Some Properties of Continuous Yao Graph

Abstract

Given a set S of points in the plane and an angle $0 < \theta \le 2\pi$, the continuous Yao graph $cY (\theta )$ with vertex set S and angle $\theta$ defined as follows. For each $p, q \in S$, we add an edge from p to q in $cY (\theta )$ if there exists a cone with apex p and angular diameter $\theta$ such that q is the closest point to p inside this cone.

In this paper, we prove that for $0<\theta <\pi /3$ and $t\ge \frac{1}{1-2\sin (\theta /2)}$, the continuous Yao graph $cY(\theta )$ is a $\mathcal {C}$-fault-tolerant geometric t-spanner where $\mathcal {C}$ is the family of convex regions in the plane. Moreover, we show that for every $\theta \le \pi$ and every half-plane h, $cY(\theta )\ominus h$ is connected, where $cY(\theta )\ominus h$ is the graph after removing all edges and points inside h from the graph $cY(\theta )$. Also, we show that there is a set of n points in the plane and a convex region C such that for every $\theta \ge \frac{\pi }{3}$, $cY(\theta )\ominus C$ is not connected.

Given a geometric network G and two vertices x and y of G, we call a path P from x to y a self-approaching path, if for any point q on P, when a point p moves continuously along the path from x to q, it always get closer to q. A geometric graph G is self-approaching, if for every pair of vertices x and y there exists a self-approaching path in G from x to y. In this paper, we show that there is a set P of n points in the plane such that for some angles $\theta$, Yao graph on P with parameter $\theta$ is not a self-approaching graph. Instead, the corresponding continuous Yao graph on P is a self-approaching graph. Furthermore, in general, we show that for every $\theta >0$, $cY(\theta )$ is not necessarily a self-approaching graph.

1 Introduction

Let S be a set of n points in $\mathbb {R}^d$ and let $t \ge 1$ be a real number. A geometric graph is an edge-weighted graph on $S\subseteq \mathbb {R}^d$ such that the weight of each edge is the Euclidean distance between its endpoints. A geometric graph G with vertex set S is called a t-spanner for S, if for each two points p and q in S, there exists a path Q in G between p and q whose length is at most t times |pq|, the Euclidean distance between p and q. The length of a path is defined to be the sum of the lengths, or weight, of all edges on the path. The path Q is called a t-spanner path (or t-path) between p and q. We denote the length of path Q by |Q|. The stretch factor (or dilation) of G is the smallest value of t for which G is a t-spanner. The t-spanners were introduced by Peleg and Schäffer [14] in the scope of distributed computing and, then, by Chew [6] in the scope of computational geometry. The t-spanners are applicable in many scopes such as graph theory, network topology design, distributed systems, robotics. We refer the reader to [5, 8, 12, 16] for reading about the t-spanners and their applications.

The problem of efficient construction of a t-spanner for a given point set and a constant $t>1$ has been studied extensively. One can see the major algorithms for building spanners in the book by Narasimhan and Smid [13].

The Yao graph used by Andrew Yao to construct Euclidean minimum spanning tree on high-dimensional Euclidean space [17]. For a set S of points in the plane, the Yao graph $Y_k$, for $k \ge 2$, is defined as follows. At each point $u\in S$, we draw k cones with apex at u and angle $\frac{2\pi }{k}$. For each point $u\in S$ and cone C, we add an edge between u and the closest point to u in C. If one chooses k sufficiently large, the Yao graph becomes a t-spanner.

In 2014, Barba et al. [3] introduced the continuous Yao graph as a variation of Yao graph. The continuous Yao graph $cY(\theta )$ with vertex set S and angle $\theta$ defined as follows: For each $p, q \in S$, we add an edge from p to q in $cY (\theta )$ if there exists a $\theta$ -cone, a cone with aperture $\theta$, with apex at p such that q is the closest point to p inside this $\theta$-cone. In continuous Yao graph, for each $u\in S$, we rotate a $\theta$-cone with apex u around u continuously, and connect u to the closest point inside the $\theta$-cone during this rotation. They showed that $cY(\theta )$ has stretch factor at most $1/(1-2sin(\theta /4))$ for $0<\theta <2\pi /3$. Unlike Yao graphs that always have a linear number of edges for any constant k, in the worst case, continuous Yao graphs may have a quadratic number of edges. Since $cY(\theta )\subseteq cY(\gamma )$ for any $\theta \ge \gamma$, the continuous Yao graphs are useful in potential applications that require scalability. Moreover, when some rotations apply on the input point set the continuous Yao graphs are invariant [3].

