From HCA graphs to HCA matrices

Figure5.1: Overview of the canonical representation algorithm for HCA graphs.

5.2.1 From HCA graphs to HCA matrices

Letµ= (µ_i,j)_i̸=j∈V be a quadratic matrix. The elements ofVare called the verticesofµ and it is assumed thatV is linearly ordered. Another quadratic matrixλ= (λ_i,j)_{i̸=j∈U is} isomorphictoµ(writtenλ∼= µ) if there is a bijectionσ: U→ Vsuch thatλ_i,j =µ_σ(i),σ(j)

for alli ̸= j ∈ U. Note that two graphs are isomorphic if and only if their adjacency matrices are isomorphic.

Anintersection matrixis a matrixµ= (µ_u,v)_u̸=v∈V with entriesµ_u,v ∈ {di,cs,cd,cc,ov} that satisfies (a)µu,v =cd⇔µv,u =csand (b) µu,v =µv,u in all other cases. The idea is that intersection matrices describe the intersection structure between the arcs of an arc systemA. When two arcs Aand Boverlap (i.e., A≬B), it will be helpful to distinguish the case where A contains both extreme points of B, which is called circle cover and denoted by A B, from the case whereAcontains only one extreme point of B, which is calledstrict overlapand denoted by A B.

Recall that an arc systemAissharpif every point on the circle V(A)is the extreme point of exactly one arc inA. The following notation was introduced by Lin and Szwarcfiter.

Definition 5.2.1 [LS09]. Let A be an arc system with V(A) ∈ A/ where each arc has

A matrix µ is a(Helly) circular-arc matrix if there is a (Helly) arc system A such that µ∼= µA.

Note that a matrixµis an interval matrix as defined in Section4.6if and only if there is an interval systemI such thatµ∼=µ_I.

Definition 5.2.2 [LS09]. Given a graph G without twins and universal vertices, its neighborhood matrixλ_G = (λu,v)_u̸=v∈V(G) is defined by the entries

Note that λ_G can be viewed as an enriched adjacency matrix: Pairs of nonadjacent vertices are marked withdiand the entries for adjacent pairs are differentiated into the four other categories. Theunderlying graphof an intersection matrixµ= (µ_u,v)_u̸=v∈V is denoted byGµ; it consists of the verticesVand the edges{

{u,v}^⏐⏐µu,v̸=di} .

An arc representationα: V(G)→ A_{of a graph}GisnormalizedifAis sharp andαis an isomorphism between the neighborhood matrixλ_Gand the intersection matrix µA, i.e.,λ_u,v =µ_α(u),α(v) for allu,v∈V(G). Normalized arc representations were introduced by Hsu, who provides a linear-time algorithm that transforms any arc representation of a CA graph with certain properties into a normalized representation.

5.2 Canonical representation of Helly circular-arc graphs Lemma 5.2.3[Hsu95, Corollary2.3]. Any CA graph G without twins and universal vertices has a normalized arc representation.

Proof sketch. Letα: V(G) → A be an arc representation of G. It can be assumed that Ais sharp, as the proof of Lemma4.6.1 can easily be adapted from the interval to the circular-arc setting.

Hsu’s algorithm modifiesαin two stages to ensure λ_G =µ_A. In the first stage, all arcs of Aare extended as far as possible. This eliminatestype1violations, where two vertices u,v ∈ V(G) exist withλ_u,v = cc ̸= ov = µ_α(u),α(v). To be precise, the algorithm splits the circle intoblocksof consecutive points, such that each block is a maximal sequence either of start points or of end points. For each start point of an arc A= [a⁻,a⁺]∈ A, the preceding block E⊂V(A)of end points is split into the setE₁ of end points whose arcs intersect with Aand the remaining points E₂. Then the points inEare reordered so that those in E₁ come before those inE2, and the start pointa⁻of Ais moved between E₁andE₂. Afterwards, the end point of each arc is treated symmetrically.

The second stage of the algorithm reorders the points within each block to get as many containments as possible. This resolvestype2violations, where two vertices u,v∈V(G) exist withλ_u,v=cd̸=ov=µ_α(u),α(v). In each block of start points, the points are ordered by how far the corresponding arcs extend beyond this block, placing those first whose arcs extend furthest. After that, each block of end points is reordered by how far the corresponding arcs extend beyond this block, this time placing those points first whose arcs extend least.

It is not hard to see that this algorithm does not change the intersection graph ofA and resolves all violations of λ_G =µA[for details see Hsu95, p.415].

All normalized arc representations have a property that is calledstableby Joeris et al., who use their characterization of HCA graphs by forbidden induced subgraphs to prove that every stable arc representation of an HCA graphGyields an HCA model [JLM⁺11, Theorem4.1]. This implies the following.

Lemma5.2.4. Let G be an HCA graph without twins and universal vertices. For any normalized arc representationα: V(G)→ Aof G, the resulting arc modelAis Helly.

Anarc representation of a CA matrixλ= (λ_u,v)_u̸=v∈V is a bijectionαfrom Vto some arc system Athat has satisfies the intersection structure prescribed byλ, i.e.,αmust be an isomorphism fromλto the intersection matrix µA.

LetO = (O_u)_u∈V be a family of equivalence relations such that each O_u partitions {v ∈ V|λ_u,v = ov} into at most two equivalence classes. As in the interval case, an arc representation αofλis calledO-respecting, if for any three verticesu,v,w∈Vwith λ_u,v= λ_u,w=ov, the arcsα(v)andα(w)contain the same extreme point of α(u)if and only if (v,w)∈O_u.

