Maximum cut on interval graphs of interval count four is NP-complete
Celina M. H. de Figueiredo, Alexsander A. de Melo, Fabiano S. Oliveira, Ana Silva
MMaximum cut on interval graphs of interval count five is NP -complete Celina M. H. de Figueiredo a , Alexsander A. de Melo a , Fabiano S. Oliveira b , Ana Silva c a. Universidade Federal do Rio de Janeirob. Universidade do Estado do Rio de Janeiroc. Universidade Federal do Cear´a Abstract
The computational complexity of the
MaxCut problem restricted to interval graphs has been open since the 80’s, being one of theproblems considered by Johnson on his
Ongoing Guide to NP -completeness , and has been settled as NP-complete only recently byAdhikary, Bose, Mukherjee and Roy. On the other hand, many flawed proofs of polynomiality for MaxCut on the more restrictiveclass of proper/unit interval graphs (or graphs with interval count 1) have been presented along the years, and the classification ofthe problem is still not known. In this paper, we present the first NP -completeness proof for MaxCut when restricted to intervalgraphs with bounded interval count, namely graphs with interval count 5.
Keywords: maximum cut, interval graphs, interval lengths, interval count, NP -complete. A cut is a partition of the vertex set of a graph into two disjoint parts and the maximum cut problem (denoted MaxCut for short) aims to determine a cut with the maximum number of edges for which each endpoint isin a distinct part. The decision problem
MaxCut is known to be NP -complete since the seventies [8], andonly recently its restriction to interval graphs has been announced to be hard [1], settling a long-standing openproblem that appeared in Johnson’s Ongoing Guide to NP -completeness [9].An interval model is a family of closed intervals of the real line. A graph is an interval graph if thereexists an interval model, for which each interval corresponds to a vertex of the graph, such that distinctvertices are adjacent in the graph if and only if the corresponding intervals intersect. The interval countof an interval graph is the smallest number of interval lengths used by an interval model of the graph [7].Published attempts to polynomial-time algorithms for MaxCut when restricted to graphs having intervalcount 1 (also known as indifference, proper interval or unit interval graphs) [3, 6] were subsequently proved tobe wrong [4, 10]. In this paper, we give the first classification that bounds the interval count, namely, we provethat
MaxCut is NP -complete when restricted to interval graphs of interval count 5. This opens the search fora full polynomial/ NP -complete dichotomy classification in terms of the interval count. Notice that it can stillhappen that the problem is hard even on graphs of interval count 1.Next, we establish basic definitions and notation. Section 2 describes our reduction and Section 3 discussesthe interval count of the interval graph constructed in [1]. Due to space restrictions, proofs of statementsmarked with ( (cid:63) ) have been moved to the Appendix. In this work, all graphs considered are simple. For missing definitions and notation of graph theory, we referto [5]. And for a comprehensive study of interval graphs, we refer to [11]. This project was partially supported by CNPq and FAPERJ Emails: [email protected] , [email protected] , [email protected] , [email protected] a r X i v : . [ c s . CC ] D ec e Figueiredo et al. Let G be a graph. Let X and Y be two disjoint subsets of V ( G ). We let E G ( X, Y ) be the set of edgesof G with an endpoint in X and the other endpoint in Y , i.e. E G ( X, Y ) = { uv ∈ E ( G ) : u ∈ X, v ∈ Y } .For every subset S ⊆ V ( G ), we let S X = S ∩ X and S Y = S ∩ Y . A cut of G is a partition of V ( G ) intotwo parts A, B ⊆ V ( G ), denoted by [ A, B ]. Let [
A, B ] be a cut of G . The edge set E G ( A, B ) is called the cut-set of G associated with [ A, B ]. For each two vertices u, v ∈ V ( G ), we say that u and v are in a same partof [ A, B ] if either { u, v } ⊆ A or { u, v } ⊆ B ; otherwise, we say that u and v are in opposite parts of [ A, B ].Denote by mc ( G ) the maximum size of a cut-set of G , i.e. mc ( G ) = max {| E G ( A, B ) | : [ A, B ] is a cut of G } .The MaxCut problem has as input a graph G and a non-negative integer k , and it asks whether mc ( G ) ≥ k .Let I ⊆ R be a closed interval of the Real line. We let (cid:96) ( I ) and r ( I ) denote respectively the minimum andmaximum points of I , which we will call the left and the right endpoints of I , respectively. We denote a closedinterval I by [ (cid:96) ( I ) , r ( I )]. In this work, we assume all intervals are closed, unless mentioned otherwise. The length of an interval I is defined as | I | = r ( I ) − (cid:96) ( I ). An interval model is a finite multiset M of intervals.The interval count of an interval model M , denoted by ic ( M ), is defined as the number of distinct lengthsof the intervals in M , i.e. ic ( M ) = |{| I | : I ∈ M}| . Let G be a graph and M be an interval model. An M -representation of G is a bijection φ : V ( G ) → M such that, for every two distinct vertices u, v ∈ V ( G ), wehave that uv ∈ E ( G ) if and only if φ ( u ) ∩ φ ( v ) (cid:54) = ∅ . If such an M -representation exists, we say that M isan interval model of G . We note that a graph may have either no interval model or arbitrarily many distinctinterval models. A graph is called an interval graph if it has an interval model. The interval count of aninterval graph G , denoted by ic ( G ), is defined as the minimum interval count over all interval models of G ,i.e. ic ( G ) = min { ic ( M ) : M is an interval model of G } . An interval graph is called a unit interval graph if itsinterval count is equal to 1; these are also called proper interval graphs, and indifference graphs.Note that, for every interval model M , there exists a unique (up to isomorphism) graph that admits an M -representation. Thus, for every interval model M = { I , . . . , I n } , we let G M be the graph with vertexset V ( G M ) = { , . . . , n } and edge set E ( G M ) = { ij : I i , I j ∈ M , I i ∩ I j (cid:54) = ∅ , i (cid:54) = j } . Since G M is uniquelydetermined (up to isomorphism) from M , in what follows we may make an abuse of language and use graphterminologies to describe properties related to the intervals in M .For each positive integer a ∈ N + , we let [ a ] = { , , . . . , a } . For each three positive integers a, b, c ∈ N + , wewrite a ≡ b c to denote that a modulo b is equal to c modulo b . The following theorem is the main contribution of this work:
Theorem 2.1
MaxCut is NP -complete on interval graphs of interval count . This result is a stronger version of that of Adhikary et al. [1]. In order to prove Theorem 2.1, we present apolynomial-time reduction from
MaxCut on cubic graphs, which is known to be NP -complete [2]. Since ourproof is based on that of Adhikary et al., we start by presenting some important properties of their key gadget. The interval graph constructed in the reduction of [1] is strongly based on two types of gadgets, which theycalled V-gadgets and E-gadgets. But in fact, they are the same except for the amount of intervals of certainkinds. In this subsection, we present a generalization of such gadgets, rewriting their key properties to suit ourpurposes. In order to discuss the interval count of the reduction of [1], we describe it in details in Section 3.Let x and y be two positive integers. An ( x, y ) -grained gadget is an interval model H = LS ∪ LL ∪ RS ∪ RL that satisfies the properties presented next. The intervals belonging to LS (resp. LL ) are called the left short (resp. left long ) intervals of H . Analogously, the intervals belonging to RS (resp. RL ) are called the rightshort (resp. right long ) intervals of H . Below are the properties satisfied by H (see Figure 1):(i) |LS| = |RS| = x and |LL| = |RL| = y ;(ii) Each pair of long intervals intersect. More formally, for each pair I, I (cid:48) ∈ LL ∪ RL , I ∩ I (cid:48) (cid:54) = ∅ ;(iii) Left short intervals intersect only left long intervals. More formally, for each I ∈ LS and each I (cid:48) ∈ H \ { I } , I ∩ I (cid:48) (cid:54) = ∅ if and only if I (cid:48) ∈ LL ;(iv) Right short intervals intersect only right long intervals. More formally, for each I ∈ RS and each I (cid:48) ∈H \ { I } , I ∩ I (cid:48) (cid:54) = ∅ if and only if I (cid:48) ∈ RL .When multiple grained gadgets are in context, we shall write LS ( H ), LL ( H ), RS ( H ) and RL ( H ) to referto the sets LS , LL , RS and RL of H , respectively.Note that, if H is an ( x, y )-grained gadget, then G H is a split graph such that LS ∪ RS is an independentset of size 2 x , LL ∪ RL is a clique of size 2 y , N G H ( LS ) = LL and N G H ( RS ) = RL . Moreover, note that the2 e Figueiredo et al. intervals belonging to LL are true twins in G H ; similarly, the intervals belonging to RL are true twins in G H . Fig. 1. General structure of an ( x, y )-grained gadget.
Let H be an ( x, y )-grained gadget and I be an interval such that I (cid:54)∈ H . We say that: I covers H if, foreach I (cid:48) ∈ H , we have I ⊇ I (cid:48) (see Figure 2a); I weakly intersects H to the left (resp. right ) if, for each I (cid:48) ∈ H ,we have I ∩ I (cid:48) (cid:54) = ∅ if and only if I (cid:48) ∈ LL ( H ) (resp. I (cid:48) ∈ RL ( H )) (see Figures 2b and 2c); and that I stronglyintersects H to the left (resp. right ) if, for each I (cid:48) ∈ H , we have I ∩ I (cid:48) (cid:54) = ∅ if and only if I (cid:48) ∈ LS ( H ) ∪ LL ( H )(resp. I (cid:48) ∈ RS ( H ) ∪ RL ( H )) (see Figures 2d and 2e). (a) Coveringintersection (b) Weak intersectionto the left (c) Weak intersectionto the right (d) Strong intersectionto the left (e) Strong intersectionto the rightFig. 2. Interval I ∈ M \ {H} covering H (a), weakly intersecting H to the left (b) and to the right (c), and strongly intersecting H to the left (d) and to the right (e). Let M be an interval model and H be an ( x, y )-grained gadget such that H ⊆ M . Let c M ( H ) denotethe number of intervals of M that cover H ; wl M ( H ) (resp. wr M ( H )) denote the number of intervals of M that weakly intersect H to the left (resp. right); and sl M ( H ) (resp. sr M ( H ))denote the number of intervals of M that strongly intersect H to the left (resp. right). We say that M respects the structure of H if, for eachinterval I ∈ M \ H , exactly one of the following conditions is satisfied: (i) I does not intersect H ; (ii) I covers H ; (iii) I weakly/strongly intersects H to the left/to the right. The following lemma tells that the left (right)long and short intervals must be in opposite parts of any maximum cut. Lemma 2.2 ( (cid:63) ) Let x and y be positive integers, H be an ( x, y ) -grained gadget and M be an interval modelthat respects the structure of H . For every maximum cut [ A, B ] of G M , the following properties hold: (i) if y + sl M ( H ) + c M ( H ) ≡ and x > y − wl M ( H ) + sl M ( H ) + c M ( H ) , then LS ( H ) ⊆ A and LL ( H ) ⊆ B , or vice versa; (ii) if y + sr M ( H ) + c M ( H ) ≡ and x > y − wr M ( H ) + sr M ( H ) + c M ( H ) , then RS ( H ) ⊆ A and RL ( H ) ⊆ B , or vice versa. Now, we want now to add conditions that, together with the ones from the previous lemma, ensure that theleft long intervals will be put opposite to the right long intervals. Based on Lemma 2.2, we say that ( H , M ) is well-valued if Conditions (i) and (ii) hold, in addition to the following one y > y · wr M ( H ) + ( x − y ) · (cid:0) sr M ( H ) + c M ( H ) (cid:1) . (1)Let [ A, B ] be a maximum cut of G M . We say that H is A -partitioned by [ A, B ] if LS ( H ) ∪ RL ( H ) ⊆ A ,and RS ( H ) ∪ LL ( H ) ⊆ B . Define B -partitioned analogously. Lemma 2.3 ( (cid:63) ) Let x and y be positive integers, H be an ( x, y ) -grained gadget, M be an interval model and [ A, B ] be a maximum cut of G M . If M respects the structure of H and ( H , M ) is well-valued, then H is either A -partitioned or B -partitioned by [ A, B ] . Slightly different versions of these lemmas are presented in [1], but we present our own proofs for complete-ness and consistency with our notation. 3 e Figueiredo et al.
