ; v0 ; r0 Þ where r0 ðv0 Þ ¼ rðvÞ  1: & As for Theorem 2.1, we first prove Theorem 2.2 in the case when G is loopless and NðGÞ is independent. Theorem 2.23. Let G ¼ ðV ; EÞ be a loopless graph with at least two edges, v r : V ðGÞ-Zþ ; and let f be an r-degree specification, where f ðvÞ ¼ ðf1v ; f2v ; y; frðvÞ Þ v v v and f1 Xf2 X?XfrðvÞ for all vAV : Suppose that NðGÞ is independent. Then G has a non-separable f-detachment if and only if (a) G is 2-edge connected, (b) fiv X2 for all vAV and all 1piprðvÞ; (c) eðX þ v; V  X  vÞ  f1v XrðX þ vÞ þ bðX þ vÞ  2 for all vANðGÞ and all X DN2 ðG; rÞ  v: Proof. We first prove necessity. Suppose that G has a non-separable f -detachment H: It is easy to see that conditions (a) and (b) must hold for G: Choose vANðGÞ and X DN2 ðG; rÞ  v: Let C1 ; C2 ; y; Cb be the components of G  ðX þ vÞ; where b ¼ bðX þ vÞ; and let Ci0 be the subgraph of H induced by the pieces of all vertices of Ci : Let v1 be the piece of v in H of degree f1v : Let S be the set of all pieces of vertices of X in H and all pieces of v other than v1 : Then jSj ¼ rðX þ vÞ  1: Since G is loopless and NðGÞ is independent, we have eðX þ v; V  X  vÞ  f1v edges in H joining the subgraphs C10 ; C20 ; y; Cb0 and vertices in S: Since H is non-separable H  v1 is

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connected and hence we must have eðX þ v; V  X  vÞ  f1v Xb þ rðX þ vÞ  2: Thus (c) holds for G: We next prove sufficiency. We proceed by contradiction. Suppose that ðG; r; f Þ satisfies (a)–(c) and that G does not have a non-separable f -detachment. We first use Theorem 2.12 to show that G has a non-separable r-detachment. Since ðG; rÞ satisfies (a) and (b), it also satisfies Theorem 2.12 (a) and (b). Furthemore, for yAN1 ðG; rÞ and X DN2 ðG; rÞ we have f1y ¼ dðyÞ; rðX þ yÞ ¼ rðX Þ þ 1; and, since G is loopless and NðGÞ is independent, eðX þ y; V  X  yÞ ¼ eðX ; V  X Þ þ dðyÞ: Since ðG; rÞ satisfies (c), it follows that ðG; rÞ also satisfies Theorem 2.12(c). Hence G has a nonseparable r-detachment. Applying Lemma 2.20, G has a non-separable h-detachment, H for some r-degree specification h satisfying hv1 Xhv2 Xhv3 ¼ ? ¼ hvrðvÞ ¼ 2 for all vAV : We may suppose P that H has been chosen such that yðhÞ ¼ vAV hv1 is as large as possible. If hv1 Xf1v for all vAV then hðvÞXrðvÞ f ðvÞ for all vAV and by Lemma 2.21, G has a non-separable f -detachment. Hence we may suppose that hv1 pf1v  1 for some vAV : Necessarily we must have vAN2 ðG; rÞ and hv2 Xf2v þ 1X3: Let v1 ; v2 ; y; vrðvÞ be the pieces of v in H; where dH ðvi Þ ¼ hvi for 1piprðvÞ: If H 0 ¼ Hðv2 z-v1 zÞ is non-separable for some zAGH ðv2 Þ; then H 0 is a non-separable h0 -detachment of G with yðh0 Þ4yðhÞ: This contradicts the choice of H: Applying Lemma 2.16, we deduce that bH ðfv1 ; v2 gÞ ¼ dH ðv2 Þ ¼ hv2 : Claim 2.24. bH ðfv1 ; v2 ; y; vrðvÞ gÞ ¼ hv2 þ rðvÞ  2: Proof. We have already shown that bH ðfv1 ; v2 gÞ ¼ dH ðv2 Þ ¼ hv2 and hence the claim holds for rðvÞ ¼ 2: Suppose rðvÞX3 and choose i; 3piprðvÞ: Let z be a neighbour of v2 in H such that z; vi belong to different components of H  fv1 ; v2 g: If H 00 ¼ Hðv2 z-vi zÞðvi z0 -v1 z0 Þ is non-separable for some z0 AGH ðvi Þ; then H 00 is a nonseparable h00 -detachment of G with yðh00 Þ4yðhÞ: This contradicts the choice of H: Applying Lemma 2.19, we deduce that each edge of H incident to vi is a cut-edge of H  fv1 ; v2 g for all 3piprðvÞ: Since hvi ¼ 2 for 3piprðvÞ; it follows that H  fv1 ; v2 ; y; vrðvÞ g has hv2 þ rðvÞ  2 components. & Let L be the graph obtained from H by contracting fv1 ; v2 y; vrðvÞ g back to the single vertex v: Then L is an c-detachment of G for the r0 -degree specification for G defined by r0 ðvÞ ¼ 1; cðvÞ ¼ dG ðvÞ; and r0 ðuÞ ¼ rðuÞ and cðuÞ ¼ hðuÞ for uAV  v: Furthermore L is a block-star centered on v and bL ðvÞ ¼ hv2 þ rðvÞ  2: Claim 2.25. Let L0 be an c-detachment of G such that L0 is a block-star centered on v. Then bL0 ðvÞXbL ðvÞ: Proof. Suppose bL0 ðvÞpbL ðvÞ  1: By Lemma 2.22, we can detach v in L0 into rðvÞ pieces v01 ; v02 ; yv0rðvÞ such that the resulting graph H n is 2-edge-connected and satisfies dH n ðv02 Þ ¼ bL0 ðvÞ  rðvÞ þ 2; dH n ðv0i Þ ¼ 2 for 3piprðvÞ; and neither v01 nor v02 is a cut-vertex of H n : Using Claim 2.24 and the fact that bL0 ðvÞpbL ðvÞ  1;