For a given HCA graph G, the following two lemmas allow to compute a family O_G = (O_u)_u∈V(G)such that the neighborhood matrixλ_GofGadmits anO_G-respecting arc representation.

Lemma 5.2.5. Letα: V(G)→ A be a normalized HCA representation of a graph G. Also, let u,v,w∈V(G)withα(u) α(v),α(u) α(w)andα(v) α(w). Then it holds that

α(u)∩α(v)⊆α(w)⇔N[u]∩N[v]⊆N[w].

Proof. The forward direction follows from the Helly property and α(u)∩α(v) ̸= _∅.

Indeed, let x ∈ N[u]∩N[v]. Thus α(x) intersects both α(u) and α(v). By the Helly property, α(_x) intersects evenα(_u)∩α(_v). By assumption, the latter set is contained inα(w), implyingx ∈N[w]as desired.

Now suppose that α(u)∩α(v) ⊈ ^α(w). We may assume without loss of generality thatα(u)contains the start point ofα(v); this will be denoted by α(u)≺ α(v)_{. As the} Helly property precludes the caseα(v)≺ α(w) ≺ α(u) (see Fig.5.2(a)), it follows that either α(u) ≺ α(w) and α(v) ≺ α(w) or α(w) ≺ α(u) and α(w) ≺ α(v). Assume the former relation (see Fig.5.2(b)); the latter case is symmetric. Becauseαis normalized and α(v)⊈ ^α(w), there exists a vertexx∈ N[v]\N[w]. Its arcα(x)must contain a point in α(v)\α(w), which is a subset ofα(u). Thusx is also a neighbor ofu and witnesses that N[u]∩_N[v]⊈N[w]_.

(a)

α(w)

α(u) α(v)

(b)

α(u) ^α(v) α(w)

Figure5.2: Proof of Lemma5.2.5

For a graphG, define the familyO_G = (O_u)_u∈V(G), whereO_u is an equivalence relation on{v∈_V(G)|λ_u,v=ov}_{, by}

(v₁,v₂)∈O_u⇔v₁ =v₂∨λ_v1,v2 ∈ {cd,cs} ∨⁽λ_v1,v2 =ov∧N[v₁]∩N[v₂]⊈^N[u]⁾. Lemma 5.2.6. Let λ_G = (λ_u,v)_u̸=v∈V(G) be the neighborhood matrix of an HCA graph G without twins and universal vertices. Then any sharp arc representationα: V(G) → Aof λ_G is O_G-respecting.

Proof. Letu,v₁,v₂∈ V(G)withλ_u,v1 = λ_u,v2 =ov; it has to be shown that(v₁,v₂)∈O_u if and only ifα(v₁)andα(v2)contain the same extreme point ofα(u).

Ifλ_v1,v2 =cd(and thus(v₁,v₂)∈O_u), it follows thatα(v₁)⊊^α(v₂). Thus the extreme point of α(u) contained in α(v₁) is also contained in α(v₂). The case λ_v1,v2 = cs is symmetric.

In caseλ_v1,v2 = di(where(v₁,v₂)∈/ O_u), it follows that α(v₁)and α(v₂)are disjoint and consequently cannot contain the same extreme point ofα(u). Similarly,λ_v1,v2 =cc (where(v₁,v2)∈/Ou) impliesα(v₁) α(v2), and asα(u)strictly overlaps both these arcs, its extreme points must lie inα(v₁)\α(v₂)andα(v₂)\α(v₁), respectively.

For the remaining case λ_v1,v2 = ov, the condition (v₁,v₂) ∈ O_u holds if and only if N[v₁]∩N[v₂]⊈N[u]_{. By Lemma}⁵_.²_.⁵, this is equivalent toα(v₁)∩α(v₂)⊈^α(u)_{, which} in turn holds exactly ifα(v₁)andα(v₂)contain the same extreme point of α(u).

Using these facts, the first step of the representation algorithm for HCA graphs can be described.

5.2 Canonical representation of Helly circular-arc graphs Lemma5.2.7. There is a logspace reduction from computing canonical HCA representations for HCA graphs to computing canonical sharp (O-respecting) arc representations of vertex-colored HCA matrices.

Proof. First consider the case that the input graph G is twin-free and has no univer-sal vertex. In this case, the reduction computes the neighborhood matrix λ_G (along with the familyO_G), queries the oracle for a sharp (O_G-respecting) arc representation α_G: V(G) → A_{G of} λ_G, and returns α_G as HCA representation of G. By definition, a function α: V(G) → A is a normalized arc representation of G if and only if it is a sharp arc representation of λ_G. By Lemma5.2.3, such an αexists, it is O_G-respecting by Lemma 5.2.6, and the resulting arc model A is Helly by Lemma5.2.4. Regarding the canonicity, observe thatG∼= His equivalent toλ_G∼=λ_H. Thus, ifλ_G ∼=λ_H implies A_G=A_H, then so does G∼= H.

If the input graphGcontains twins, apply the above algorithm to its quotient graphG^′, which has one vertex for each twin class[v]ofGand has the edge{

[u],[v]^}whenever Ghas the edge{

u,v}

(note that{u^′,v^′} ∈E(G)if and only if{u,v} ∈E(G)for allu^′ ∈ [u] and v^′ ∈ [v], so E(G^′) is well-defined). Let α_G^′ be the computed HCA representation of G^′. Then define a representation α ofG by α(v) = α^′(

[v]⁾. To preserve canonicity, color each vertex [v]of G^′ with the size of the twin class[v]; thenG∼= His equivalent to G^′ ∼= H^′. It remains to observe that universal vertices can be removed before computing the neighborhood matrix, adding arcs for them afterwards.