Constrained intersection structure

Another way to constrain an interval representation of an interval graph is to prescribe the intersection structure between all pairs of intervals. This can be done in form of an interval matrixµ= (µu,v)_u̸=v∈V with entriesµu,v∈ {di,ov,cd,cs}. A function α: V → I, whereI is an interval system, is aninterval representationofµif it holds for allu̸= v∈V that

α(u)∩α(v) =_∅⇔µ_u,v =di, α(u)≬^α(v)⇔µ_u,v =ov, α(u)⊊^α(v)⇔µu,v =cd, and α(u)⊋^α(v)⇔µ_u,v =cs.

Note thatµspecifies anunderlying graph G_µ on the vertex set V(G_µ) =V that contains the edges E(G_µ) =^{{u,v} ∈^(V₂^{) ⏐}⏐µ_u,v ̸=di}

. Any interval representation ofµis also one of G_µ; on the other hand, an interval representation of G_µ is only one of µ if it additionally satisfies the restrictions given by the non-dientries ofµ.

LetO= (O_u)_u∈V be a family of equivalence relations such that eachO_upartitions the set{v ∈ V|µ_u,v = ov}ofov-neighbors of u into at most two equivalence classes. An interval representationαofµis calledO-respectingif for any three verticesu,v,w∈ V withµ_u,v= µ_u,w =ov, the intervalsα(v)andα(w)contain the same extreme point ofα(u) if and only if(v,w)∈O_u.

This section gives a logspace reduction from computingO-respecting interval repre-sentations of interval matrices to computing(ℓ_,s)-respecting interval representations of interval graphs.

4.6 Constrained intersection structure As a first step, the following lemma observes that an (O-respecting) interval represen-tationαof an interval matrixµcan be required to be sharp, i.e., that every point of the resulting interval systemα(µ)is the extreme point of exactly one interval.

Lemma 4.6.1. Letµ = (µ_u,v)_u̸=v∈V be an interval matrix and let O = (Ou)_u∈V be a family of equivalence relations. Given any interval representation α: V → I of µ, a sharp interval representationα^′: V → I^′ ofµcan be obtained in logspace. Moreover, ifαis O-respecting, then so isα^′.

Proof. Intuitively, the sharp interval representationα^′ is obtained from αby dropping all points that are not an extreme point of an interval and by individualizing coinciding extreme points as outlined in Figure4.11: Place the start points in descending order of interval length, followed by the end points in ascending order of interval length. Note that these modifications preserve overlaps and inclusions, so α^′ is an (O-respecting) interval representation of µ. Clearly,α^′ can be computed in logspace. The following equivalences follow directly from the definition of α^′. so α^′ is like α an interval representation of µ. Moreover, if α is O-respecting, then so is α^′. To see that α^′ is sharp, observe that for all v ∈ V, it holds that l(v),r(v) ∈ {0, . . . , 2· |V| −1}, and the above equivalences implyl(u)̸=r(v)for allu,v∈Vas well as l(u)̸=l(v)_andr(u)̸=r(v)_{for all}u ̸=v ∈V. (Note thatu̸= vimplies α(u)̸= α(v)_, as all inclusions allowed byµare proper.)

The key observation is that the entries of an interval matrixµ= (µ_u,v)_u̸=v∈V together with the equivalence relations in O = (O_u)_u∈V already determine the lengths of all intervals and their pairwise intersections in any sharp interval representation ofµ. LetGµ

be the underlying graph. Define the functionsℓ_µ: V → N^{+and s}µ,O: E(Gµ) → N^+as The next lemma shows thatℓ_{µ and}s_µ,O reflect the lengths of the intervals and pairwise intersections in any sharpO-respecting interval representation of µ.

Lemma4.6.2. Letµ= (µu,v)_u̸=v∈V be an interval matrix and let O= (Ou)_u∈Vbe a family of equivalence relations. Then any sharp O-respecting interval representationα:V → I ofµis an (ℓ_µ,s_µ,O)-respecting interval representation of the underlying graph G_µ.

Proof. Letαbe a sharpO-respecting interval representation ofµ. This directly implies thatαis also an interval representation of the underlying graphG_µ.

