Point visibility graphs preserving collinearities

We first describe constructions of point visibility graphs, such that all their geometric realizations preserve some fixed subsets of collinear points.

To do so, we introduce the following notations. The open segment betweenp and q is denoted by pq. We will often call pq thesight line between p and q, since p and q see each other if and only ifpq∩P =∅. We call two sight linesp1q1 andp2q2 non-crossing if p1q1 ∩p2q2 = ∅. For each point p all other points of G lie on deg(p) many rays R^p₁, . . . , R^p_deg(p) originating from p.

Preliminary observations

In the realization of a PVG, the point p sees exactly deg(p) many vertices, hence all other points lie on deg(p) rays of originp.

5.2. Point visibility graphs preserving collinearities

p p

q q

Figure 5.1.: (Lemma 5.1) Left: a point sees points on consecutive rays with small angle.

Right: a vertex of deg(q) = 1 in G[N(p)] lies on the boundary of an empty halfspace.

Lemma 5.1. Let q∈N(p) be a degree-one vertex in G[N(p)]. Then all points of Glie in one of the closed half-spaces defined by the line`(p, q). Furthermore, the neighbor of q lies on the ray that forms the smallest angle with qp.

Proof. If the angle between two consecutive rays is smaller than π, then every vertex on one ray sees every vertex on the other ray. Hence one of the angles of a ray next to the ray ofq in the circular order is at least π and the neighbour ofq lies on the other neighbouring ray.

Corollary 5.2. IfG[N(p)] is an induced path, then the order of the path and the order of the rays the points lie on coincide.

Proof. By Lemma 5.1 the two endpoints of the path lie on rays on the boundary of empty half-spaces. Thus, all other rays form angles which are smaller thanπ, and thus they see their two neighbors of the path on their neighboring rays.

Observation 5.3. Let q, q 6=p, be a point that sees all points of N(p). Thenq is the second point (not includingp) on one of the rays emerging fromp.

Proof. Assumeq is not the second point on one of the rays. Thenq cannot see the first point on its ray which is a neighbor ofp.

Observation 5.4. Let q, q 6=p, be a point that is not the second point on one of the rays from p and sees all but one of the neighbors of p. Then q lies on the ray of the neighbor it does not see.

Fans and generalized fans

We have enough tools to show the uniqueness of a PVG obtained from the following construction, which is depicted in Figure 5.2. Consider a setS of segments between two lines`and `⁰ intersecting in a pointp. For each intersection of a pair of segments, con-struct a ray of originpand going through this intersection point. Add two segmentss₁

`

`⁰

s2

s1

a b

c d

p e

Figure 5.2.: A fan: a vertex is placed on each intersection of two lines/segments.

and s₂ between ` and `⁰, such the first intersection point on` and `⁰ lie on s₁ and the second ons₂.

We now put a point on each intersection of the segments and rays and construct the PVG of this set of points. We call this graph the fan of S and denote it by fan(S). Since we have the choice of the position of the segments s₁ and s₂, we can avoid any collinearity between a point on s1 or s2 and points on other segments, except for the obvious collinearities on one ray. Thus, every point sees all points on s1 except for the one of the ray it lies on.

Lemma 5.5. All realizations of a fan preserve collinearities between points that lie on one segment and between points that lie on one ray.

Proof. We first show that the distribution of the points onto the rays ofpis unique. By construction the points on s2 see all the points on s1, which are exactly the neighbors ofp. Thus, by Observation 5.3, the points froms2 are the second points of a ray. Since there is exactly one point for each ray ons2, all the other points are not second points on a ray. By construction each of the remaining points sees all but one point of s1. Observation 5.4 gives a unique ray a point lies on. The order of the rays is unique by Corollary 5.2. On each ray the order of the points is as constructed, since the PVG of points on one ray is an induced path. Now, we have to show that the points originating from one segment are still collinear. Consider three consecutive rays R1, R2, R3. We consider a visibility between a point p1 on R1 and one point p3 on R3 that has to be blocked by a point on R2. Let p2 be the original blocker from the construction. For each point onR₂ that lies closer top, there is a sight line blocked by this point, and for each point that lies further away fromp there is a sight line blocked by this point. For each of these points we pick one sight line that corresponds to an original segment and p₁p₃. This set of sight lines is non-crossing, since the segments only intersect on rays by assumption. So we have a set of non-crossing sight lines and the same number of blockers available. Since the order on each ray is fixed, and the sight lines intersectR₂ in a certain order, the blocker for each sight line is uniquely determined and has to be the original blocker. By transitivity of collinearity all points from the segments remain

5.3. ∃R-completeness of PVG recognition

collinear.

The fan is the tool we need to show that PVG recognition is complete in ∃R. In the proof for this fact, which is published in [CH15], we used a generalized fan for this reduction.

5.3. ∃R -completeness of PVG recognition

In this section we show that the recognition of PVGs is complete in∃R.

Theorem 5.6. The recognition of point visibility graphs is ∃R-complete.

` `⁰

p

s1

s2

Figure 5.3.: Construction of a fan from a pseudoline arrangement A (black) with a given allowable sequence.

Proof. The idea of the proof is to reduce the stretchability of a pseudoline arrangement with given allowable sequence. Therefore, let L be a simple pseudoline arrangement with allowable sequenceA. We construct the following graph Gusing the fan construc-tion: We draw the wiring diagram of the pseudoline arrangement in the plane using x-monotone curves as shown in black in Figure 5.3, such that the order ofx-coordinates

of the crossings agrees with the allowable sequence. We place the point p, where the rays of the fan are originating from, on the intersection point of a vertical line and the line at infinity. The lines `and `⁰ that form the boundary of the fan are vertical lines that are placed to the left and to the right of all intersection points of the pseudoline arrangement.

We defineGas the fan of this construction: We add two horizontal segmentss₁ ands₂ that are spanned between `and`⁰ above the pseudolines. We add a vertical ray from p through each of the intersection points of the pseudoline arrangement. We denote the set of vertical rays by V. On each intersection point of a ray with any other line we place a point. The only non-visibilities appear between non-consecutive points on one ray or pseudoline.

We show thatGis a PVG if and only ifLis stretchable withAas allowable sequence. If Lis stretchable with allowable sequenceA, we can repeat the construction of the fan as described above with the realizationRofLto obtain a PVG realization. The described non-visibilities (the collinearities) are preserved in this realization by Lemma 5.5. Thus, it remains to show that there is a realization, such that all visibilities between each pair of points that do not lie on the same ray of the fan and not on the same line of the arrangement is not blocked. We show that this is possible by perturbing the realization of the fan. We do this again by dualizing the arrangement of lines and rays. Since

D(V)

r

q t

D(V)

r

q

t

s

Figure 5.4.: Perturbingq. Left: Movingqaway from`(q, t). Right: Movingqalong`(q, t)to perturb another point onD(V).

the points of the PVG lie on the intersection points in this arrangement we have to guarantee that three intersection points that do not lie on one line are not collinear.

This dualizes to three lines spanned by three pairs of points of the order type that intersect in one point. We use the mirrored dual mapD:L → P, x=Ay+B7→(A, B) to map the realization R of L to an order type O. The vertical rays, which we add in

5.4. Point visibility graphs on a grid