One of the useful properties of a network is fault tolerance that is after one or more vertices or edges fail, the remaining graph is still a good network of alive vertices. In particular, a graph $G=(S, E)$ is called k-vertex fault-tolerant t-spanner [10] for S, denoted by (k, t)-VFTS for a given real number $t\ge 1$ and non-negative positive integer k, if for each set $S'\subseteq S$ with cardinality of at most k, the graph $G\backslash S'$ is a t-spanner for $S\backslash S'$. Also, G is called a k-edge fault-tolerant t-spanner [10] for S, denoted by (k, t)-EFTS, if for each set $E'\subseteq E$ with cardinality at most k and for each pair of points p and q in S, the graph $G\backslash E'$ contains a path P between p and q with $|P|\le t |P_S|$ where $P_S$ is the shortest path between p and q in the graph $K_S\backslash E'$ in which that $K_S$ is Euclidean complete graph on S. Levcopoulos et al. [10] for the first time considered the problem of constructing fault-tolerant spanners in Euclidean spaces efficiently. They proposed three algorithms that construct k-vertex fault-tolerant spanners. Some other works on the fault tolerant spanners have been done [7, 11].

In 2009, Abam et al. [1] introduced the concept of region-fault tolerant spanner for planar point sets. For a fault region F and a geometric graph G on a point set S, assume $G\ominus F$ is the remaining graph after removing the vertices of G that lie inside F and all edges that intersect F. For a set ${\mathcal {F}}$ of regions in the plane, an $\mathcal {F}$-fault tolerant t-spanner is a geometric graph G on S such that for any region $F \in \mathcal {F}$, the graph $G \ominus F$ is a t-spanner for $\mathcal {G}_c(S)\ominus F$, where $\mathcal {G}_c(S)$ is the complete geometric graph on S. They showed for any set of n points in the plane and any family $\mathcal {C}$ of convex regions, one can construct a $\mathcal {C}$-fault tolerant spanner of size $O(n\log n)$ in $O(n\log ^2 n)$ time.

In 2013, Alamdari et al. [2] introduced the concept of self-approaching and increasing-chord graph drawings. The problem is that, we are given a graph and we need to check if the graph has an self-approaching or increasing-chord embedding in the Euclidean space.

A geometric graph is self-approaching, if for every pair s and t of vertices of the graph, there exists a self-approaching path from s to t, denoted by st-path. A path from s to t is a self-approaching path if for each q on the path, not only the vertices, but any place of the path, if a point p starts at s and moves toward q, it always get closer to q in its movement. Also, a graph G is called increasing-chord if, for each pair u and v of its vertices, there exists a path between u and v such that the path is self-approaching both from u to v and from v to u. Obviously, an increasing-chord graph is a self-approaching graph.

In the geometric context, the position of the vertices of the graph is fixed, so we just want to know whether a given geometric graph is self-approaching or increasing-chord. One of the interesting properties of these graph is the following. It is known that the stretch factor (or dilation) of any self-approaching graph is at most 5.3332 [9] and the stretch factor of any increasing-chord graph is at most 2.094 [15].

Our Results. In this paper, we prove the following results. For the rest of the paper S denotes a set of n points in the plane.

• 1. The continuous Yao graph $cY(\theta )$ on S is a $\mathcal {C}$-fault-tolerant geometric t-spanner of S, where $0<\theta <\pi /3$ and $t\ge \frac{1}{1-2\sin (\theta /2)}$.

• 2. For any $\theta \le \pi$ the graph $cY(\theta )\ominus h$ on S is connected, where h is an arbitrary half-plane in the plane.

• 3. There is a set of n points in the plane and a convex region C such that for every $\theta \ge \frac{\pi }{3}$, $cY(\theta )\ominus C$ is not connected.