In this subsection, we formally present our construction. Recall that we are making a reduction from
MaxCut on cubic graphs. So, consider a cubic graph G on n vertices and m edges. Intuitively, we consider an orderingof the edges of G , and we divide the real line into m regions, with the j -th region holding the informationabout whether the j -th edge is in the cut-set. For this, each vertex u will be related to a subset of intervalstraversing all the m regions, bringing the information about which part u belongs to. We first describe thegadget related to the vertices.Let n and m be positive integers. An ( n, m ) -escalator is an interval model D = (cid:83) i ∈ [ n ] ( H i ∪ · · · ∪H m +1 i ∪ { L i , . . . , L mi } ) such that, for each i ∈ [ n ], we have that H i , . . . , H m +1 i are ( p, q )-grained gadgetsand L i , . . . , L mi are intervals, called link intervals , satisfying the following conditions (see Figure 3):(i) The grained gadgets are mutually disjoint. More formally, for each pair j, j (cid:48) ∈ [ m + 1] with j (cid:54) = j (cid:48) , each I ∈ H ji and each I (cid:48) ∈ H j (cid:48) i , we have that I ∩ I (cid:48) = ∅ ;(ii) for each j ∈ [ m ], intervals L j − i , L ji are true twins in G D , weakly intersect H ji to the right and weaklyintersect H j +1 i to the left;(iii) if i (cid:54) = 1, then for each j ∈ [ m ], intervals L j − i − , L ji − cover H ji , and intervals L j − i , L ji cover H j +1 i − .In Subsection 2.3, we shall choose suitable values for p and q . Fig. 3. General structure of a region of the ( n, m )-escalator. The rectangles represent the ( p, q )-grained gadgets H ji . Now, let π V = ( v , . . . , v n ) be an ordering of V ( G ) and π E = ( e , . . . , e m ) be an ordering of E ( G ). Weconstruct from G = ( G, π V , π E ) an interval model M ( G ) of constant interval count such that, for each positiveinteger k , the size mc ( G ) of a maximum cut-set of G satisfies mc ( G ) ≥ k if and only if mc ( G M ( G ) ) ≥ f ( G, k ),where f is a suitable positive function that will be defined later on. This construction consists of two steps,which are described next.First, create an ( n, m )-escalator D = (cid:83) i ∈ [ n ] ( H i ∪ · · · ∪ H m +1 i ∪ { L i , . . . , L mi } ). Second, for each edge e j = v i v i (cid:48) ∈ E ( G ), with i < i (cid:48) , create a ( p (cid:48) , q (cid:48) )-grained gadget E j and intervals C j , C j , C j , C j satisfying thefollowing conditions (see Figure 4):(i) for each I ∈ D and each I (cid:48) ∈ E j , I ∩ I (cid:48) (cid:54) = ∅ if and only if I ∈ { L j − h , L jh : h ∈ [ n ] } , and in this case, I covers E j ;(ii) the intervals C j and C j are true twins in G M ( G ) , weakly intersect H ji to the right and weakly intersect E j to the left;(iii) the intervals C j and C j are true twins in G M ( G ) , weakly intersect H ji (cid:48) to the right and strongly intersect E j to the left.In Subsection 2.3, we shall choose suitable values for p (cid:48) and q (cid:48) . As before, consider a cubic graph G on n vertices and m = 3 n/ π V = ( v , . . . , v n ) be anordering of V ( G ), π E = ( e , . . . , e m ) be an ordering of E ( G ) and G = ( G, π V , π E ). We are ready to give anoutline of the proof that mc ( G ) ≥ k if and only if mc ( G M ( G ) ) ≥ f ( G, k ), where f is defined at the end of thesubsection. As it is usually the case in this kind of reduction, constructing an appropriate cut of the reductiongraph G M ( G ) , given a cut of G , is an easy task. On the other hand, constructing an appropriate cut [ X, Y ]of G , from a given a cut [ A, B ] of the reduction graph G M ( G ) , requires that the intervals in M ( G ) behaveproperly with respect to [ A, B ] (that is, how they are partitioned by the cut) so that [
X, Y ] can be inferred,a task achieved with the help of Lemmas 2.2 and 2.3. In order to use these lemmas, we choose next suitable4 e Figueiredo et al.
Left short intervals
Fig. 4. General structure of the constructed interval model M ( G ), highlighting the intersections between the intervals of the( n, m )-escalator D , the intervals of the ( p (cid:48) , q (cid:48) )-grained gadget E j , and the intervals C j , C j , C j , C j . In this illustration, gadget E j related to the edge e j = v i v i (cid:48) , with i < i (cid:48) , is depicted so that all the intersections are seen. values for p, q, p (cid:48) , q (cid:48) , and we observe that M ( G ) respects the structure of the involved grained gadgets. Afterensuring that each grained gadget is well behaved, to ensure that the behaviour of H i can be used to decidein which part of [ X, Y ] we should put v i , it is necessary that all gadgets related to v i agree with one another.In other words, for each v i , we want that the behaviour of the first gadget H i influence the behaviour of thesubsequent gadgets H i , . . . , H m +1 i , as well as the behaviour of the gadgets related to edges incident to v i . Thisis done by choosing the following values for our floating variables: q = 28 n + 1, p = 2 q + 7 n , q (cid:48) = 18 n + 1 and p (cid:48) = 2 q (cid:48) + 5 n .These values indeed satisfy Conditions (i) and (ii) of Lemma 2.2, and Equation (1). As previously said,the idea behind this choice of values is to store information about v i in the gadgets H i , . . . , H m +1 i . Now, given e j = v i v i (cid:48) , i < i (cid:48) , a final ingredient is to ensure that E j is influenced only by intervals C j and C j , which inturn are influenced by vertex v i (cid:48) in a way that the number of edges in the cut-set of G M ( G ) increases when theedge v i v i (cid:48) is in the cut-set of G . These ideas are captured in the definitions below.Given v i ∈ V ( G ) and a cut [ A, B ] of G M ( G ) , we say that the gadgets of v i alternate in [ A, B ] if, for every j ∈ [ m ], we get that H ji is A -partitioned if and only if H j +1 i is B -partitioned. Also, we say that [ A, B ] is alternating partitioned if the gadgets of v i alternate in [ A, B ], for every v i ∈ V ( G ), and the following holds forevery edge e j = v i v i (cid:48) ∈ E ( G ), i < i (cid:48) :(i) If H ji is A -partitioned by [ A, B ], then { C j , C j } ⊆ B ; otherwise, { C j , C j } ⊆ A ; and(ii) If H ji (cid:48) is A -partitioned by [ A, B ], then { C j , C j } ⊆ B and E j is A -partitioned by [ A, B ]; otherwise, { C j , C j } ⊆ A and E j is B -partitioned by [ A, B ].The following lemma is a key element in our proof.
Lemma 2.4 ( (cid:63) ) If [ A, B ] is a maximum cut of G M ( G ) , then [ A, B ] is an alternating partitioned cut. Now, if [
A, B ] is an alternating partitioned cut of G M ( G ) , we let Φ( A, B ) = [
X, Y ] be the cut of G suchthat, for each vertex v i ∈ V ( G ), we have v i ∈ X if and only if H i is A -partitioned by [ A, B ]. Note that [
X, Y ] iswell-defined and uniquely determined by [
A, B ]. On the other hand, given a cut [
X, Y ] of G , there is a uniquealternating partitioned cut [ A, B ] = Φ − ( X, Y ) of G M ( G ) such that [ X, Y ] = Φ(
A, B ). Therefore, it remains torelate the sizes of these cut-sets. Basically we can use the good behaviour of the cuts in G M ( G ) to prove thatthe size of [ A, B ] grows as a well-defined function on the size of Φ(
A, B ). More formally, we can prove that thefunction f previously referred to is given by (recall that k is part of the input on the original problem): f ( G, k ) = (cid:0) n / n (cid:1) (2 pq + q ) + 3 n/ p (cid:48) q (cid:48) + ( q (cid:48) ) ) + 6 nq ( n + 1)+ (cid:0) n + 3 n (cid:1) ( n − p + q ) + 3 n ( p (cid:48) + q (cid:48) ) + 3 n (( k + 1) q (cid:48) + p (cid:48) ) + 4 k .5 e Figueiredo et al. Consider a cubic graph G on n vertices and m = 3 n/ π V , π E of the vertex set and edgeset of G . Denote the triple ( G, π V , π E ) by G . We want to prove that the interval count of our constructedinterval model M ( G ) is at most 5. But observe that the construction of M ( G ) is actually not unique, sincethe intervals are not uniquely defined; e.g., given such a model, one can obtain a model satisfying the sameproperties simply by adding (cid:15) > G that satisfies the desired conditions and has interval count 5.Consider our constructed interval model M ( G ), and denote S j = E j ∪ (cid:83) (cid:96) ∈ [4] C (cid:96)j ∪ (cid:83) i ∈ [ n ] ( H ji ∪ { L ji ∪ L j − i } )for each j ∈ [ m ]. We show how to accommodate S within [0 , n −
7] in such a way that the same pattern canbe adopted in the subsequent regions of M ( G ) too, each time starting at multiples of 6 n −
5. More specifically,letting t = 6 n − S j will be accommodated within [ t · ( j − , n − t · ( j − e = v h v h (cid:48) , with h < h (cid:48) . Below, we say exactly which interval of the line corresponds to each interval I ∈ S . • For each i ∈ [ n ], the left long intervals of H i are equal to [2 i − , i − /
2] and the left short intervals areany choice of q distinct points within the open interval (2 i − , i − / H i are equal to [2 i − / , i −
1] and the right short intervals are any choice of q distinct points withinthe open interval (2 i − / , i − • C and C are equal to [2 h − , h + 2 n − • C and C are equal to [2 h (cid:48) − , h (cid:48) + 4 n − • The left long intervals of E are equal to [2 n, n − • The left short intervals of E are any choice of q (cid:48) distinct points in the open interval (4 n − , n − • The right long intervals of E are equal to [6 n − , n − /
2] and the right short intervals are any choiceof q (cid:48) distinct points within the open interval (6 n − , n − / • For each i ∈ [ n ], intervals L i , L i are equal to [2 i − , i + 6 n − Fig. 5. The above figure represents the intervals in S ∪ (cid:83) i =1 H i of a graph on 4 vertices. We consider e to be equal to v v .Each colour represents a different interval size. The short intervals are contained in the dotted (open) intervals. Vertical linesmark the endpoints of the intervals in S \ L , while the blue vertical line marks the beginning of the intervals in S . The possible lengths of an interval are (see Figure 5):(i) 0: short intervals of all grained gadgets (dots in Figure 5);(ii) 1 /
2: left long and right long intervals of each H i , and right long intervals of E (red intervals in Figure 5);(iii) 2 n −
1: intervals C and C (blue intervals in Figure 5);(iv) 4 n −
6: intervals C , C , and left long intervals of E (green intervals in Figure 5);(v) 6 n −
6: intervals L i and L i , for every i ∈ [ n ] (orange intervals in Figure 5).Now, let M (cid:48) ( G ) be the interval model where each S j is defined exactly as S , except that we shift all theintervals to the right in a way that point 0 now coincides with point t · ( j − I in S j corresponding to the copy of an interval [ (cid:96), r ] in S is defined as [ (cid:96) + t · ( j − , r + t · ( j − m + 1)-th grained gadgets to be at the end of this model, using the same sizes ofintervals as above; i.e., H m +1 i is within the interval [2 i − t · m, i − t · m ].We have shown above that M (cid:48) ( G ) has interval count 5. The following lemma shows that the above chosenintervals satisfy the properties imposed in Subsections 2.1 and 2.2 on our constructed interval model M ( G ). Lemma 2.5 ( (cid:63) ) Let G be a cubic graph. Then, there exists an interval model M ( G ) with interval count 5 for G = ( G, π V , π E ) , for every ordering π V and π E of the vertex set and edge set of G , respectively. e Figueiredo et al. We provided in Section 2 a reduction from the
MaxCut problem having as input a cubic graph G into that of MaxCut in an interval graph G (cid:48) having ic ( G (cid:48) ) ≤
5. Although our reduction requires the choice of orderings π V and π E of respectively V ( G ) and E ( G ) in order to produce the resulting interval model, we have establishedthat we are able to construct an interval model with interval count 5 regardless of the particular choices for π V and π E (Lemma 2.5). Our reduction was based on that of [1], strengthened in order to control the intervalcount of the resulting model. This section is dedicated to discuss the interval count of the original reductionas presented in [1]. First, we establish that the original reduction yields an interval model corresponding toa graph G (cid:48) such that ic ( G (cid:48) ) = O ( (cid:112) | V ( G (cid:48) ) | ). Second, we exhibit an example of a cubic graph G for whicha choice of π V and π E yields a model M (cid:48) with interval count Ω( (cid:112) | V ( G (cid:48) ) | ), proving that this bound is tightfor some choices of π V and π E . For bridgeless cubic graphs, we are able in Lemma 3.1 to decrease the upperbound by a constant factor, but to the best of our knowledge O ( (cid:112) | V ( G (cid:48) ) | ) is the tightest upper bound.Before we go further analysing the interval count of the original reduction, it is worthy to note that a tightbound on the interval count of a general interval graph G as a function of its number of vertices n is still open.It is known that ic ( G ) ≤ (cid:98) ( n + 1) / (cid:99) and that there is a family of graphs for which ic ( G ) = ( n − / G , an interval graph G (cid:48) is defined through the constructionof one of its models M , described as follows:(i) let π V = ( v , v , . . . , v n ) and π E = ( e , e , . . . , e m ) be arbitrary orderings of V ( G ) and E ( G ), respectively;(ii) for each v i ∈ V ( G ), e j ∈ E ( G ), let G ( v i ) and G ( e j ) denote respectively a ( p, q )-grained gadget and a( p (cid:48) , q (cid:48) )-grained gadget, where: • q = 200 n + 1, p = 2 q + 7 n , and • q (cid:48) = 10 n + 1, p (cid:48) = 2 q (cid:48) + 7 n ;(iii) for each v k ∈ V ( G ), insert G ( v k ) in M such that G ( v i ) is entirely to the left of G ( v j ) if and only if i < j .For each e k ∈ E ( G ), insert G ( e k ) in M entirely to the right of G ( v n ) and such that G ( e i ) is entirely to theleft of G ( e j ) if and only if i < j ;(iv) for each e j = ( v i , v i (cid:48) ) ∈ E ( G ), with i < i (cid:48) , four intervals I i,j , I i,j , I i (cid:48) ,j , I i (cid:48) ,j are defined in M , called link intervals, such that: • I i,j and I i,j (resp. I i (cid:48) ,j and I i (cid:48) ,j ) are true twin intervals that weakly intersect G ( v i ) (resp. G ( v i (cid:48) )) tothe right; • I i,j and I i,j (resp. I i (cid:48) ,j and I i (cid:48) ,j ) weakly intersect (resp. strongly intersect) G ( e j ) to the left.By construction, therefore, I i,j and I i,j (resp. I i (cid:48) ,j and I i (cid:48) ,j ) cover all intervals in grained gadgets associatedto a vertex v (cid:96) with (cid:96) > i (resp. (cid:96) > i (cid:48) ) or an edge e (cid:96) with (cid:96) < j .Note that the number of intervals is invariant under the particular choices of π V and π E and, therefore, so isthe number of vertices of G (cid:48) . Let n (cid:48) = | V ( G (cid:48) ) | . Since G is cubic, m = 3 n/
