Pseudo-Triangle Visibility Graph: Characterization and Reconstruction

The visibility graph of a simple polygon $\mathcal{P}$ is a graph $\mathcal{G}(V,E)$ where $V$ is the vertices of $\mathcal{P}$ and an edge $(u,v)$ exists in $E$ if and only if the line segment $uv$ lies completely inside $\mathcal{P}$, i.e they are visible from each other. Based on this definition, each pair of adjacent vertices on the boundary of the polygon are assumed to be visible from each other. This implies that we have always a Hamiltonian cycle in a visibility graph which determines the order of vertices on the boundary of the corresponding polygon.

Computing the visibility graph of a given simple polygon has many applications in computer graphics, computational geometry and robotics. There are several efficient polynomial time algorithms for this problem. Asano et al. [1] and Welzl [2] proposed $\mathcal{O}(n^{2})$ time algorithms for computing the visibility graph of a simple polygon of $n$ vertices. This was then improved to $\mathcal{O}(m+n\log\log n)$ by Hershberger[3] where $m$ is the number of edges in the visibility graph. The term $n\log\log n$ is due to the time required for triangulating a simple polygon. Using the $\mathcal{O}(n)$ time triangulation algorithm of Chazelle [4] reduces the time complexity of Hershberger’s result to $\mathcal{O}(m+n)$ which is optimal.

This concept has been studied in reverse as well: Characterizing a visibility graph problem is to determine whether a give graph is isomorphic to the visibility graph of some simple polygon, and the reconstruction problem is to build such a simple polygon. Everett showed that these probleme are in PSPACE[5], and this is the only result known about the complexity of these problems. These problems have been solved only for special cases of spiral and tower polygons. These results are obtained by Everett and Corneil [6] for spiral polygons and by Colley et al. [7] for tower polygons. In spiral polygons there is at most one concave chain (Fig. 1a) and the boundary of a tower polygon is composed of two concave chains and a single edge (Fig. 1b).

Although there is a bit progress on this type of reconstruction problem, there have been plenty of studies on characterizing visibility graphs [8, 7, 9, 10, 6, 11]. In 1988, Ghosh introduced three necessary conditions for visibility graphs and conjectured their sufficiency[11]. In 1990, Everett proposed a counter-example graph disproving Ghosh’s conjecture[5]. She also refined Ghosh’s third necessary condition to a new stronger condition[12]. In 1992, Abello et al. built a graph satisfying Ghosh’s conditions and the stronger version of the third condition which was not the visibility graph of any simple polygon, disproving the sufficiency of these conditions[13]. In 1997, Ghosh added his fourth necessary condition and conjectured that this condition along with his first two conditions and the stronger version of the third condition are sufficient for a graph to be a visibility graph. This was also disproved by a counter-example from Streinu in 2005[14].

There are also several results about other versions of the visibility graph like point visibility graph[15, 16, 17], pseudo-polygon visibility[18, 19, 20, 21] and terrain visibility[22]. Excellent surveys on visibility graph results can be found in [23, 24, 25].

In this paper, we solve the reconstruction problem for pseudo-triangles. A pseudo-triangle is a simple polygon whose boundary is composed of three concave chains, called side-chains, where each pair shares one convex vertex (called a corner). Let $\mathcal{P}$ be a pseudo-triangle formed by the concave side-chains $AB=[A,\ldots,B]$, $AC=[A,\ldots,C]$, and $BC=[B,\ldots,C]$ where $A$, $B$, and $C$ are the corners (Fig. 1c). According to this notation, a concave side-chain joining corner vertices $X$ and $Y$ is denoted by $XY$ where $X$ and $Y$ are in $\{A,B,C\}$.

Let $\mathcal{H}=\ <A,\ldots,C,\ldots,B,\ldots,A>$ be the Hamiltonian cycle of the visibility graph of $\mathcal{P}$ which indicates the order of vertices on the boundary of $\mathcal{P}$. Here, we use the same notation for a vertex on the boundary of $\mathcal{P}$ and its corresponding vertex in the visibility graph and $\mathcal{H}$. For a given pair of Hamiltonian cycle $\mathcal{H}$ and visibility graph $\mathcal{G}(V,E)$, we introduce a set of necessary properties on $\mathcal{H}$ and $\mathcal{G}$ when this pair belongs to a pseudo-triangle and prove that these properties are sufficient as well.

Having these properties, we propose a linear-time algorithm for reconstructing a pseudo-triangle $\mathcal{P}=<A,\ldots,C,\ldots,B,\ldots,A>$ with $\mathcal{G}(V,E)$ as its visibility graph. Moreover, we propose algorithms for verifying the properties on a given pair of $\mathcal{H}$ and $\mathcal{G}$. These characterizing algorithms run in linear time in terms of the size of $\mathcal{G}$. Therefore, in this paper we solve the characterizing and reconstructing problems for another class of polygons called pseudo-triangles.