To see that αisℓ_µ-respecting, consider anyv∈V. As αis sharp, the cardinality of the intervalα(v)is exactly the number of extreme points contained in it. Clearly,α(v)must contain both of its own extreme points. For each other vertexw∈ V\ {v}, the matrix this time together with the equivalence relations in O, determine the cardinality of α(u)∩α(v). If µu,v ∈ {cd,cs}, then the length of the contained interval equals the length of the intersection. The case µ_u,v = ov is more complicated and illustrated in Figure4.12(b). The intersectionα(u)∩α(v)contains exactly one extreme point for each ofα(u)andα(v). The contribution of every other vertexw∈V\ {u,v}is determined by the matrix entriesµ_u,wandµ_v,wand the equivalence relationO_w. Indeed, if at least one of these entries isdiorcd, then no extreme point of α(w)lies inα(u)∩α(v); if both entries

Figure4.12: Proof of Lemma 4.6.2: The contribution of different types of neighbors to (a)ℓ_µ(v)and (b) s_µ,O(

{u,v}⁾in caseµu,v =ov.

4.7 Completeness results the other iscs, then exactly one extreme point ofα(w)lies inα(u)∩α(v). If both entries areov, there are two cases: Ifα(w)overlaps bothα(u)andα(v)from the same side, i.e.

if (_u,v)∈Ow, then one extreme point ofα(_w)_{lies in}α(_u)∩α(_v), otherwise none. Thus

⏐

⏐α(u)∩α(v)^⏐⏐=s_µ,O(

{u,v}⁾.

These two lemmas allow to compute interval representations of interval matrices.

Theorem 4.6.3. Given an interval matrix µ = (µ_u,v)_u̸=v∈V and a family O = (O_u)_u∈V of equivalence relations, a sharp O-respecting interval representation α: V → I ofµcan be com-puted in logspace; if this is not possible, the algorithm detects this. Moreover, for any sharp O-respecting interval representation α^′: V → I^′ of µ, there is a hypergraph isomorphism φ fromI toI^′ with φ(

α(v)⁾ =α^′(v)for all v∈V.

Proof. Ifµadmits a sharpO-respecting interval representation, Lemma4.6.2shows that the underlying graph G_µ admits an (ℓ_µ,s_µ,O)-respecting interval representation. Note that the graph Gµand the functionsℓ_µ ands_µ,O can easily be computed fromµandOin logspace. By Theorem4.5.5, an(ℓ_µ,s_µ,O)-respecting interval representationα₀:V → I₀ of G_µcan be computed in logspace; if the algorithm detects that there is no suchα₀, the input µand O is rejected. Moreover, the resulting interval model I₀ is unique up to isomorphism. It is not hard to see that α₀is also an interval representation ofµ. Indeed, µ_u,v = diis equivalent to {u,v} ∈/ E(G_µ)and thus to α₀(u)∩α₀(v) = _∅; and likewise µu,v = cd is equivalent to ℓ_µ(u) = s_µ,O(

{u,v}⁾ < ℓ_µ(v) and thus to α0(u) ⊊ ^α0(v). Further, Lemma 4.5.3 implies that α₀ is O-respecting. Note though, that while the interval model I₀ must be isomorphic to a sharp interval model of µ, it does not need to be sharp itself; for example,I₀might be {

[1, 4],[3, 4]^}. However, Lemma4.6.1 allows to transformα₀into a sharp interval representationα: V → I ofµin logspace. By Lemma4.6.2,αis(ℓ_µ,s_µ,O)-respecting, so by the uniqueness part of Theorem4.5.5there is a hypergraph isomorphism φfrom I toI₀ withφ(α(v)) =α₀(v)for allv∈V. Applying the same argument to an arbitrary sharpO-respecting interval representationα^′: V→ I^′ of µproves themoreoverpart of the theorem.

Combining this with the logspace algorithm of Theorem4.2.6for canonical representa-tion of interval hypergraphs gives the following corollary.

Corollary 4.6.4. Canonical O-respecting interval representations of interval matrices can be