2. By construction, n (cid:48) = n (2 p + 2 q ) + m (2 p (cid:48) + 2 q (cid:48) ) + 4 m = 1200 n + 90 n + 25 n + 21 n. and thus n = Θ( √ n (cid:48) ). Since the set of intervals covered by any link interval depends on π V and π E , distinctsequences yield distinct resulting graphs G (cid:48) having distinct interval counts. Let U be the set of all possibleinterval models that can be obtained over all possible orderings π V , π E , and G (cid:48) min be the interval graphcorresponding to a model M min such that ic ( M min ) = min { ic ( M ) : M ∈ U} . Therefore, the NP -completeness result derived from the original reduction can be strengthened to state that MaxCut is NP -complete for interval graphs having interval count at most ic ( G (cid:48) min ), and we show next that ic ( G (cid:48) min ) = O ( √ n (cid:48) ). Moreover, we will also show that there actually exists M (cid:48) ∈ U for which ic ( M (cid:48) ) = Ω( √ n (cid:48) ).First, let us show an upper bound on ic ( M min ). Note that • the intervals of all gadgets G ( v i ) and G ( e j ) can use only two interval lengths (one for all short intervals,other for all the long intervals); • for each e j = v i v i (cid:48) ∈ E ( G ), with i < i (cid:48) , both intervals I i,j and I i,j may be coincident in any model, andtherefore may have the same length. The same holds for both intervals I i (cid:48) ,j and I i (cid:48) ,j .Therefore, ic ( M min ) ≤ m + 2 = 3 n + 2 = Θ( √ n (cid:48) ). Second, we show that there is a model M (cid:48) , defined in terms7 e Figueiredo et al. of particular orderings π V , π E for which ic ( M (cid:48) ) = Ω( √ n (cid:48) ). Consider the cubic graph G depicted in Figure 6(a)which consists of an even cycle ( v , v , . . . , v n ) with the addition of the edges ( v i , v i + n ) for all 1 ≤ i ≤ n/
2. Forthe ordering π V = ( v n , v n − , . . . , v ) and any ordering π E in which the first n edges are the edges of the cycle( v , v , · · · , v n ), in this order, the reduction yields the model M of Figure 6(b) for which there is the chain I , ⊂ I , ⊂ · · · ⊂ I n,n of nested intervals, which shows that ic ( M (cid:48) ) ≥ n , and thus ic ( M (cid:48) ) = Ω( √ n (cid:48) ). (a) (b)Fig. 6. (a) A cubic graph G , and (b) the resulting model M (cid:48) for which ic ( M (cid:48) ) = Ω( √ n (cid:48) ). It can be argued from the proof of NP -completeness for MaxCut when restricted to cubic graphs [2] that,in fact, the constructed cubic graph may be assumed to have no bridges. This fact was not used in the originalreduction of [1]. In an attempt to obtain a model M having fewer lengths for bridgeless cubic graphs, we havederived Lemma 3.1. Although the absolute number of lengths in this new upper bound has decreased by aconstant factor, it is still Θ( n ) = Θ( √ n (cid:48) ). Lemma 3.1 ( (cid:63) ) Let G be a cubic bridgeless graph with n = | V ( G ) | . There exist particular orderings π V of V ( G ) and π E of E ( G ) such that: (i) there is a resulting model M produced in the original reduction of MaxCut such that ic ( M ) ≤ n/ . (ii) for all such resulting models M , we have that ic ( M ) ≥ if G is not a Hamiltonian graph. As a concluding remark, we note that the interval count of the interval model M (cid:48) produced in the originalreduction is highly dependent on the assumed orderings of V ( G ) and E ( G ), and may achieve ic ( M (cid:48) ) = Ω( √ n (cid:48) ).Our reduction enforces that ic ( M (cid:48) ) = 5 which is invariant for any such orderings. References [1] Ranendu Adhikary, Kaustav Bose, Satwik Mukherjee, Bodhayan Roy, Complexity of maximum cut on interval graphs,Available as preprint arXiv:2006.00061v2, 2020.[2] Piotr Berman, Marek Karpinski, On some tighter inapproximability results, In: Wiedermann J., van Emde Boas P., NielsenM. (eds.) International Colloquium on Automata, Languages, and Programming, ICALP 1999, Lect. Notes Comput. Sci. vol1644, pp. 200– 209. Springer, 1999.[3] Hans L. Bodlaender, Ton Kloks, Rolf Niedermeier, Simple max-cut for unit interval graphs and graphs with few P s, Electron.Notes Discret. Math. 3:19–26, 1999.[4] Hans L. Bodlaender, Celina M. H. de Figueiredo, Marisa Gutierrez, Ton Kloks, Rolf Niedermeier, Simple Max-Cut for Split-Indifference Graphs and Graphs with Few P ’s, In: Ribeiro, C.C., Martins, S.L. (eds.) Experimental and Efficient Algorithms,Third International Workshop, WEA 2004, Lect. Notes Comput. Sci. vol. 3059, pp. 87– 99. Springer, 2004.[5] Adrian Bondy, Uppaluri S. R. Murty, Graph Theory, Graduate Texts in Mathematics, Springer London, 2008.[6] Arman Boyaci, Tinaz Ekim, Mordechai Shalom, A polynomial-time algorithm for the maximum cardinality cut problem inproper interval graphs, Inf. Process. Lett. 121: 29–33, 2017.[7] M´arcia R. Cerioli, Fabiano de S. Oliveira, Jayme L. Szwarcfiter, The interval count of interval graphs and orders: a shortsurvey, J. Braz. Comp. Soc. 18: 103–112, 2012.[8] Michael Garey, David S. Johnson, Larry Stockmeyer, Some simplified NP -complete graph problems, Theor. Comput. Sci.1:237–267, 1976.[9] David S. Johnson, The NP-completeness column: an ongoing guide, J. Algorithms 6 (3): 434–451, 1985.[10] Jan Kratochv´ıl, Tom´as Masar´ık, Jana Novotn´a, U-bubble model for mixed unit interval graphs and its applications: Themaxcut problem revisited, Available as preprint arXiv:2002.08311, 2020.[11] Peter C. Fishburn, Interval Orders and Interval Graphs. Wiley, New York, 1985. e Figueiredo et al. A Proofs omitted in Subsection 2.1
Restatement of Lemma 2.2
Let x and y be positive integers, H be an ( x, y )-grained gadget and M be aninterval model that respects the structure of H . For every maximum cut [ A, B ] of G M , the following propertieshold:(i) if y + sl M ( H ) + c M ( H ) ≡ x > y − wl M ( H ) + sl M ( H ) + c M ( H ), then LS ( H ) ⊆ A and LL ( H ) ⊆ B , or vice versa;(ii) if y + sr M ( H ) + c M ( H ) ≡ x > y − wr M ( H ) + sr M ( H ) + c M ( H ), then RS ( H ) ⊆ A and RL ( H ) ⊆ B , or vice versa. Proof.
Assume that H = LS ∪ LL ∪ RS ∪ RL . (i). First, we prove that, if y + sl M ( H ) + c M ( H ) ≡
1, then theleft short intervals of H are in a same part of [ A, B ]. Let SL be the set of intervals in M that strongly intersect H to the left, and let C be the set of intervals in M that cover H . Since y + sl M ( H ) + c M ( H ) ≡
1, either |LL A | + |SL A | + |C A | > |LL B | + |SL B | + |C B | or |LL A | + |SL A | + |C A | < |LL B | + |SL B | + |C B | . Assume withoutloss of generality that the latter inequality holds. Since M respects the structure of H , N G M ( LS ) = LL∪SL∪C .Thus, | N G M ( LS ) A | < | N G M ( LS ) B | . As a result, we obtain that LS ⊆ A , otherwise [ A, B ] would not be amaximum cut of G M . Now, provided that LS ⊆ A and supposing x > y − wl M ( H ) + sl M ( H ) + c M ( H ),we prove that LL ⊆ B . Let WL be the set of intervals in M that weakly intersect H to the left. Note that,for every interval I ∈ LL , N G M ( I ) = LS ∪ ( LL \ { I } ) ∪ WL ∪ SL ∪ C ∪ RL . Consequently, since LS ⊆ A and x > y − wl M ( H ) + sl M ( H ) + c M ( H ), we obtain that, for every interval I ∈ LL , | N G M ( I ) A | = x (cid:122)(cid:125)(cid:124)(cid:123) |LS| + | ( LL \ { I } ) A | + |WL A | + |SL A | + |C A | + |RL A | > | ( LL \ { I } ) B | + |WL B | + |SL B | + |C B | + |RL B | = | N G M ( I ) B | .Therefore, LL ⊆ B , otherwise [ A, B ] would not be a maximum cut of G M . (ii). The reasoning concerning theright intervals is analogous. (cid:50) Restatement of Lemma 2.3
Let x and y be positive integers, H be an ( x, y )-grained gadget, M be an intervalmodel and [ A, B ] be a maximum cut of G M . If M respects the structure of H and ( H , M ) is well-valued, then H is either A -partitioned or B -partitioned by [ A, B ]. Proof.
Assume that H = LS ∪ LL ∪ RS ∪ RL . For the sake of contradiction, suppose that M respects thestructure of H and ( H , M ) is well-valued, but H is neither A -partitioned nor B -partitioned by [ A, B ]. Then,based on Lemma 2.2, we can assume without loss of generality that
LL ∪ RL ⊆ A and LS ∪ RS ⊆ B . Considerthe cut [ A (cid:48) , B (cid:48) ] of G M defined as follows: A (cid:48) = ( A \ RL ) ∪ RS and B (cid:48) = ( B \ RS ) ∪ RL . Note that H is B (cid:48) -partitioned by [ A (cid:48) , B (cid:48) ]. We prove that this change increases the number of edges in the cut-set, more formally,that z = | E G M ( A (cid:48) , B (cid:48) ) \ E G M ( A, B ) | − | E G M ( A, B ) \ E G M ( A (cid:48) , B (cid:48) ) | >
0, contradicting the hypothesis that [
A, B ]is a maximum cut of G M . Let WR and SR be the sets of intervals in M that weakly intersect and stronglyintersect H to the right, respectively. Also, let C be the set of intervals in M that cover H . Note that the edgesbetween RS and RL continue to be in the cut-set; so it suffices to count the edges between these intervals andthe rest. Also observe that N G M ( RL ) \ ( RL ∪ RS ) =
LL ∪ WR ∪ SR ∪ C and N G M ( RS ) \ RL = SR ∪ C .Therefore, z = |RL| · (cid:0) | ( N G M ( RL ) \ ( RL ∪ RS )) A | − | ( N G M ( RL ) \ ( RL ∪ RS )) B | (cid:1) + |RS| · (cid:0) | ( N G M ( RS ) \ RL ) B | − | ( N G M ( RS ) \ RL ) A | (cid:1) = |RL| · (cid:0) |LL| + |WR A | + |SR A | + |C A | − |WR B | − |SR B | − |C B | (cid:1) + |RS| · (cid:0) |SR B | + |C B | − |SR A | − |C A | (cid:1) = y + y · (cid:0) |WR A | − |WR B | (cid:1) + ( y − x ) · (cid:0) |SR A | + |C A | (cid:1) + ( x − y ) · (cid:0) |SR B | + |C B | (cid:1) .This implies that z ≤ y ≤ y · (cid:0) |WR B | − |WR A | (cid:1) + ( x − y ) · (cid:0) |SR A | + |C A | (cid:1) + ( y − x ) · (cid:0) |SR B | + |C B | (cid:1) ≤ y · |WR| + ( x − y ) · (cid:0) |SR A | + |C A | − |SR B | − |C B | (cid:1) ≤ y · wr M ( H ) + ( x − y ) · (cid:0) sr M ( H ) + c M ( H ) (cid:1) . e Figueiredo et al. This is not the case since H is compatible with M , and we get the desired contradiction, that is, that z > H is either A -partitioned or B -partitioned by [ A, B ]. (cid:50) B Proof of Theorem 2.1
Let G be a cubic graph, π V = ( v , . . . , v n ) be an ordering of V ( G ), π E be an ordering of E ( G ), and G =( G, π V , π E ). Recall that q = 28 n + 1, p = 2 q + 7 n , q (cid:48) = 18 n + 1 and p (cid:48) = 2 q (cid:48) + 5 n . Before we start, we givea more precise definition of alternating partitioned cut.A cut [ A, B ] of G M ( G ) is called alternating partitioned if the following properties hold (below, we assume i < i (cid:48) whenever we write e j = v i v i (cid:48) ):(I) for each i ∈ [ n ] and each j ∈ [ m + 1], H ji is either A -partitioned or B -partitioned by [ A, B ];(II) for each j ∈ [ m ], with e j = v i v i (cid:48) , if H ji is A -partitioned by [ A, B ], then { C j , C j } ⊆ B ; otherwise, { C j , C j } ⊆ A ;(III) for each j ∈ [ m ], with e j = v i v i (cid:48) , if H ji (cid:48) is A -partitioned by [ A, B ], then { C j , C j } ⊆ B and E j is A -partitioned by [ A, B ]; otherwise, { C j , C j } ⊆ A and E j is B -partitioned by [ A, B ];(IV) for each i ∈ [ n ] and each j ∈ [ m ], if H ji is A -partitioned by [ A, B ], then { L j − i , L ji } ⊆ B and H j +1 i is B -partitioned by [ A, B ]; otherwise, { L j − i , L ji } ⊆ A and H j +1 i is A -partitioned by [ A, B ]. Lemma B.1
Let G be a cubic graph on n vertices and m = 3 n/ edges, π V = ( v , . . . , v n ) be an ordering of V ( G ) , π E = ( e , . . . , e m ) be an ordering of E ( G ) and G = ( G, π V , π E ) . For each i ∈ [ n ] and each j ∈ [ m + 1] ,we get that M ( G ) respects the structure of H ji and ( H ji , M ( G )) is well-valued. Additionally, for each j ∈ [ m ] ,we get that M ( G ) respects the structure of E j and ( E j , M ( G )) is well-valued. Proof.