• 4. There is a set P of n points in the plane such that for some angles $\theta$, Yao graph on P with parameter $\theta$ is not a self-approaching graph. Instead, the corresponding continuous Yao graph on P is a self-approaching graph.

• 5. For every $\theta >0$, $cY(\theta )$ is not necessarily a self-approaching graph.

2 $cY(\theta )$ is Fault-Tolerant

In this section, we show that the continuous Yao graph $cY(\theta )$ for $0<\theta <\pi /3$ and $t\ge \frac{1}{1-2\sin (\theta /2)}$ is a $\mathcal {C}$-fault-tolerant geometric t-spanner where $\mathcal {C}$ is the family of convex regions in the plane. Furthermore, we show that for every $\theta \le \pi$ and every half-plane h, $cY(\theta )\ominus h$ is connected. Moreover, we show that there is a set of n points such that for a some convex region C, $cY(\theta )\ominus C$ is not connected for every $\theta \ge \frac{\pi }{3}$. We need the following lemmas.

Lemma 1

[4]. Let a, b and c be three points such that $|ac| \le |ab|$ and $\angle bac \le \alpha < \pi$. Then

$$\begin{aligned} |bc| \le |ab|-(1-2 sin(\alpha /2)) |ac|. \end{aligned}$$

Lemma 2

[1]. A geometric graph G on S is a $\mathcal {C}$-fault tolerant t-spanner if and only if it is an $\mathcal {H}$-fault-tolerant t-spanner, where $\mathcal {H}$ is the family of all half-planes.

Now, we prove the following theorem:

Theorem 1

Let $\theta$ be a real number with $0<\theta <\pi /3$ and let t be a real number with $t\ge \frac{1}{1-2\sin (\theta /2)}$. For any point set S, the continuous Yao graph $cY(\theta )$ is a $\mathcal {C}$-fault-tolerant geometric t-spanner.

Proof

By Lemma 2, it is sufficient to prove that $cY(\theta )$ is an $\mathcal {H}$-fault-tolerant geometric t-spanner.

Let h be an arbitrary half-plane in $\mathcal {H}$. We must show that for each pair of points $p,q\in S$ outside h, there is a t-path between p and q in $cY(\theta )\ominus h$. The proof is by induction on the rank of distance |pq|.

For the base step, suppose that the pair p and q is the closest pair in $cY(\theta )\ominus h$. Without loss of generality, assume the distance between p and h is less than of equal to the distance between q and h. Since p and q are outside h and $\theta <\pi /2$, there is a $\theta$-cone $C_p$ with apex at p completely outside h such that $q\in C_p$. Since pair p and q is the closest pair, by the construction of $cY(\theta )$, the edge (p, q) must be in $cY(\theta )\ominus h$. Note that if both p and q have same distance to h, then we can choose both p and q as apex, but we have to choose the point such that the corresponding $\theta$-cone does not intersect h.

For the induction hypothesis step, suppose for each pair $u,v\in S$ outside h with $|uv|<|pq|$ there is a t-path between u and v connecting them in $cY(\theta )\ominus h$.

Now we consider the induction step. Since p and q are outside h, there is a $\theta$-cone $C_p$ with apex at p such that $q\in C_p$ (here we assumed that p is closer than q to h).

Fig. 1.

Illustrating of the proof of Theorem 1.

Suppose that r is the closest point to p inside the cone $C_p$ (see Fig. 1). Since $\theta <\pi /3$, $1-2\sin (\theta /2)>0$, and also since $|pr|\le |pq|$, by Lemma 1 we have $|rq|<|pq|$. Therefore, by the induction hypothesis, there is a t-path Q between r and q in $cY(\theta )\ominus h$. Now consider the path $P:=\{(p,r)\}\cup Q$. Clearly the path P connects p and q, and P is in $cY(\theta )\ominus h$. Now by Lemma 1, we have

$$\begin{aligned} |P|= & {} |pr|+|Q|\\\le & {} |pr|+t|rq|\\\le & {} |pr|+t\left( |pq|-(1-2\sin (\theta /2))|pr|\right) \\= & {} t|pq|+\left( 1-t (1-2\sin (\theta /2))\right) |pr|\\\le & {} t|pq| \left( \text {since}\;t\ge \frac{1}{1-2\sin (\theta /2)}\right) . \end{aligned}$$

Thus P is a t-path in $cY(\theta )\ominus h$ between p and q. This completes the proof.    $\square$

Note that even though $cY(\theta )$ is a $\mathcal {C}$-fault tolerant spanner, it is not necessarily connected. It is because in fault-tolerant spanners, we compare the graph with the complete graph. So if the complete graph after removing points in some regions becomes disconnected, then some of fault-tolerant spanners of the points set might be disconnected. In the following, we show that for every half-plane h in the plane, the graph $cY(\theta )\ominus h$ is a connected graph for every $\theta \le \pi$.