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we have dH n ðv02 ÞobL0 ðvÞ  rðvÞ þ 2 ¼ hv2 : Since dH n ðv0i Þ ¼ 2 ¼ hvi for 3piprðvÞ; we deduce that dH n ðv01 Þ4hv1 : Since H n is 2edge-connected and dH n ðv0i Þ ¼ 2 for 3piprðvÞ; v0i is not a cut-vertex of H n for 1piprðvÞ: Thus, if H n had a cut-vertex x; then x would belong to some component of L0  v; and x would be a cut-vertex of L0 distinct from v: Hence H n is a nonseparable hn -detachment of G for some r-degree specification hn for G satisfying yðhn Þ4yðhÞ: This contradicts the choice of H: & It follows from Claim 2.25 that we may apply Lemma 2.6 to ðG; L; cÞ: Let Si and Wi be the subsets of V ðLÞ defined as in Lemma 2.6 (but with respect to v). Since the sets Si are pairwise disjoint and L is finite, we may choose i such that Siþ1 ¼ |: Let X 0 ¼ ðWi  vÞ and X ¼ fxAV ðGÞ: some piece of x in L belongs to X 0 g: By Lemma 2.6, every edge x1 u of L  v with x1 AX 0 is a cut-edge in L  v: Thus the graph we get from L  v by contracting each component of L  X 0  v to a single vertex is a forest F with bL ðvÞ components and jX 0 j þ bL ðX 0 þ vÞ vertices. Using the facts that X þ vDNðGÞ; and NðGÞ is an independent set of vertices in G; we deduce that F has eL ðX 0 ; V ðLÞ  X 0 Þ edges. Thus eL ðX 0 ; V ðLÞ  X 0 Þ ¼ bL ðX 0 þ vÞ þ jX 0 j  bL ðvÞ:

ð5Þ

We have eL ðX 0 ; V ðLÞ  X 0 Þ ¼ eG ðX þ v; V ðGÞ  X  vÞ  dL ðvÞ; jX 0 j ¼ rðX Þ; bL ðvÞ ¼ hv2 þ rðvÞ  2 ¼ dG ðvÞ  hv1  ðrðvÞ  2Þ; and dL ðvÞ ¼ dG ðvÞ: Furthermore, for each uAV ðGÞ  X  v; all pieces of u in L belong to the same component of L  X 0  v; since Siþ1 ¼ |: Thus bG ðX þ vÞ ¼ bL ðX 0 þ vÞ: Substituting into (5) we obtain eG ðX þ v; V ðGÞ  X  vÞ ¼ rðX þ vÞ þ bG ðX þ vÞ þ hv1  2: Since hv1 pf1v  1; this contradicts the fact that G satisfies (c). ˆ EÞ ˆ be obtained from G by subdividing every edge Proof of Theorem 2.2. Let Gˆ ¼ ðV; ˆ ˆ ˆ is independent. Extend r to rˆ and f of G: Then G is loopless, NðGÞ ¼ NðGÞ and NðGÞ ˆ ¼ f ðvÞ for all vAV ðGÞ; rˆðvÞ ¼ 1 and fðvÞ ˆ ¼ 2 for to fˆ by putting rˆðvÞ ¼ rðvÞ and fðvÞ ˆ ˆ ˆ all vAV  V : Then N1 ðG; rˆÞ ¼ N1 ðG; rÞ; N2 ðG; rˆÞ ¼ N2 ðG; rÞ: We shall show that conditions (a)–(c) of Theorem 2.2 hold for ðG; r; f Þ if and only if conditions (a), (b), ˆ Clearly Theorem 2.2(a) and (b) hold for ˆ rˆ; fÞ: and (c) of Theorem 2.23 hold for ðG; ˆ Furthermore for ˆ rˆ; fÞ: ðG; r; f Þ if and only if Theorem 2.23(a) and (b) hold for ðG; ˆ rˆÞ; we have f v ¼ fˆv ; rðX Þ ¼ rˆðX Þ; and ˆ and X DN2 ðG; rÞ ¼ N2 ðG; vANðGÞ ¼ NðGÞ 1 1 eG ðX þ v; V  X  vÞ þ eG ðX þ vÞ  bG ðX þ vÞ ¼ eGˆ ðX þ v; Vˆ  X  vÞ  bGˆ ðX þ vÞ: Thus Theorem 2.2(c) holds for ðG; r; f Þ; if and only if Theorem 2.23(c) holds for ˆ & ˆ rˆ; fÞ: ðG; We close this section by noting that our proofs of Theorems 2.1 and 2.2 are constructive and give rise to polynomial algorithms which either construct the specified detachment or construct a certificate that shows it does not exist.

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3. Some corollaries and open problems Our first corollary extends Euler’s Theorem. Corollary 3.1. Let G ¼ ðV ; EÞ be a 2-edge-connected graph and r : V -Zþ such that dðvÞX2rðvÞ for all vAV and rðvÞX2 for all vANðGÞ: Let f be an r-degree specification v for G such that f ðvÞ ¼ ðf1v ; f2v ; y; frðvÞ Þ and 2pfiv pJdðvÞ=2n  rðvÞ þ 2 for all vAV and all 1piprðvÞ: Then G has a non-separable f-detachment. Proof. Theorem 2.1 implies that G has a non-separable r-detachment (conditions (c) and (d) of Theorem 2.1 hold vacuously for G since N1 ðGÞ ¼ |). The existence of a non-separable f -detachment now follows from Lemmas 2.20 and 2.21. (This corollary can also be derived from Theorem 2.2.) & The more difficult direction of Euler’s theorem follows from Corollary 3.1 by taking rðvÞ ¼ dðvÞ=2 in a graph in which all vertices have even degree. Our next corollary is a result of Hakimi [1] which characterises the degree sequences of non-separable graphs. Corollary 3.2. Let d1 Xd2 X?Xdn X2 be integers with nX2: Then there exists a nonseparable graph with degree sequence ðd1 ; d2 ; y; dn Þ if and only if (a) d1 þ d2 þ ? þ dn is even, and (b) d1 pd2 þ d3 þ ? þ dn  2n þ 4:

Proof. We first prove necessity. Suppose there exists a non-separable graph H with this degree sequence and let vi AV ðHÞ have degree di for 1pipn: Clearly (a) holds. Since H is non-separable, H  v1 is connected. Thus jEðH  v1 ÞjXn  2: Hence d1 ¼ eðV  v1 ; v1 Þpd2 þ d3 þ ? þ dn  2n þ 4: Sufficiency follows by applying Theorem 2.2 to the graph G consisting of a single vertex v incident to ðd1 þ d2 þ ? þ dn Þ=2 loops, by setting rðvÞ ¼ n and f ðvÞ ¼ ðd1 ; d2 ; y; dn Þ: & Our next result considers the case when we only want to detach one vertex in a graph. The special case when dðvÞ is even and rðvÞ ¼ dðvÞ=2 gives a ‘splitting off’ result for non-separable graphs. Corollary 3.3. Let G ¼ ðV ; EÞ be a graph, uAV ; and r : V -Zþ such that rðuÞ ¼ mX2 and rðvÞ ¼ 1 for all vAV  u: Let f be an r-degree specification for G where f ðuÞ ¼ ðf1 ; f2 ; y; fm Þ; f1 Xf2 X?Xfm X2; and f ðvÞ ¼ dðvÞ for vAV  u: Then G has a nonseparable f-detachment if and only if (a) G is 2-edge-connected, (b) eðvÞ ¼ 0 and bðvÞ ¼ 1 for all vAV  u;

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(c) f2 þ f3 þ ? þ fm XbðuÞ þ eðuÞ þ m  2; and (d) eðu; V  v  uÞ þ eðuÞXm þ bðfu; vgÞ  1 for all vAV  u:

Proof. The necessity of conditions (a)–(d) is easy to see. To prove sufficiency, we suppose that G satisfies (a)–(d) and use Theorem 2.2 to deduce that G has a non-separable f -detachment. It is easy to see that conditions (a) and (b) of Theorem 2.2 hold for G: To see that condition (c) of Theorem 2.2 holds, let vANðGÞ and X DN2 ðGÞ  v: Since N2 ðGÞ ¼ fug we have ðv; X ÞAfðv; |Þ; ðu; |Þ; ðv; fugÞg: Condition (c) of Theorem 2.2 holds for each of these three alternatives since conditions (b)–(d) of the corollary hold for G: Note that when ðv; X Þ ¼ ðu; |Þ we have dðuÞ ¼ f1 þ f2 þ ? þ fm ¼ eðu; V  uÞ þ 2eðuÞ and when ðv; X Þ ¼ ðv; fugÞ we have eðX þ v; V  X  vÞ þ eðX þ vÞ  f1v ¼ eðfu; vg; V  fu; vgÞ þ eðfu; vgÞ  dðvÞ ¼ eðu; V  v  uÞ þ eðuÞ; since eðvÞ ¼ 0 by condition (b) of the corollary.

&

We next consider non-separable simple detachments. Corollary 3.4. Let G ¼ ðV ; EÞ be a graph and r : V -Zþ : Then G has a non-separable simple r-detachment if and only if (a) G is 2-edge connected, (b) dðvÞX2rðvÞ for all vAV ; (c) eðX ; V  X  yÞ þ eðX ÞXrðX Þ þ bðX þ yÞ  1 for all X DN2 ðG; rÞ; and (d) eðuÞprðuÞðrðuÞ  1Þ=2 and eðu; vÞprðuÞrðvÞ for all u; vAV :