By construction, it is immediate that M ( G ) respects the structures of H ji for each i ∈ [ n ] and each j ∈ [ m + 1], and M ( G ) respects the structures of E j for each j ∈ [ m ]. Thus, it only remains to prove that thepairs ( H ji , M ( G )) and ( E j , M ( G )) are well-valued.Let i ∈ [ n ] and j ∈ [ m + 1], and consider H ji = LS ∪ LL ∪ RS ∪ RL . First, we show that ( H ji , M ( G )) iswell-valued. Note that there is no interval in M ( G ) that strongly intersect H ji to the left or to the right. Thereare at most four intervals in M ( G ) that weakly intersect H ji to the right, namely the intervals L j − i , L ji , theintervals C j , C j if i = min( e j (cid:48) ), and the intervals C j , C j if i = max( e j (cid:48) ). Moreover, if j = 1, then there is nointerval in M ( G ) that weakly intersects to the left; otherwise, if j >
1, then there are exactly two intervals in M ( G ) that weakly intersect to the left, namely L j − i and L j − i . We also note that there are always an evennumber of intervals in M ( G ) that cover H ji , and that there are at most 2( n + 1) such intervals. Indeed, H ji is covered by the intervals L j − , L j , . . . , L j − i − , L ji − , by the intervals L j − i +1 , L j − i +1 , . . . , L j − n , L j − n if j > C j , C j if i > min( e j (cid:48) ), and by the intervals C j , C j if i > max( e j (cid:48) ). Finally, we remark that,by definition, q is odd. Thus, we have that0 ≡ sl M ( G ) ( H ji ) ≡ sr M ( G ) ( H ji ) (cid:54)≡ q + c M ( G ) ( H ji ) ≡ p = 2 q + 7 n > q + 2 n + 5 ≥ q − { wl M ( H ji ) + sl M ( G ) ( H ji ) , wr M ( G ) ( H ji ) + sr M ( G ) ( H ji ) } + c M ( H ji ).Finally, since q = 28 n + 1, we have that q = 784 n + 56 n + 1 > n + 168 n + 14 n + 16 n + 6= 2 qn + 6 q + 14 n + 14 n = 4 q + ( p − q )2 n (+1) ≥ q · wr M ( G ) ( H ji ) + ( p − q ) · (cid:0) sr M ( G ) ( H ji ) + c M ( G ) ( H ji ) (cid:1) .10 e Figueiredo et al. Therefore, the pair ( H ji , M ( G )) is well-valued.Now, let j ∈ [ m ], and consider E = LS ∪LL∪RS ∪RL . We prove that ( E j , M ( G )) is well-valued. Note thatthere is no interval in M ( G ) that weakly/strongly intersects E j to the right. There are exactly two intervalsin M ( G ) that weakly intersect E j to the left, namely the intervals C j and C j . There are exactly two intervalsin M ( G ) that strongly intersect E j to the left, namely the intervals C j and C j . We also note that there areexactly 2 n intervals in M ( G ) that cover E j , namely the intervals L j − , L j , . . . , L j − n , L jn . Finally, we remarkthat, by definition, q (cid:48) is odd. Thus, we have that0 ≡ sl M ( G ) ( E j ) ≡ sr M ( G ) ( E j ) (cid:54)≡ q + c M ( G ) ( E j ) ≡ p (cid:48) = 2 q (cid:48) + 5 n > q (cid:48) + 2 n + 3 = 2 q − { wl M ( E j ) + sl M ( G ) ( E j ) , wr M ( G ) ( E j ) + sr M ( G ) ( E j ) } + c M ( E j ).Finally, since q (cid:48) = 18 n + 1, we have that( q (cid:48) ) = 324 n + 36 n + 1 > n + 10 n + 2 n = 2 q (cid:48) n + 10 n = 2 p (cid:48) n + 2 q (cid:48) n ≥ q · wr M ( G ) ( E j ) + ( p (cid:48) − q (cid:48) ) · (cid:0) sr M ( G ) ( E j ) + c M ( G ) ( E j ) (cid:1) .Therefore, the pair ( E j , M ( G )) is well-valued. (cid:50) Recall that if [
A, B ] is an alternating partitioned cut of G M ( G ) , we let Φ( A, B ) = [
X, Y ] be the cut of G defined as follows: for each vertex v i ∈ V ( G ), v i ∈ X if and only if H i is A -partitioned by [ A, B ]. We remarkthat [
X, Y ] is well-defined and uniquely determined by [
A, B ]. On the other hand, given a cut [
X, Y ] of G ,there is a unique alternating partitioned cut [ A, B ] = Φ − ( X, Y ) of G M ( G ) such that [ X, Y ] = Φ(
A, B ).For each cubic graph G on n vertices and each positive integer k , recall that f ( G, k ) = (cid:0) n / n (cid:1) (2 pq + q ) + 3 n/ p (cid:48) q (cid:48) + ( q (cid:48) ) ) + 6 nq ( n + 1)+ (cid:0) n + 3 n (cid:1) ( n − p + q ) + 3 n ( p (cid:48) + q (cid:48) ) + 3 n (( k + 1) q (cid:48) + p (cid:48) ) + 4 k .Before we present the proof of Lemma 2.4, we need the following. Note that this lemma tells us that if[ X (cid:48) , Y (cid:48) ] has a cut-set bigger than [ X, Y ] = Φ(
A, B ), then [ A (cid:48) , B (cid:48) ] = Φ − ( X (cid:48) , Y (cid:48) ) has a cut-set bigger than[ A, B ]. Also, given an edge e j , if e j = v i v i (cid:48) with i < i (cid:48) , in what follows we denote i by min( e j ) and i (cid:48) bymax( e j ). Lemma B.2
Let G be a cubic graph on n vertices, π V = ( v , . . . , v n ) be an ordering of V ( G ) , π E =( e , . . . , e n/ ) be an ordering of E ( G ) , G = ( G, π V , π E ) , [ A, B ] be an alternating partitioned cut of G M ( G ) and [ X, Y ] = Φ(
A, B ) . If k = | E G ( X, Y ) | , then f ( G, k ) ≤ | E G M ( G ) ( A, B ) | < f ( G, k (cid:48) ) for any integer k (cid:48) > k . Proof.
Since [
A, B ] is an alternating partitioned cut of G M ( G ) , we shall count the edges in the cut-set E G M ( G ) ( A, B ) according to the following three types of intervals incident to these edges: the edges in thecut-set that have an endpoint in a ( p, q )-grained gadget; the edges in the cut-set that have an endpoint ina ( p (cid:48) , q (cid:48) )-grained gadget; and the edges in the cut-set that have both endpoints in a link interval and/or aninterval of the type C (cid:96)j .First, we count the edges in the cut-set that have an endpoint in a ( p, q )-grained gadget. The possiblecombinations are as follows.(1.1) Edges within ( p, q )-grained gadgets related to vertices. More formally, (cid:91) i ∈ [ n ] ,j ∈ [ m +1] E G M ( G ) (( H ji ) A , ( H ji ) B ).Since each such gadget is either A -partitioned or B -partitioned and m = 3 n/
2, that there are exactly11 e Figueiredo et al. ( n + n )(2 pq + q ) such edges.(1.2) Edges between link intervals L j − i and L ji , and the ( p, q )-gadgets related to vertices. More formally, (cid:91) i ∈ [ n ] ,j ∈ [ m ] (cid:16) E G M ( G ) (( H ji ) A , { L j − i , L ji } B ) ∪ E G M ( G ) ( { L j − i , L ji } A , ( H ji ) B ) ∪ E G M ( G ) (( H j +1 i ) A , { L j − i , L ji } B ) ∪ E G M ( G ) ( { L j − i , L ji } A , ( H j +1 i ) B ) (cid:17) .Because intervals L j − i and L ji intersect exactly H ji and H j +1 i for every i ∈ [ n ], and these intersectionsare not shared, there are exactly m · n · (2 q + 2 q ) = 6 n q such edges.(1.3) Edges between intervals C j , . . . , C j and the ( p, q )-grained related to the vertices incident to edge e j . Moreformally, (cid:91) j ∈ [ m ] (cid:16) E G M ( G ) (( H j min( e j ) ) A , { C j , C j } B ) ∪ E G M ( G ) ( { C j , C j } A , ( H j min( e j ) ) B ) ∪ E G M ( G ) (( H j max( e j ) ) A , { C j , C j } B ) ∪ E G M ( G ) ( { C j , C j } A , ( H j max( e j ) ) B ) (cid:17) .Writing e j as v i v i (cid:48) , i < i (cid:48) , because C j , C j are opposite to the left long intervals of H ji , the same holdingfor C j , C j and H ji (cid:48) , note that there are exactly n (2 q + 2 q ) = 6 nq such edges.(1.4) Edges between ( p, q )-grained gadgets related to vertices, and link intervals covering them. More formally, (cid:91) i ∈ [ n ] ,j ∈ [ m ] (cid:16) E G M ( G ) (( H ji +1 , ∪ · · · ∪ H jn ∪ H j +11 ∪ · · · ∪ H j +1 i − ) A , { L j − i , L ji } B ) ∪ E G M ( G ) ( { L j − i , L ji } A , ( H ji +1 , ∪ · · · ∪ H jn ∪ H j +11 ∪ · · · ∪ H j +1 i − ) B ) (cid:17) .Note that, because each ( p, q )-grained gadget is either A -partitioned or B -partitioned, and since L j − i and L ji are in the same part, if they cover H j (cid:48) i (cid:48) , then we count exactly 2( p + q ) edges for each covered gadget.Since L j − i and L ji cover H j (cid:48) i (cid:48) if and only if ( i (cid:48) , j (cid:48) ) ∈ { (1 , j ) , . . . , ( i − , j ) , ( i + 1 , j + 1) , . . . , ( n, j + 1) } ,we get that they cover exactly n − mn such pairs of linkvertices, we get that there are exactly mn ( n − p + 2 q ) = 3 n ( n − p + q ) such edges.(1.5) Edges between intervals C j , . . . , C j and ( p, q )-grained gadgets covered by them. More formally, (cid:91) j ∈ [ m ] (cid:16) E G M ( G ) (( H j min( e j )+1 , . . . , H jn ) A , { C j , C j } B ) ∪ E G M ( G ) ( { C j , C j } A , ( H j min( e j )+1 , . . . , H jn ) B ) ∪ E G M ( G ) (( H j max( e j )+1 , . . . , H jn ) A , { C j , C j } B ) ∪ E G M ( G ) ( { C j , C j } A , ( H j max( e j )+1 , . . . , H jn ) B ) (cid:17) .Since the graph G is cubic, observe that, given a vertex v i ∈ V ( G ) incident to edges e j , e j , e j , we getthat the grained gadgets covered by some interval in I = (cid:83) { C j , . . . , C j : j ∈ { j , j , j }} are exactly thegrained gadgets of the type H ji (cid:48) , for every j ∈ { j , j , j } and every i (cid:48) ∈ { i +1 , . . . , n } . Also, these are goingto be covered by exactly two intervals, C j , C (cid:48) j ∈ { C j , . . . , C j } . Finally, because each ( p, q )-grained gadgetis either A -partitioned or B -partitioned, and since C j , C (cid:48) j are in the same part, for each j ∈ { j , j , j } ,we get that there are exactly (cid:80) i ∈ [ n ] n − i )( p + q ) = 3 n ( n − p + q ) such edges.Second, we count the edges in the cut-set that have an endpoint in a ( p (cid:48) , q (cid:48) )-grained gadget. The possiblecombinations are as follows. 