Since a tower polygon is a special case of a pseudo-triangle, we use the tower reconstruction algorithm [7] as a sub-routine in our algorithm to build the initial part of the polygon.

Our motivation in solving this problem for pseudo-triangles is that every polygon can be partitioned into pseudo-triangles. Then, an idea for solving a general reconstruction problem is to handle these steps:

• •

  Recognize a pseudo-triangle decomposition for the target polygon from $\mathcal{G}(V,E)$ and $\mathcal{H}$.

• •

  Reconstruct each pseudo-triangle separately.

• •

  Attach the reconstructed pseudo-triangles satisfying the pseudo-triangle decomposition and the visibility constraints.

In Section 2, we briefly describe the tower reconstruction algorithm [7] for reconstructing tower polygons which is used as a sub-routine in our algorithm. In Section 3, we introduce a set of necessary conditions(properties) of the visibility graph of pseudo-triangles and in Section 4, we prove sufficiency of these conditions by proposing a reconstruction algorithm. Finally, we analyze the running time of the reconstruction algorithm and the algorithms required to check the properties.

A strong ordering on a bipartite graph $\mathcal{G}(V,E)$ with partitions $U$ and $W$ is a pair, $<_{U}$ and $<_{W}$, of orderings on respectively (resp.) $U$ and $W$ such that if $u<_{U}u^{\prime}$, $w<_{W}w^{\prime}$, and there are edges $(u,w^{\prime})$ and $(u^{\prime},w)$ in $E$, the edges $(u,w)$ and $(u^{\prime},w^{\prime})$ also exist in $E$.

The following theorem by Colley et al. [7] indicates the main property of the visibility graph of a tower polygon and guarantees the existence of a tower polygon consistent with such a visibility graph.

The outline of the reconstruction algorithm proposed by Colley et al. [7] is as follows. As input, it takes the corner vertex $A=u_{0}=v_{0}$ and a connected bipartite graph $\mathcal{G}(V,E)$ with vertices partitioned into two independent sets $AB=\{u_{1},\ldots,u_{m}\}$ and $AC=\{v_{1},\ldots,v_{n}\}$ having strong ordering.

In the first step, the position of the corner $A$ and the vertices $u_{1}$ and $v_{1}$ are determined as in Fig. 2. In a middle step, suppose that the positions of the vertices $u_{0},\ldots,u_{j-1}$ and $v_{0},\ldots,v_{k-1}$ have been determined and the directions of the half-lines from $u_{j-1}$ and $v_{k-1}$ which respectively contain $u_{j}$ and $v_{k}$, where $(u_{j},v_{k})\in E$, are also known. To complete such a middle step, position of $u_{j}$ and the half-line from $u_{j}$ which contains $u_{j+1}$ (where $u_{j+1}$ is visible from $v_{k}$) must be determined. For this purpose, $u_{j}$ is located somewhere on its containing half-line horizontally below the vertex $v_{c}$ which $v_{c}$ has the minimum index among vertices of $AC$ which are visible from $u_{j+1}$. Then, the containing half-line of $u_{j+1}$ will be the half-line on the supporting line of $u_{j}$ and $s_{j+1}$ downward from $u_{j}$. Here, $s_{j+1}$ is a point on $v_{c-1}v_{c}$ with an $\epsilon$ distance below $s_{j}$, when $s_{j}$ lies on $v_{c-1}v_{c}$. If $s_{j}$ does not lie on $v_{c-1}v_{c}$, then $s_{j+1}$ is a point on $v_{c-1}v_{c}$ with an $\epsilon$ distance below $v_{c-1}$.

According to this construction, $s_{j}$ will be the intersection of chain $AC$ and the supporting line of $u_{j}$ and $u_{j-1}$. Similarly, $r_{j}$ will be the intersection of $AB$ and the supporting line of $v_{j}$ and $v_{j-1}$ (Fig. 2). We say “will” because we first fix position of $s_{j}$ (resp. $r_{j}$) from which position of vertex $u_{j}$ (resp. $v_{j}$ ) is determined. We will use this notation once again in Section 4.

In this section, we describe a set of properties that a pair of $\mathcal{H}$ and $\mathcal{G}$ must have to be the Hamiltonian cycle and visibility graph of a pseudo-triangle.