yAN1 ðG; rÞ

and

Proof. Necessity of (a)–(c) follows from Theorem 2.1 while necessity of (d) is obvious. To see sufficiency we use Theorem 2.1 to deduce that G has a nonseparable r-detachment H: We may assume that H has as few parallel edges as possible. Suppose that eH ðu1 ; v1 ÞX2 for two vertices u1 and v1 of H: Let u1 and v1 be pieces in H of the vertices u and v; respectively, of G; (allowing the possibility that u ¼ v). Then (d) implies that there exist distinct pieces ui of u and vj of v in H such that eH ðui ; vj Þ ¼ 0: Then H  u1 v1 þ ui vj has one less parallel edge than H: & It is an open, and perhaps difficult, problem to characterize when a graph has a non-separable simple detachment for some given degree specification. A graph G ¼ ðV ; EÞ is said to be k-connected if jV jXk þ 1 and G  U is connected for all UDV ðGÞ with jUjpk  1: Thus, if jV jX3; then G is non-separable if and

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only if G is 2-connected and loopless. Our next result characterises when a graph has a 2-connected r-detachment. Corollary 3.5. Let G ¼ ðV ; EÞ be a graph and r : V -Zþ such that rðV ÞX3: Then G has a 2-connected r-detachment if and only if (a) G is 2-edge connected, (b) dðvÞX2rðvÞ for all vAV ; (c) eðX ; V  X  yÞ þ eðX ÞXrðX Þ þ bðX þ yÞ  1 X DN2 ðG; rÞ:

for

all

yAN1 ðG; rÞ

and

Proof. This follows easily by applying Theorem 2.1 to ðG 0 ; rÞ where G 0 is the graph obtained from G by deleting all loops incident to vertices in N1 ðG; rÞ: & By Theorem 1.3, condition (c) of Corollary 3.5 is equivalent to the statement ‘‘G  y has a connected rjV y -detachment for all yANðGÞ with rðyÞ ¼ 1’’. It is conceivable that Corollary 3.5 extends to k-connectivity as follows. Conjecture 3.6. Let kX2 be an integer, G ¼ ðV ; EÞ be a graph, and r : V -Zþ such that rðV ÞXk þ 1: Then G has a k-connected r-detachment if and only if (a) G is k-edge connected, (b) dðvÞXkrðvÞ for all vAV ; (c) G  y has a ðk  rðyÞÞ-connected rjV y -detachment for all yAV with rðyÞpk  1:

Using Theorem 1.1, it can be seen that the truth of this conjecture for j-connected detachments for all 2pjpk would be equivalent to the truth of the following conjecture. Conjecture 3.7. Let k be a positive integer, G ¼ ðV ; EÞ be a graph, and r : V -Zþ such that rðV ÞXk þ 1: Then G has a k-connected r-detachment if and only if (a) G  Y is ðk  rðY ÞÞ-edge connected for all Y DV with rðY Þpk  2; (b) dðvÞ  eðv; Y ÞXðk  rðY ÞÞrðvÞ for all vAV and all Y DV  v with rðY Þpk  2; (c) eðX ; V  X  Y Þ þ eðX ÞXrðX Þ þ bðX ,Y Þ  1 for all Y DV with rðY Þpk  1 and all X DV  Y : Conjecture 3.7 is true for k ¼ 1; 2 by Theorem 1.1, and Corollary 3.5, respectively.

Acknowledgments We thank the two anonymous referees for their very useful comments.

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References [1] S.L. Hakimi, On realizability of a set of integers as degrees of the vertices of a linear graph. I, J. Soc. Indust. Appl. Math. 10 (1962) 496–506. [2] C.St.J.A. Nash-Williams, Detachments of graphs and generalised Euler trails, in: I. Anderson (Ed.), Surveys in Combinatorics 1985, London Mathematical Society, Lecture Note Series, Vol. 103, Cambridge University Press, Cambridge, 1985, pp. 137–151. [3] C.St.J.A. Nash-Williams, Connected detachments of graphs and generalized Euler trails, J. London Math. Soc. 31 (1985) 17–29.