12 e Figueiredo et al. (2.1) Edges within ( p (cid:48) , q (cid:48) )-grained gadgets related to edges. More formally, (cid:91) j ∈ [ m ] E G M ( G ) ( E Aj , E Bj ).Note that there are exactly n (2 p (cid:48) q (cid:48) + ( q (cid:48) ) ) such edges.(2.2) Edges between ( p (cid:48) , q (cid:48) )-grained gadgets related to edges and the link intervals covering them. More formally, (cid:91) j ∈ [3 n/ (cid:16) E G M ( G ) ( E Aj , { L j − , L j , . . . , L j − n , L jn } B ) ∪ E G M ( G ) ( { L j − , L j , . . . , L j − n , L jn } A , E Bj ) (cid:17) .As before, one can see that we can count exactly p (cid:48) + q (cid:48) edges, for each interval covering E j . Since thereare exactly 2 n intervals covering E j , we get that there are exactly 3 n ( p (cid:48) + q (cid:48) ) such edges.(2.3) Edges between ( p (cid:48) , q (cid:48) )-grained gadget E j and intervals C j , . . . , C j . More formally, (cid:91) j ∈ [ m ] (cid:16) E G M ( G ) ( E Aj , { C j , . . . , C j } B ) ∪ E G M ( G ) ( { C j , . . . , C j } A , E Bj ) (cid:17) .To count the size of this set, for each j ∈ [ m ], let α j = | E G M ( G ) ( E Aj , { C j , C j } B ) ∪ E G M ( G ) ( { C j , C j } A , E Bj ) | and β j = | E G M ( G ) ( E Aj , { C j , C j } B ) ∪ E G M ( G ) ( { C j , C j } A , E Bj ) | .Write e j as v i v i (cid:48) , i < i (cid:48) , and note that Conditions (II) and (III) tell us that C j , C j (resp. C j , C j ) areopposite to the right long intervals of H ji (resp. H ji (cid:48) ), and that the left long intervals of E j are oppositeto C j , C j . This means that C j , C j are opposite to the left long intervals of E j if and only if the rightlong intervals of H ji and H ji (cid:48) are in opposite parts. In other words, for each j ∈ [ m ], either α j = 0 or α j = 2 q (cid:48) ; moreover, α j = 2 q (cid:48) if and only if v i and v i (cid:48) are in distinct parts of [ X, Y ], where i = min( e j ) and i (cid:48) = max( e j ). Additionally, since C j , C j strongly intersect E j and are opposite to its left long intervals, weget β j = 2( p (cid:48) + q (cid:48) ), for each j ∈ [ m ]. It follows that there are exactly n (2 kq (cid:48) +2( p (cid:48) + q (cid:48) )) = 3 n (( k +1) q (cid:48) + p (cid:48) )such edges (recall that k = | E G ( X, Y ) | ).Third, we count the edges in the cut-set that have both endpoints in a link interval and/or an interval ofthe type C (cid:96)j for some (cid:96) ∈ { , . . . , } and j ∈ [ m ].(3.1) Edges between intervals C j , C j and C j , C j . More formally, (cid:91) j ∈ [ m ] (cid:16) E G M ( G ) ( { C j , C j } A , { C j , C j } B ) ∪ E G M ( G ) ( { C j , C j } A , { C j , C j } B ) (cid:17) .Similarly to the last case, for each j ∈ [ m ], if c j = | E G M ( G ) ( { C j , C j } A , { C j , C j } B ) ∪ E G M ( G ) ( { C j , C j } A , { C j , C j } B ) | ,then either c j = 0 or c j = 4; and c j = 4 if and only if v i and v i (cid:48) are in distinct parts of [ X, Y ], where i = min( e j ) and i (cid:48) = max( e j ). It follows that there are exactly 4 k such edges.(3.2) Edges between pairs of intervals L j − , L j , . . . , L j − n , L jn . More formally, (cid:91) j ∈ [ m ] E G M ( G ) ( { L j − , L j , . . . , L j − n , L jn } A , { L j − , L j , . . . , L j − n , L jn } B ).13 e Figueiredo et al. Denote { L j − , L j , . . . , L j − n , L jn } by L j , and note that the maximum number of edges between L Aj and L Bj occurs when each subset has size n . We then get that there are at most (cid:80) j ∈ [ m ] n = mn = n suchedges.(3.3) Edges between intervals L j − , L j , . . . , L j − n , L jn and intervals in C j , . . . , C j . More formally, (cid:91) j ∈ [ m ] (cid:16) E G M ( G ) ( { L j − , L j , . . . , L j − n , L jn } A , { C j , . . . , C j } B ) ∪ E G M ( G ) ( { C j , . . . , C j } A , { L j − , L j , . . . , L j − n , L jn } B ) (cid:17) .Note that, if i = min( e j ), then the intervals L j − i , L ji and C j , C j belong to a same part of [ A, B ];analogously, if i (cid:48) = max( e j ), then L j − i (cid:48) , L ji (cid:48) and C j , C j belong to a same part of [ A, B ]. Therefore,each interval in { C j , · · · , C j } is incident to at most 2( n −
1) edges in the cut-set that have an interval in { L j − , L j , . . . , L j − n , L jn } as endpoint. We then get that there are at most (cid:80) j ∈ [ m ] n −
1) = 8 m ( n −
1) =12 n ( n −
1) = 12 n − n such edges.(3.4) Edges between link intervals in consecutive regions of the escalator. More formally, (cid:91) j ∈{ ,...,m } (cid:16) E G M ( G ) ( { L j − , L j − , . . . , L j − n , L j − n } A , { L j − , L j , . . . , L j − n , L jn } B ) ∪ E G M ( G ) ( { L j − , L j , . . . , L j − n , L jn } A , { L j − , L j − , . . . , L j − n , L j − n } B ) (cid:17) .Note that, for each i ∈ [ n ], the only link intervals in { L j − , L j − , . . . , L j − n , L j − n } that are adjacentto L j − i , L ji are the intervals L j − i +1 , L j − i +1 , . . . , L j − n , L j − n . Therefore, each interval in { L j − i , L ji } isincident to at most 2( n − i ) edges in the cut-set that have an interval in { L j − , L j − , . . . , L j − n , L j − n } as endpoint. Then, one can verify there are at most (cid:88) j ∈{ ,...,m } (cid:88) i ∈ [ n ] n − i ) = (cid:88) j ∈{ ,...,m } n ( n −
1) = 2( m − n ( n −
1) = 3 n ( n − − n ( n −
1) = 3 n − n + 2 n such edges.(3.5) Finally, edges between intervals C j , . . . , C j and link intervals in the previous regions of the escalator.More formally, (cid:91) j ∈{ ,...,m } (cid:16) E G M ( G ) ( { L j − , L j − , . . . , L j − n , L j − n } A , { C j , . . . , C j } B ) ∪ E G M ( G ) ( { C j , . . . , C j } A , { L j − , L j − , . . . , L j − n , L j − n } B ) (cid:17) .Using an argument similar to the one in item (1.5), one can verify there are at most (cid:80) i ∈ [ n ] n − i ) =6 n − n such edges.Therefore, summing up the number of edges in the cut-set E G M ( G ) ( A, B ) according to three types describedabove, except for the edges described in Cases (3.2)–(3.5) which, as we have seen, do not give exact values, weobtain that | E G M ( G ) ( A, B ) | ≥ (cid:18) n n (cid:19) (2 pq + q ) + 3 n p (cid:48) q (cid:48) + ( q (cid:48) ) ) + 6 nq ( n + 1)+ (3 n + 3 n )( n − p + q ) + 3 n ( p (cid:48) + q (cid:48) ) + 3 n (( k + 1) q (cid:48) + p (cid:48) ) + 4 k = f ( G, k ).On the other hand, note that the number of edges in Cases (3.2)–(3.5) is upper bounded by n + 13 n − n .14 e Figueiredo et al. Thus, since q (cid:48) > n + 13 n − n , we have: f ( G, k ) ≤ | E G M ( G ) ( A, B ) | ≤ f ( G, k ) + 9 n n − n < f ( G, k ) + q (cid:48) . As a result, because there is a factor kq (cid:48) in f ( G, k ), we obtain that f ( G, k (cid:48) ) > | E G M ( G ) ( A, B ) | for any k (cid:48) > k . (cid:50) The proof of Lemma 2.4 will employ the following definition and result:Let H be an ( x, y )-grained gadget, M be an interval model that respects H and such that ( M , H ) is well-valued, and let I ⊆ M \ H . We say that H is indifferent to I in M if, for every maximum cut [ A, B ] of G M ,the number of edges in the related cut-set incident to H ∪ I does not depend on whether H is A -partitionedor B -partitioned. More formally, H is indifferent to I in M if, for every maximum cut [ A, B ] of G M , we havethat | E G M (( H ∪ I ) A , ( H ∪ I ) B ) | = | E G M (( H ∪ I ) A (cid:48) , ( H ∪ I ) B (cid:48) ) | ,where A (cid:48) = ( A \ H A ) ∪ H B and B (cid:48) = ( B \ H B ) ∪ H A . Lemma B.3
Let x and y be positive integers, H be an ( x, y ) -grained gadget, M be an interval model thatrespects the structure of H and such that ( H , M ) is well-valued, and let C ⊆ M \ H . If every interval in C covers H , then H is indifferent to C in M . Proof.
Assume that every interval in C covers H . It follows from Lemma 2.3 that, for every maximum cut [ A, B ]of G M , H is either A -partitioned or B -partitioned by [ A, B ]. This implies that, if [
A, B ] is a maximum cut of G M , then |H A | = |H B | . Consequently, for every maximum cut [ A, B ] of G M , | N G M ( C ) ∩H A | = | N G M ( C ) ∩H B | ,since every interval in C covers H . Therefore, H is indifferent to C in M . (cid:50) Finally, below, we restate and prove Lemma 2.4. It is worth mentioning that this result is a key element,and the most involved component, of the proof of Theorem 2.1.
Restatement of Lemma 2.4
If [
A, B ] is a maximum cut of G M ( G ) , then [ A, B ] is an alternating partitionedcut.
Proof.