Any sub-sequence $<v_{i},\ldots,v_{j}>$ on the Hamiltonian cycle is called a chain and is denoted by $[v_{i},\ldots,v_{j}]$. A vertex $v_{a}$ on a chain $[v_{i},\ldots,v_{j}]$ is a blocking vertex for the invisible pair $(v_{i},v_{j})$ if there is no visible pair of vertices $v_{l}$ in $[v_{i},\ldots,v_{a-1}]$ and $v_{k}$ in $[v_{a+1},\ldots,v_{j}]$. Ghosh showed that for every invisible pair of vertices $(u,v)$ in a visibility graph, there is at least one blocking vertex in $[u,\ldots,v]$ or $[v,\ldots,u]$. Furthermore, every vertex on the shortest Euclidean path between $u$ and $v$(inside the corresponding polygon) is a blocking vertex for this pair [11]. Note that in a pseudo-triangle the shortest Euclidean path between two invisible vertices turns in only one direction (i.e. clockwise or counterclockwise).

Let $AB$, $AC$, and $BC$ be the side-chains of a pseudo-triangle. The order of vertices in these chains is defined with respect to one of their corner vertices. For a vertex $u$ in chain $XY$, $Ind^{X}(u)$ is equal to the number of vertices in chain $[X,\ldots,u]$ minus one. According to this definition, $Ind^{A}(A)$ is zero and $Ind^{B}(A)$ is $k-1$ where $k$ is the number of vertices in chain $AB$. Then, based on a given vertex indexing we refer to the previous and next vertices of a given vertex on a side-chain. For a vertex $v$ in chain $XY$ with $Ind^{X}(v)=i$, we use $N^{X}(v,j)$ to refer to the vertex $u\in XY$ with $Ind^{X}(u)=i+j$. Similarly, $P^{X}(v,j)$ is the vertex $u\in XY$ with $Ind^{X}(u)=i-j$. For the sake of brevity, we use $N^{X}(v)$ instead of $N^{X}(v,1)$, and $P^{X}(v)$ instead of $P^{X}(v,1)$. Note that in this notation, $j$ can be a positive or negative natural number. For corner vertices which belong to two side-chains we use $N$ or $P$ notation only when the target chain is known from the context. For a vertex $u$, $FV^{X}(u,XY)$ is a vertex on chain $XY$ with minimum index that is visible from $u$ when the indices start from corner vertex $X$. Similarly, $LV^{X}(u,XY)$ is a vertex on chain $XY$ with maximum index that is visible from $u$ when the indices start from corner vertex $X$. We have used $FV$ and $LV$ respectively as abbreviation for first visible and last visible. Fig. 3 depicts this notation.

Consider a pseudo-triangle $\mathcal{P}$ with side-chains $AB$, $AC$, and $BC$, and $\mathcal{G}$ and $\mathcal{H}$ as its visibility graph and Hamiltonian cycle, respectively. Assume that the method described in Lemma 3.1, identifies only two corners of $\mathcal{P}$. Without loss of generality, assume that $A$ is the unidentified vertex. This means that there is a subchain on $BC$ which blocks the visibility of adjacent vertices of $A$ on chains $AB$ and $AC$. Then, there is no visibility edge between a vertex from $AB$ and a vertex of $AC$. By removing the edges of the Hamiltonian cycle from the visibility graph, two isolated vertices $B$ and $C$ and a connected bipartite graph, with parts $S$ and $T$, is obtained where $S$ consists of vertices of chains $AB$ and $AC$ except the isolated vertices $B$ and $C$ and $T$ consists of vertices of $BC$ except $B$ and $C$. By adding the isolated vertex $B$ to $T$, and the boundary edge $e$ to this bipartite graph that connects $B$ to its adjacent vertex on $AB$, we will have a single isolated vertex $C$ and a bipartite graph with strong ordering. Then, according to Theorem 2.1 this bipartite graph corresponds to a tower polygon with base edge $e$ and $\mathcal{G}$ and $\mathcal{H}$ as its visibility graph and Hamiltonian cycle, respectively. Fig. 5 shows how such a pseudo-triangle can be interpreted as a tower polygon.

Therefore, we have the following property about the pair of $\mathcal{H}$ and $\mathcal{G}$ of a pseudo-triangle.

From this property, we assume for the remainder of this section that the method described in the proof of Lemma 3.1 identifies all three corners. Because otherwise, we can use the tower polygon algorithm to decide whether the given pair of $\mathcal{H}$ and $\mathcal{G}$ belong to a tower polygon( which is a special case pseudo-triangle) and obtain the answer.

An interval of a chain with endpoints $p$ and $q$ is the set of points on this chain connecting $p$ to $q$. Note that in this definition, the endpoints of an interval are not necessarily vertices of the chain. For example, for points $p$ on edge $(u_{i},u_{i+1})$ and $q$ on edge $(u_{j},u_{j+1})$ of a chain $AB$ where $i<j$, the interval defined by $p$ and $q$ is the chain $[p,u_{i+1},\ldots,u_{j},q]$.