Let [
A, B ] be a maximum cut of G M ( G ) . We prove that the Properties (I)–(IV) — required for a cutof G M ( G ) to be alternating partitioned — are satisfied by [ A, B ].(I) It follows from Fact B.1 that, for each i ∈ [ n ] and each j ∈ [ m + 1], M ( G ) respects the structure of H ji and ( M ( G ) , H ji ) is well-valued; the same holds for M ( G ) and E j for every j ∈ [ m ]. Consequently, byLemma 2.3, for each i ∈ [ n ] and each j ∈ [ m + 1], H ji is either A -partitioned or B -partitioned by [ A, B ].(II) Let j ∈ [ m ], and write e j = v i v i (cid:48) , with i < i (cid:48) . Suppose H ji is A -partitioned (the case in which H ji is B -partitioned is analogous). For the sake of contradiction, suppose that { C j , C j } (cid:54)⊆ B . Then, considerthe cut [ A (cid:48) , B (cid:48) ] of G M ( G ) defined as follows: A (cid:48) = A \ { C j , C j } and B (cid:48) = B ∪ { C j , C j } . We show that z = | E G M ( G ) ( A (cid:48) , B (cid:48) ) \ E G M ( G ) ( A, B ) | − | E G M ( G ) ( A, B ) \ E G M ( G ) ( A (cid:48) , B (cid:48) ) | >
0, contradicting the hypothesis that[
A, B ] is a maximum cut of G M ( G ) . Note that N G M ( G ) ( { C j , C j } ) = RL ( H ji ) ∪ { C j , . . . , C j } ∪ { L j − i (cid:48) , L j − i (cid:48) : i (cid:48) ∈ { i + 1 , . . . , n } , j > }∪ { L j − i (cid:48) , L ji (cid:48) : i (cid:48) ∈ [ n ] } ∪ ( H ji +1 ∪ · · · ∪ H jn ) ∪ LL ( E j ).In particular, we have that the intervals C j and C j cover the gadgets H ji +1 , . . . , H jn . Consequently, it followsfrom Lemma B.3 that H ji +1 , . . . , H jn are indifferent to { C j , C j } . Thus, one can verify that z = |{ C j , C j } A | · (cid:16) |RL ( H ji ) | + |{ C j , C j } A | − |{ C j , . . . , C j } B | + |{ L j − i (cid:48) , L j − i (cid:48) : i (cid:48) ∈ { i + 1 , . . . , n } , j > } A |− |{ L j − i (cid:48) , L j − i (cid:48) : i (cid:48) ∈ { i + 1 , . . . , n } , j > } B | + |{ L j − i (cid:48) , L ji (cid:48) : i (cid:48) ∈ [ n ] } A | − |{ L j − i (cid:48) , L ji (cid:48) : i (cid:48) ∈ [ n ] } B | + |LL ( E j ) A | − |LL ( E j ) B | (cid:17) ≥ min { q − − n − i ) − n − q (cid:48) , q − − n − i ) − n − q (cid:48) } . Hence, since q > q (cid:48) + 2(2 n − i ) + 3, we obtain that z >
0, which contradicts the hypothesis that [
A, B ] is a15 e Figueiredo et al. maximum cut of G M ( G ) . Therefore, { C j , C j } ⊆ B .(III) Let j ∈ [ m ]. Assume that i = max( e j ), H ji is A -partitioned (the case in which H ji is B -partitionedis analogous. For the sake of contradiction, suppose that { C j , C j } (cid:54)⊆ B . Then, consider the cut [ A (cid:48) , B (cid:48) ] of G M ( G ) defined as follows: A (cid:48) = A \ { C j , C j } and B (cid:48) = B ∪ { C j , C j } . We show that z = | E G M ( G ) ( A (cid:48) , B (cid:48) ) \ E G M ( G ) ( A, B ) | − | E G M ( G ) ( A, B ) \ E G M ( G ) ( A (cid:48) , B (cid:48) ) | >
0, contradicting the hypothesis that [
A, B ] is a maximumcut of G M ( G ) . Note that N G M ( G ) ( { C j , C j } ) = RL ( H ji ) ∪ { C j , . . . , C j } ∪ { L j − i (cid:48) , L j − i (cid:48) : i (cid:48) ∈ { i + 1 , . . . , n } , j > }∪ { L j − i (cid:48) , L ji (cid:48) : i (cid:48) ∈ [ n ] } ∪ ( H ji +1 ∪ · · · ∪ H jn ) ∪ LL ( E j ) ∪ LS ( E j ).Thus, similarly to the proof of property (II), one can verify that z ≥ min { q − − n − i ) − n + q (cid:48) − p (cid:48) , q − − n − i ) − n + 2 q (cid:48) − p (cid:48) } .Consequently, since q > p (cid:48) − q (cid:48) + 2(2 n − i ) + 3, we obtain that z >
0, which contradicts the hypothesis that[
A, B ] is a maximum cut of G M ( G ) . Therefore, { C j , C j } ⊆ B .Now, we prove that E j is A -partitioned by [ A, B ]. For the sake of contradiction, suppose that this is not thecase. Then, by Lemma B.1 and Lemma 2.3, E j is B -partitioned by [ A, B ]. Consider the cut [ A (cid:48)(cid:48) , B (cid:48)(cid:48) ] of G M ( G ) defined as follows: A (cid:48)(cid:48) = ( A \ E Aj ) ∪ E Bj and B (cid:48)(cid:48) = ( B \ E Bj ) ∪ E Aj . Note that, H ji is A (cid:48)(cid:48) -partitioned by [ A (cid:48)(cid:48) , B (cid:48)(cid:48) ], { C j , C j } ⊆ B (cid:48)(cid:48) and E j is A (cid:48)(cid:48) -partitioned by [ A (cid:48)(cid:48) , B (cid:48)(cid:48) ]. We show that z = | E G M ( G ) ( A (cid:48)(cid:48) , B (cid:48)(cid:48) ) \ E G M ( G ) ( A, B ) | −| E G M ( G ) ( A, B ) \ E G M ( G ) ( A (cid:48)(cid:48) , B (cid:48)(cid:48) ) | >
0, contradicting the hypothesis that [
A, B ] is a maximum cut of G M ( G ) .Note that N G M ( G ) ( E j ) = { L j − i (cid:48) , L ji (cid:48) : i (cid:48) ∈ [ n ] } ∪ { C j , . . . , C j } .In particular, we have that the intervals L j − , L j , . . . , L j − n , L jn cover the gadget E j . Consequently, it followsfrom Lemma B.3 that E j is indifferent to { L j − i (cid:48) , L ji (cid:48) : i (cid:48) ∈ [ n ] } . Moreover, we have that the intervals C j and C j weakly intersect E j to the left, and the intervals C j and C j strongly intersect E j to the left. Thus, since { C j , C j } ⊆ B , one can verify that z = |{ C j , C j } A | · |LL ( E j ) | − |{ C j , C j } B | · |LL ( E j ) | − |{ C j , C j }| · |LL ( E j ) | + |{ C j , C j }| · |LS ( E j ) |≥ − q (cid:48) − q (cid:48) + 2 p (cid:48) = 2 p (cid:48) − q (cid:48) . Hence, since p (cid:48) > q (cid:48) , we obtain that z >
0, which contradicts the hypothesis that [
A, B ] is a maximum cut of G M ( G ) . Therefore, E j is A -partitioned by [ A, B ].(IV) Let i ∈ [ n ] and j ∈ [ m ]. Assume that H ji is A -partitioned (the case in which H ji is B -partitioned isanalogous). In this case, we want to ensure that { L j − i , L ji } ⊆ B and that H j +1 i is B -partitioned. The generalidea is to switch intervals of part in order to make this true. However, we will possibly need to switch intervalsinside more than one region all at once. Generally, for the desired condition to hold for every j (cid:48) ∈ { j, . . . , m } ,we must have that, if j (cid:48) has the same parity as j , then { L j (cid:48) − i , L j (cid:48) i } ⊆ B and H j (cid:48) +1 i is A -partioned, while theopposite must occur if j (cid:48) has different parity from j . The definitions of agreement below capture this notion.For each j (cid:48) ∈ { j, . . . , m } , we say that { L j (cid:48) − i , L j (cid:48) i } partially agrees (resp. agrees ) with H ji with respectto [ A, B ] if either j (cid:48) − j ≡ { L j (cid:48) − i , L j (cid:48) i } ∩ B (cid:54) = ∅ (resp. { L j − i , L ji } ⊆ B ), or j (cid:48) − j ≡ { L j (cid:48) − i , L j (cid:48) i } ∩ A (cid:54) = ∅ (resp. { L j − i , L ji } ⊆ A ). Similarly, for each j (cid:48) ∈ { j, . . . , m } , we say that { L j (cid:48) − i , L j (cid:48) i } partially disagrees (resp. disagrees ) with H ji with respect to [ A, B ] if either j (cid:48) − j ≡ { L j (cid:48) − i , L j (cid:48) i }∩ A (cid:54) = ∅ (resp. { L j − i , L ji } ⊆ A ), or j (cid:48) − j ≡ { L j (cid:48) − i , L j (cid:48) i } ∩ B (cid:54) = ∅ (resp. { L j − i , L ji } ⊆ B ).Additionally, for each j (cid:48) ∈ { j + 1 , . . . , m + 1 } , we say that H j (cid:48) i agrees ( disagrees ) with H ji with respect to[ A, B ] if either j (cid:48) − j ≡ H j (cid:48) i is A -partitioned by [ A, B ] (resp. B -partitioned), or j (cid:48) − j ≡ H j (cid:48) i is B -partitioned by [ A, B ] (resp. A -partitioned).We prove that { L j − i , L ji } and H j +1 i agree with H ji with respect to [ A, B ]. For the sake of contradiction,suppose that { L j − i , L ji } (cid:54)⊆ B or H j +1 i is A -partitioned by [ A, B ]. Now, as already mentioned, we want to16 e Figueiredo et al. switch intervals of part in order to satisfy the desired condition. The general idea in what follows is to takethe closest subsequent region to H ji that agrees or partially agrees with it, say the r -th region; then, to switchintervals of part in such a way as to ensure that all grained gadgets and link intervals between the j -th andthe r -th regions, themselves included, agree with H ji . For this, we define the following indices.Let l be the least integer in { j + 1 , . . . , m } such that { L l − i , L li } partially agrees with H ji with respect to[ A, B ] (see Figures B.1 and B.3), if it exists; otherwise, let l = m + 2. Similarly, let h be the least integer in { j + 1 , . . . , m + 1 } such that H j (cid:48) i agrees with H ji with respect to [ A, B ] (see Figures B.2 and B.4), if it exists;otherwise, let h = m + 2. We remark that, for each j (cid:48) ∈ { j + 1 , . . . , h − } , H j (cid:48) i disagrees with H ji with respectto [ A, B ]. Moreover, for each j (cid:48) ∈ { j + 1 , . . . , l − } , { L j (cid:48) − i , L j (cid:48) i } disagrees with H ji with respect to [ A, B ].We want to switch of part the intervals which are in between the ( j + 1)-th and the min { l, h } -th regions,themselves included. For this, we formally define next the subset S comprising all such intervals. It isworth mentioning that, besides the grained-gadgets H j (cid:48) i and the link intervals L j (cid:48) − i , L j (cid:48) i , possibly some ofthe grained-gadgets E j (cid:48) and some of the intervals C j (cid:48) , . . . , C j (cid:48) must belong to S . Indeed, it follows fromProperties (II)–(III) that, if e j (cid:48) is incident to v i and i = min( e j (cid:48) ), then C j (cid:48) , C j (cid:48) are influenced by H j (cid:48) i ; and, if e j (cid:48) is incident to v i and i = max( e j (cid:48) ), then C j (cid:48) , C j (cid:48) and E j (cid:48) are influenced by H j (cid:48) i .Thus, let S ⊆ M ( G ) be the subset of intervals defined as follows: if l < h or l = h = m + 2 (see Figures B.1and B.2b), then (below l (cid:48) = min { l, m } ) S =( H j +1 i ∪ · · · ∪ H l (cid:48) i ) ∪{ L j (cid:48) − i , L j (cid:48) i : j (cid:48) ∈ { j + 1 , . . . , l (cid:48) }} ∪ { C j (cid:48) , C j (cid:48) : j (cid:48) ∈ { j + 1 , . . . , l (cid:48) } , i = min( e j (cid:48) ) }∪{ C j (cid:48) , C j (cid:48) : j (cid:48) ∈ { j + 1 , . . . , l (cid:48) } , i = max( e j (cid:48) ) } ∪ { I ∈ E j (cid:48) : j (cid:48) ∈ { j + 1 , . . . , l (cid:48) } , i = max( e j (cid:48) ) } ;otherwise, if l ≥ h and h < m (see Figures B.2a and B.3), then S =( H j +1 i ∪ · · · ∪ H h − i ) ∪{ L j (cid:48) − i , L j (cid:48) i : j (cid:48) ∈ { j + 1 , . . . , h − }} ∪ { C j (cid:48) , C j (cid:48) : j (cid:48) ∈ { j + 1 , . . . , h − } , i = min( e j (cid:48) ) }∪{ C j (cid:48) , C j (cid:48) : j (cid:48) ∈ { j + 1 , . . . , h − } , i = max( e j (cid:48) ) } ∪ { I ∈ E j (cid:48) : j (cid:48) ∈ { j + 1 , . . . , h − } , i = max( e j (cid:48) ) } .Now, let [ A (cid:48) , B (cid:48) ] be the cut of G M ( G ) defined as follows: A (cid:48) = A \ ( { L j − i , L ji } ∪ S A ) ∪ S B and B (cid:48) = B \ S B ∪ { L j − i , L ji } ∪ S A .One can verify that, for each j (cid:48) ∈ { j, . . . , min { h, l } − } , { L j (cid:48) − i , L j (cid:48) i } and H j (cid:48) +1 i agree with H ji withrespect to [ A (cid:48) , B (cid:48) ]. Moreover, we have that Properties (I)-(III) still hold for [ A (cid:48) , B (cid:48) ], provided that they holdfor [ A, B ]. It remains to show that z = | E G M ( G ) ( A (cid:48) , B (cid:48) ) \ E G M ( G ) ( A, B ) | − | E G M ( G ) ( A, B ) \ E G M ( G ) ( A (cid:48) , B (cid:48) ) | > A, B ] is a maximum cut of G M ( G ) . In order to prove this, we note that, foreach j (cid:48) ∈ [ m + 1], N G M ( G ) ( H j (cid:48) i ) \ H j (cid:48) i = { L j (cid:48) − i (cid:48) , L j (cid:48) − i (cid:48) : i (cid:48) ∈ { i + 1 , . . . , n } , j (cid:48) > } ∪ { L j (cid:48) − i (cid:48) , L j (cid:48) i (cid:48) : i (cid:48) ∈ [ i ] }∪ { C j (cid:48) , C j (cid:48) : i ≥ min( e j (cid:48) ) } ∪ { C j (cid:48) , C j (cid:48) : i ≤ max( e j (cid:48) ) } ; (B.1)17 e Figueiredo et al. Furthermore, we note that, for each j (cid:48) ∈ [ m ], N G M ( G ) ( { L j (cid:48) − i , L j (cid:48) i } ) = { L j (cid:48) − i (cid:48) , L j (cid:48) − i (cid:48) : i (cid:48) ∈ { i + 1 , . . . , n } , j (cid:48) > } ∪ { L j (cid:48) − i (cid:48) , L j (cid:48) i (cid:48) : i (cid:48) ∈ [ n ] }∪ { L j (cid:48) +1 i (cid:48) , L j (cid:48) +2 i (cid:48) : i (cid:48) ∈ [ i − , j (cid:48) < n/ } ∪ RL ( H j (cid:48) i ) ∪ LL ( H j (cid:48) +1 i ) ∪ ( H j (cid:48) i +1 ∪ · · · ∪ H j (cid:48) n ) ∪ ( H j (cid:48) +11 ∪ · · · ∪ H j (cid:48) +1 i − ) ∪ { C j (cid:48) , . . . , C j (cid:48) } , (B.2) N G M ( G ) ( { C j (cid:48) , C j (cid:48) } ) = RL ( H j (cid:48) i ) ∪ { C j (cid:48) , . . . , C j (cid:48) } ∪ { L j (cid:48) − i (cid:48) , L j (cid:48) − i (cid:48) : i (cid:48) ∈ { i + 1 , . . . , n } , j (cid:48) > }∪ { L j (cid:48) − i (cid:48) , L j (cid:48) i (cid:48) : i (cid:48) ∈ [ n ] } ∪ ( H j (cid:48) i +1 ∪ · · · ∪ H j (cid:48) n ) ∪ LL ( E j (cid:48) ), (B.3) N G M ( G ) ( { C j (cid:48) , C j (cid:48) } ) = RL ( H j (cid:48) i ) ∪ { C j (cid:48) , . . . , C j (cid:48) } ∪ { L j (cid:48) − i (cid:48) , L j (cid:48) − i (cid:48) : i (cid:48) ∈ { i + 1 , . . . , n } , j (cid:48) > }∪ { L j (cid:48) − i (cid:48) , L j (cid:48) i (cid:48) : i (cid:48) ∈ [ n ] } ∪ ( H j (cid:48) i +1 ∪ · · · ∪ H j (cid:48) n ) ∪ LL ( E j (cid:48) ) ∪ LS ( E j (cid:48) ), and (B.4) N G M ( G ) ( E j (cid:48) ) = { L j (cid:48) − i (cid:48) , L j (cid:48) i (cid:48) : i (cid:48) ∈ [ n ] } ∪ { C j (cid:48) , . . . , C j (cid:48) } . (B.5)In particular, by Lemma B.3, the gadget H j (cid:48) i is indifferent to { L j (cid:48) − i +1 , L j (cid:48) − i +1 , . . . , L j (cid:48) − n , L j (cid:48) − n } and to { L j (cid:48) − , L j (cid:48) , . . . , L j (cid:48) − i − , L j (cid:48) i − } ; the gadget E j (cid:48) is indifferent to { L j (cid:48) − , L j (cid:48) , . . . , L j (cid:48) − n , L j (cid:48) n } ; the gadgets H j (cid:48) i +1 , . . . , H j (cid:48) n are indifferent to { L j (cid:48) − i , L j (cid:48) i } , to { C j (cid:48) , C j (cid:48) } if i > min( e j (cid:48) ), and to { C j (cid:48) , C j (cid:48) } if i > max( e j (cid:48) );and the gadgets H j (cid:48) , . . . , H j (cid:48) i − are indifferent to { L j (cid:48) − i , L j (cid:48) i } .Next, we define a function g in order to count the number of edges between link intervals and intervals C j (cid:48) , . . . , C j (cid:48) which are in the cut-set associated with [ A (cid:48) , B (cid:48) ] but are not in the cut-set associated with [ A, B ].More importantly, through a trivial lower bound for this function, together with some case analysis, we provethat the number of crossing edges originated from the cut [ A (cid:48) , B (cid:48) ] is strictly greater than the number of crossingedges originated from the cut [ A, B ]; in other words, we prove that z > j (cid:48) ∈ { , . . . , m } and each part X ∈ { A, B } , consider a Xj (cid:48) = |{ L j (cid:48) − i (cid:48) , L j (cid:48) − i (cid:48) : i (cid:48) ∈ { i + 1 , . . . , n } , j (cid:48) > } X | + |{ L j (cid:48) − i (cid:48) , L j (cid:48) i (cid:48) : i (cid:48) ∈ [ n ] \ { i }} X | , b Xj (cid:48) = |{ L j (cid:48) +1 i (cid:48) , L j (cid:48) +2 i (cid:48) : i (cid:48) ∈ [ i − , j (cid:48) < n/ } X | , c Xj (cid:48) = |{ L j (cid:48) − i , L j (cid:48) i } X | , and d Xj (cid:48) = |{ C j (cid:48) , C j (cid:48) : i = min( e j (cid:48) ) } X | + |{ C j (cid:48) , C j (cid:48) : i = max( e j (cid:48) ) } X | .And, for each pair ( α, β ) ∈ { ( j, min { l, n/ } ) , ( j, h − , ( j + 1 , min { l, n/ } ) , ( j + 1 , h − } , let g ( α, β ) = (cid:80) j (cid:48) ∈{ α,...,β } j (cid:48) − α ≡ (cid:0) ( a Aj (cid:48) + b Aj (cid:48) − a Bj (cid:48) − b Bj (cid:48) − c Bj (cid:48) − d Bj (cid:48) ) · c Aj (cid:48) + ( a Aj (cid:48) − a Bj (cid:48) ) · d Aj (cid:48) (cid:1) + (cid:80) α ∈{ j,...,β } j (cid:48) − α ≡ (cid:0) ( a Bj (cid:48) + b Bj (cid:48) − a Aj (cid:48) − b Aj (cid:48) − c Aj (cid:48) − d Aj (cid:48) ) · c Bj (cid:48) + ( a Bj (cid:48) − a Aj (cid:48) ) · d Bj (cid:48) (cid:1) It is not hard to check that g ( α, β ) ≥ (cid:80) j (cid:48) ∈{ α,...,β } j (cid:48) − α ≡ (cid:0) ( − a Bj (cid:48) − b Bj (cid:48) − c Bj (cid:48) − d Bj (cid:48) ) · c Aj (cid:48) + ( − a Bj (cid:48) ) · d Aj (cid:48) (cid:1) + (cid:80) j (cid:48) ∈{ α,...,β } j (cid:48) − α ≡ (cid:0) ( − a Aj (cid:48) − b Aj (cid:48) − c Aj (cid:48) − d Aj (cid:48) ) · c Bj (cid:48) + ( − a Aj (cid:48) ) · d Bj (cid:48) (cid:1) ≥ (cid:80) j (cid:48) ∈{ α,...,β } ( − n − n − β − α + 1) · ( − n − n − ≥ − n + 6 n .We split the remaining of this proof into four cases, where we specify the values for α and β , according to theconsidered case. 18 e Figueiredo et al. Case 1.
First, suppose that { L j − i , L ji } (cid:54)⊆ B , and that l < h or l = h = 3 n/ l < h or l = h = 3 n/ H j +1 i is A -partitioned by [ A, B ]. Fig. B.1. Example of the case in which { L j − i , L ji } (cid:54)⊆ B , and l < h or l = h = 3 n/ l = j + 3. Theintervals colored with blue belong to A and the intervals colored with red belong to B . (For readability, some intervals of M ( G )are omitted.) Then, based on equations (B.1)–(B.5), one can verify that z = |RL ( H ji ) | · |{ L j − i , L ji } A | + |{ L j − i , L ji } B | · |LL ( H j +1 i ) | + |{ I ∈ RL ( H li ) : l ≤ n/ }| · |{ L l − i , L li : l ≤ n/ , l − j ≡ } A | + |{ I ∈ RL ( H li ) : l ≤ n/ }| · |{ L l − i , L li : l ≤ n/ , l − j ≡ } B | + |{ L l − i , L li : l ≤ n/ − , l − j ≡ } A | · |{ I ∈ LL ( H l +1 i ) : l ≤ n/ − } A | + |{ L l − i , L li : l ≤ n/ − , l − j ≡ } B | · |{ I ∈ LL ( H l +1 i ) : l ≤ n/ − } B | + g ( j, min { l, n/ } ) ≥ q + g ( j, min { l, n/ } ) ≥ q − n + 6 n .Therefore, since q > n − n , we obtain that z > Case 2.
Suppose that { L j − i , L ji } (cid:54)⊆ B , and that l ≥ h and h < n/ (a) h = j + 1(b) h = j + 3Fig. B.2. Examples of the case in which { L j − i , L ji } (cid:54)⊆ B , l ≥ h and h < n/ A and the intervals colored with red belong to B . (For readability, some intervals of M ( G ) are omitted.) Then, based on equations (B.1)–(B.5), one can verify that z = |RL ( H ji ) | · |{ L j − i , L ji } A | + |{ L j − i , L ji } A | · |LL ( H j +1 i ) A | + |{ L j − i , L ji } B | · |LL ( H j +1 i ) B | + |{ L h − − i , L h − i : ( h − − j ≡ , h > j + 1 } A | · |LL ( H hi ) | + |{ L h − − i , L h − i : ( h − − j ≡ , h > j + 1 } B | · |LL ( H hi ) | + g ( j, h − ≥ q + g ( j, h − ≥ q − n + 6 n .Therefore, since q > n − n , we obtain that z >
0. 19 e Figueiredo et al.
Case 3.
Suppose that { L j − i , L ji } ⊆ B , H j +1 i is A -partitioned by [ A, B ], and that l < h or l = h = 3 n/ Fig. B.3. Example of the case in which { L j − i , L ji } ⊆ B , H j +1 i is A -partitioned by [ A, B ], and l < h or l = h = 3 n/ l = j + 3. The intervals colored with blue belong to A and the intervals colored with red belong to B . (For readability,some intervals of M ( G ) are omitted.) Then, based on equations (B.1)–(B.5), one can verify that z = |{ L j − i , L ji }| · |LL ( H j +1 i ) | + |{ I ∈ RL ( H li ) : l ≤ n/ }| · |{ L l − i , L li : l ≤ n/ , l − j ≡ } A | + |{ I ∈ RL ( H li ) : l ≤ n/ }| · |{ L l − i , L li : l ≤ n/ , l − j ≡ } B | + |{ L l − i , L li : l ≤ n/ − , l − j ≡ } A | · |{ I ∈ LL ( H l +1 i ) : l ≤ n/ − } A | + |{ L l − i , L li : l ≤ n/ − , l − j ≡ } B | · |{ I ∈ LL ( H l +1 i ) : l ≤ n/ − } B | + g ( j + 1 , min { l, n/ } ) ≥ q + g ( j + 1 , min { l, n/ } ) ≥ q − n + 6 n .Therefore, since q > n − n , we obtain that z > Case 4.
Suppose that { L j − i , L ji } ⊆ B , H j +1 i is A -partitioned by [ A, B ]and that l ≥ h and h < n/ h > j + 1 in this case. Fig. B.4. Example of the case in which { L j − i , L ji } ⊆ B , H j +1 i is A -partitioned by [ A, B ], l ≥ h and h < n/ h = j + 3. The intervals colored with blue belong to A and the intervals colored with red belong to B . (For readability,some intervals of M ( G ) are omitted.) Then, based on equations (B.1)–(B.5), one can verify that z = |{ L j − i , L ji }| · |LL ( H j +1 i ) | + |{ L h − − i , L h − i : ( h − − j ≡ } A | · |LL ( H hi ) | contradicting + |{ L h − − i , L h − i : ( h − − j ≡ } B | · |LL ( H hi ) | + g ( j + 1 , h − ≥ q + g ( j + 1 , h − ≥ q − n + 6 n .Therefore, since q > n − n/
2, we obtain that z > (cid:50) The following lemma, together with Lemma 2.5 that is proved next, finish the proof of Theorem 2.1.
Lemma B.4
Let G be a cubic graph on n vertices, π V = ( v , . . . , v n ) be an ordering of V ( G ) , π E =( e , . . . , e n/ ) be an ordering of E ( G ) and G = ( G, π V , π E ) . For each positive integer k , mc ( G ) ≥ k ifand only if mc ( G M ( G ) ) ≥ f ( G, k ) . e Figueiredo et al. Proof.