For each pair of vertices $u\in AB$ and $v\in AC$, the diagonal edge $(u,v)$ in the visibility graph of a pseudo-triangle specifies a tower formed by the boundary vertices $<u,\ldots,A,\ldots,v>$. The vertices of this tower satisfy the strong ordering defined earlier. This strong ordering can be derived from Property 2 and corollaries 3.7 and 3.9. Therefore, we do not specify this as a new property.

As mentioned earlier, Ghosh introduced four necessary conditions for a visibility graph of a simple polygon. It is simple to show that these conditions are derived from the properties described in this section which meas that these properties includes Ghosh’s conditions.

In this section, $\mathcal{G}(V,E)$ denotes the visibility graph of a pseudo-triangle $\mathcal{P}$ with $AB$, $AC$, and $BC$ side-chains and the order of vertices on the boundary of $\mathcal{P}$ is specified by a Hamiltonian cycle $\mathcal{H}=<A,\ldots,C,\ldots,B,\ldots,A>$ in $\mathcal{G}$. We assume that the inputs $\mathcal{G}$ and $\mathcal{H}$ satisfy the properties 1 to 7. We propose an algorithm for reconstructing a pseudo-triangle corresponding to the given pair of $\mathcal{G}$ and $\mathcal{H}$.

In order to reconstruct the pseudo-triangle $\mathcal{P}$, we divide $\mathcal{P}$ into four subpolygons $\mathcal{X}$, $\mathcal{Y}$, $\mathcal{Z}$, and $\mathcal{Z}^{\prime}$ as shown in Fig. 12 and reconstruct each one separately. For the sake of brevity, $u_{i}=N^{A}(A,i)$ on side-chain $AB$, $v_{j}=N^{A}(A,j)$ on side-chain $AC$, and $w_{k}=N^{B}(B,k)$ on side-chain $BC$ where $i,j,k\geq 0$. We assume that $AB$ and $AC$ have respectively $\alpha+1$ and $\delta+1$ vertices.

The subpolygon $\mathcal{X}$ is formed by subchains $[A,\ldots,u_{\nu}]$ and $[A,\ldots,v_{\mu}]$ and edge $(u_{\nu},v_{\mu})$ where $LV^{A}(u_{\nu},AC)=v_{\mu}$ and $LV^{A}(v_{\mu},AB)=u_{\nu}$. The vertices $u_{\nu}$ and $v_{\mu}$ are identified by walking alternatively on side-chains $AB$ and $AC$ from corner vertex $A$ towards $B$ and $C$. As a step of this trace, assume that we are at vertices $u_{i}$ and $v_{j}$ and want to go one step further on $AB$. If $u_{i}$ is the last vertex on $AB$ or $v_{j}$ does not see $u_{i+1}$ we fix $u_{i}$ as $u_{\nu}$. Otherwise, we go to $u_{i+1}$ in this step. Walking on side-chain $AC$ are done similarly. The subpolygon $\mathcal{X}$ is a tower polygon with strong ordering in its visibility graph. Note that $u_{\nu+1}$ or $v_{\mu+1}$ exists only when the side-chain $BC$ has more than one edge, otherwise, two identified adjacent corners $u_{\nu}$ and $v_{\mu}$ compose the base of a tower polygon which can be constructed by the tower reconstruction algorithm. So, we assume that $BC$ has more than one edge.

The subpolygon $\mathcal{Y}$ is identified as follows: Let $[w_{i},\ldots,w_{j}]$ be the maximum subchain of $BC$ visible from both $u_{\nu}$ and $v_{\mu}$. According to Property 4(a), this chain is nonempty and continuous. Let $LV^{B}(u_{\nu+1},BC)=w_{k}$ and $FV^{B}(v_{\mu+1},BC)=w_{l}$. From Property 4(b), $k\leq j$ and $l\geq i$ and from Property 4(c), $k\leq l$. We define $M$ and $N$ as $\max(k,i)$ and $\min(l,j)$, respectively. It is clear that chain $[w_{M},\ldots,w_{N}]$ contains at least one vertex. Then, $\mathcal{Y}$ is defined to be the polygon with $<u_{\nu},w_{M},\ldots,w_{N},v_{\mu}>$ as its boundary.

The subpolygon $\mathcal{Z}$ is formed by subchains $[u_{\nu},\ldots,B)]$ and $[B,\ldots,w_{M}]$ and edge $(u_{\nu},w_{M})$. Similarly, subchains $[v_{\mu},\ldots,C]$ and $[C,\ldots,w_{N}]$ and edge $(v_{\mu},w_{N})$ specify the subpolygon $\mathcal{Z}^{\prime}$. It is clear that $\mathcal{P}$ is the union of $\mathcal{X}$, $\mathcal{Y}$, $\mathcal{Z}$, and $\mathcal{Z}^{\prime}$.