First, suppose that mc ( G ) ≥ k . Then, there exists a cut [ X, Y ] of G such that | E G ( X, Y ) | ≥ k . Let[ A, B ] be the unique alternating partitioned cut of G M ( G ) that, for each i ∈ [ n ], satisfies the following condition:if v i ∈ X , then H i is A -partitioned; otherwise, H i is B -partitioned. One can verify that [ A, B ] = Φ − ( X, Y ).Therefore, it follows from Lemma B.2 that mc ( G M ( G ) ) ≥ | E G M ( G ) ( A, B ) | ≥ f ( G, k ). Conversely, suppose that mc ( G M ( G ) ) ≥ f ( G, k ). Then, there exists a cut [
A, B ] of G M ( G ) such that | E G M ( G ) ( A, B ) | ≥ f ( G, k ). Assumethat [
A, B ] is a maximum cut of G M ( G ) . It follows from Lemma 2.4 that [ A, B ] is an alternating partitionedcut. Consequently, by Lemma B.2, [
X, Y ] = Φ(
A, B ) is a cut of G such that | E G ( X, Y ) | ≥ k . Indeed, if | E G ( X, Y ) | < k , then | E G M ( G ) ( A, B ) | < f ( G, k ). Therefore, mc ( G ) ≥ k . (cid:50) B.1 Proof of Lemma 2.5
In what follows, given a subset of intervals I , the left endpoint of I is equal to the leftmost point of I ; moreformally, it is the point min I ∈I (cid:96) ( I ). Similarly, the right endpoint of I is equal to max I ∈I r ( I ).Recall that, assuming e = v h v h (cid:48) with h < h (cid:48) , the intervals in S are defined as below. • For each i ∈ [ n ], the left long intervals of H i are equal to [2 i − , i − /
2] and the left short intervals areany choice of q distinct points within the open interval (2 i − , i − / H i are equal to [2 i − / , i −
1] and the right short intervals are any choice of q distinct points withinthe open interval (2 i − / , i − • C and C are equal to [2 h − , h + 2 n − • C and C are equal to [2 h (cid:48) − , h (cid:48) + 4 n − • The left long intervals of E are equal to [2 n, n − • The left short intervals of E are any choice of q (cid:48) distinct points in the open interval (4 n − , n − • The right long intervals of E are equal to [6 n − , n − /
2] and the right short intervals are any choiceof q (cid:48) distinct points within the open interval (6 n − , n − / • For each i ∈ [ n ], intervals L i , L i are equal to [2 i − , i + 6 n − I in S j corresponding to the copy of an interval [ (cid:96), r ] in S is defined as[ (cid:96) + t · ( j − , r + t · ( j − H m +1 i is within the interval [2 i − t · m, i − t · m ].Below, we restate and prove Lemma 2.5. Restatement of Lemma 2.5
Let G be a cubic graph. Then, there exists an interval model M ( G ) withinterval count 5 for G = ( G, π V , π E ), for every ordering π V and π E of the vertex set and edge set of G ,respectively. Proof.
Observe that the number of true twin intervals (values p, q, p (cid:48) , q (cid:48) ) are not important here. Instead, weare only interested in the structural properties. As previously said, we show that the above chosen intervalssatisfy the properties imposed in Subsections 2.1 and 2.2 on our constructed interval model M ( G ).First, we recall the conditions on Subsection 2.1 that define grained gadgets:(i) |LS| = |RS| = x and |LL| = |RL| = y ;(ii) for each two intervals I, I (cid:48) ∈ LL ∪ RL , I ∩ I (cid:48) (cid:54) = ∅ ;(iii) for each I ∈ LS and each I (cid:48) ∈ H \ { I } , I ∩ I (cid:48) (cid:54) = ∅ if and only if I (cid:48) ∈ LL ;(iv) for each I ∈ RS and each I (cid:48) ∈ H \ { I } , I ∩ I (cid:48) (cid:54) = ∅ if and only if I (cid:48) ∈ RL .As previously said, the number of intervals is indifferent, therefore in what follows Condition i is consideredto hold. Now, consider a grained gadget H ji , for some i ∈ [ n ] and j ∈ [ m + 1]. For Condition ii, observe thatthe intersection between the left and right long intervals of H ji is exactly the point 2 i − / t · ( j − i − t · ( j − , i − / t · ( j − i − / t · ( j − , i − t · ( j − E j , for every j ∈ [ m + 1].Now, recall that, in Section 2.2, the following conditions must hold for every i ∈ [ n ]:(I) for each pair j, j (cid:48) ∈ [ m + 1] with j (cid:54) = j (cid:48) , each I ∈ H ji and each I (cid:48) ∈ H j (cid:48) i , we have that I ∩ I (cid:48) = ∅ ;(II) for each j ∈ [ m ], the intervals L j − i and L ji are true twins in G D , weakly intersect H ji to the right andweakly intersect H j +1 i to the left; 21 e Figueiredo et al. (III) if i (cid:54) = 1, then for each j ∈ [ m ], the intervals L j − i − and L ji − cover H ji , and the intervals L j − i and L ji cover H j +1 i − .So, consider i ∈ [ n ] and an arbitrary grained gadget H ji of M (cid:48) ( G ), for some j ∈ [ m + 1]. By construction,we know that:(*) H ij is contained in the interval [2 i − t · ( j − , i − t · ( j − j (cid:48) ∈ { j + 1 , . . . , m + 1 } . By the above equation, the leftmost endpoint of H j (cid:48) i is equal to (cid:96) = 2 i − t · ( j (cid:48) − H ji is equal to r = 2 i − t · ( j − (cid:96) =2 i − t · ( j −
1) + t · ( j (cid:48) − j ) − r + t · ( j (cid:48) − j ) −
1. It thus follows that (cid:96) > r since t · ( j (cid:48) − j ) ≥ t > t = 6 n − j ∈ [ m ]. By construction, we know that L j − i = L ji = [2 i − t · ( j − , i + 6 n − t · ( j − H ji to theright. Also, the leftmost endpoint of H j +1 i is equal to 2 i − t · j = 2 i − t + t · ( j −
1) = 2 i + 6 n − t · ( j − L ji weakly intersects H j +1 i to the left, settling Condition (II).Finally, consider i (cid:54) = 1 and j ∈ [ m ]. Denote intervals L ji − and L ji by L and L (cid:48) , respectively. We know that L = [2 i − t · ( j − , i + 6 n − t · ( j − L (cid:48) = [2 i − t · ( j − , i + 6 n − t · ( j − L is smaller than the left endpoint of H ji . Also, the right endpoint of L can berewritten as 2 i − t · ( j −
1) + 6 n −
8, which is bigger than the right endpoint of H ji , since G is cubic andtherefore n ≥
4. It thus follows that L covers H ji . As for the second part, we first write H j +1 i − , which is equalto [2 i − t · j, i − t · j ]. Observe that the left endpoint of L (cid:48) can be rewritten as 2 i − t · j − t + 3,which is smaller than the left endpoint of H j +1 i − since t = 6 n − > n ≥
2. Similarly, the right endpoint of L (cid:48) can be rewritten as 2 i − t · j + 6 n − − t = 2 i − t · j + 1, which is bigger than the right enpoint of H j +1 i − . Condition (III) thus follows.Now, recall that, in Subsection 2.2, for every edge e j , the conditions below must be satisfied. We write e j as v h v h (cid:48) where h < h (cid:48) . Also, D denotes the escalator, i.e. the set (cid:83) i ∈ [ n ] ( H i ∪ · · · ∪ H m +1 i ) ∪ { L i , . . . , L mi } .(a) for each I ∈ D and each I (cid:48) ∈ E j , I ∩ I (cid:48) (cid:54) = ∅ if and only if I ∈ { L j − i , L ji : i ∈ [ n ] } , and in this case, I covers E j ;(b) the intervals C j and C j are true twins in G M ( G ) , weakly intersect H jh to the right and weakly intersect E j to the left;(c) the intervals C j and C j are true twins in G M ( G ) , weakly intersect H jh (cid:48) to the right and strongly intersect E j to the left.Consider j ∈ [ m ]. By construction, we know that E j is contained in [2 n + t · ( j − , n − / t · ( j − D , if we prove that E j does not intersect H jn nor H j +11 ,then it follows that I ∩ I (cid:48) = ∅ , for every I ∈ D \ { L j − h , L jh : h ∈ [ n ] } and every I (cid:48) ∈ E j . This is indeed thecase since the right endpoint of H jn is 2 n − t · ( j − E j , whilethe left endpoint of H j +11 is t · j = t + t · ( j − E j since 6 n − / t . Now, consider a link interval L = L ji for some i ∈ [ n ]. We know that the left endpoint of L is at most equal to the right endpoint of H jn , and that the right endpoint of L is at least equal to the leftendpoint of H j +11 . From what is previously said, it follows that L covers E j , and Condition (a) follows.The fact that C j and C j are true twins follows by construction; therefore, in what follows it suffices to provethe condition for C j . Now, consider again e j = v h v h (cid:48) , and recall that C j = [2 h − t · ( j − , h +2 n − t · ( j − C j weakly intersects H jh to the right. Now, observe that the right long intervalsof E j are equal to [6 n − t · ( j − , n − / t · ( j − n − t · ( j − , n − t · ( j − C j , r . Note that r is atmost 4 n − t · ( j − h ≤ n − h < h (cid:48) ), and therefore C j does not intersect any left shortinterval of E j . Also, because h ≥ r is at least 2 n + t · ( j −
1) and therefore C j intersects every leftlong interval of E j . It thus follows that C j , C j weakly intersect E j to the left, and hence Condition (b) holds.Finally, consider C j = C j = [2 h (cid:48) − t · ( j − , h (cid:48) + 4 n − t · ( j − e Figueiredo et al. and therefore we analyse only C j . By construction, one can see that C j weakly intersects H jh (cid:48) . Now, considerthe rightmost point of C j , r (cid:48) . Note that r (cid:48) is at least 4 n − t · ( j −
1) since 2 ≤ h (cid:48) , and therefore C j intersectsall left short intervals of E j . Also r (cid:48) is at most 6 n − t · ( j −
1) since h (cid:48) ≤ n , while the left endpoint of theright long intervals of E j is 6 n − t · ( j − C j strongly intersects E j to the left, andhence Condition (c) holds, finishing the proof. (cid:50) C Proofs omitted in Section 3
The proof of Lemma 3.1 will employ the following result:
Lemma C.1 (Petersen, 1891)
Every cubic bridgeless graph admits a perfect matching.
Restatement of Lemma 3.1
Let G be a cubic bridgeless graph with n = | V ( G ) | . There exist particularorderings π V of V ( G ), π E of E ( G ) such that:(i) there is a resulting model M produced in the original reduction of MaxCut such that ic ( M ) ≤ n/ M , we have that ic ( M ) ≥ G is not a Hamiltonian graph. Proof.
Let G be a cubic bridgeless graph with V ( G ) = { v , v , . . . , v n } . By Lemma C.1, G admits a perfectmatching M . Let H = G \ M . Therefore, H is 2-regular and, therefore, H consists of a disjoint union ofcycles C , C , . . . , C k , for some k ≥
1. For all 1 ≤ i ≤ k , let π iV = v i , v i , . . . , v ik i be an ordering of the verticesof C i , with k i = | C i | , such that ( v ij , v ij +1 ) ∈ E ( C i ) for all 1 ≤ j ≤ k i , where v ik i +1 = v i . Let π iE be theordering ( v i , v i ) , ( v i , v i ) , . . . , ( v ik i − , v ik i ) , ( v i , v ik i ) for all 1 ≤ i ≤ k . Let π M be any ordering of the edges of M such that ( v i , v r ) < ( v j , v s ) in π M only if v i < v j in π V . Finally, let π V be the ordering of V ( G ) obtainedfrom the concatenation of the orderings π V , π V , . . . , π kV , and π E be the ordering of E ( G ) obtained from theconcatenation of the orderings π E , π E , . . . , π kE , π M .In order to prove (ii), assume G is not a Hamiltonian graph. Therefore k >
1. Observe that there is thefollowing chain of nested intervals I ⊂ I ⊂ I ⊂ I ⊂ I , where • I is the leftmost interval in RS ( G ( v )), • I is an interval in RL ( G ( v )), • I is a link interval corresponding to both G ( v ) and G ( v v ), • I is a link interval corresponding to both G ( v ) and G ( v v k ), and • I is a link interval corresponding to both G ( v ) and G ( e ), where e is the edge of M incident to v ,since r ( I ) < r ( I ) < r ( I ) < r ( I ) < r ( I ) < (cid:96) ( I ) < (cid:96) ( I ) < (cid:96) ( I ) < (cid:96) ( I ) < (cid:96) ( I ). Thus, for all such resultingmodels M , we have that ic ( M ) ≥ M , produced by the original reduction of MaxCut considering orderings π V and π E , such that ic ( M ) ≤ n/ n = | V ( G ) | . Let L be the setof all link intervals of the grained gadgets corresponding to edges of M , that is, L = { I i,k , I i,k , I j,k , I j,k : e k =( i, j ) ∈ M } . Moreover, let L be the set of all link intervals of the grained gadgets corresponding to the edges( v i , v ik i ) of C i and the vertex v i for all 1 ≤ i ≤ k , that is, L = { I v i ,k , I v i ,k : 1 ≤ i ≤ k , e k = ( v i , v ik i ) ∈ C i } .Note that | L | = k ≤ n/ | L | = 4 · | M | = 2 n . Let L = L ∪ L . Let M (cid:48) = M \ L . We claim that ic ( M (cid:48) ) ≤
3. Since each pair of true twins I j,k , I j,k and I i,k , I i,k in L can have the same length in M , it followsfrom this claim that ic ( M ) ≤ | L | + | L | / ic ( M (cid:48) ) ≤ n/ n + 3 = 4 n/ M (cid:48)(cid:48) be the interval model obtained from M (cid:48) by removing all intervals correspondingto the grained gadgets (or, in other words, by keeping only the intervals corresponding to link intervals). It iseasily seen that M (cid:48)(cid:48) is a proper interval model, that is, no interval is properly contained in another. Therefore,the interval graph corresponding to M (cid:48)(cid:48) is a proper interval graph and M (cid:48)(cid:48) can be modified so that theirintervals have all a single length. Since it is possible to bring all grained gadgets back to M (cid:48)(cid:48) using two morelengths, we have that ic ( M (cid:48) ) ≤
3, as claimed. (cid:50)(cid:50)