Lemma 3

The continuous Yao graph $cY(\pi )$ on each set S contains CH(S), the convex hull of S.

Proof

Proof is straightforward.    $\square$

Theorem 2

For any $\theta \le \pi$ the graph $cY(\theta )\ominus h$ on S is connected, where h is a half-plane.

Proof

It is easy to see that for every $\alpha ,\beta >0$, if $\alpha \ge \beta$ then $cY(\alpha )\subseteq cY(\beta )$. So, to prove the theorem, it is sufficient to show that $cY(\pi )\ominus h$ is connected for every half-plane h.

Let h be an arbitrary half-plane in the plane. Suppose that $cY(\pi )\ominus h$ is not connected.

Since $cY(\pi )\ominus h$ is disconnected, it has more than one connected component. At least one of the connected components of $cY(\pi )\ominus h$ contains a point from CH(S). Suppose C and $C'$ are two (distinct) connected components of $cY(\pi )\ominus h$ such that at least one of them, say C, contains a point from CH(S). If both of C and $C'$ contains a point from CH(S) then by Lemma 3, part of CH(S) which lies outside h is connected which is a contradiction because we assumed that C and $C'$ are distinct connected components of $cY(\pi )\ominus h$.

Now, assume $C'$ contains no vertices on CH(S). Let x be the intersection of boundary of h and CH(S) and let $\ell$ be a line through x and tangent to $C'$ (see Fig. 2). We have two cases.

Fig. 2.

Illustrating of the proof of Theorem 2: case 1.

Case 1: $\ell \cap C'$ contains exactly one point denoted by y.

If $x\not \in S$ then clearly by the construction of graph $cY(\pi )$, the vertex y should be connected to a vertex of C that is contradiction, since we assumed that C and $C'$ are distinct connected components of $cY(\pi )\ominus h$. Now, suppose that $x\in S$. Since $\ell \cap C'$ contains exactly one point, we can with continuously rotating the line $\ell$ around the point y (in Fig. 2, we rotate $\ell$ counter-clockwise), find a line $\ell '$ such that $\ell '\cap C'$ contains exactly one point that is y, and also the point in the intersection of $\ell '$ and boundary h does not belong to S. Hence, by the construction of graph $cY(\pi )$, point y should be connected to a vertex of C. That is a contradiction with that C and $C'$ are distinct connected components of $cY(\pi )\ominus h$.

Case 2: $\ell$ is tangent to $C'$ in at least two points.

We claim that there is a line $\ell '$ that is tangent to exactly one point of S on the convex hull of $C'$ (denoted by s). Suppose that s be the farthest point of S on $\ell$ with respect to x, and suppose that r be the next point after s on the convex hull of $C'$ in counterclockwise order (see Fig. 3). Let $\ell ''$ be a line that through of s and r, and let z be the intersection of $\ell ''$ and boundary of h. Now suppose that w be a point on the segment xz ($w\ne x,z$). Let $\ell '$ be a line that through of w and s. Clearly $\ell '$ is tangent to exactly one point of S that is s on the convex hull of $C'$. Now by the construction of $cY(\pi )$, vertex s should be connected to a vertex of connected component of C. This contradicts with the assumption that C and $C'$ are distinct connected components of $cY(\pi )\ominus h$.

Fig. 3.

Illustrating of the proof of Theorem 2: case 2.

According to the above mentioned cases, $cY(\pi )\ominus h$ is connected. This completes the proof.    $\square$

In the following, we give an example such that for every $\theta \ge \frac{\pi }{3}$, $cY(\theta )\ominus C$ for some convex region C is not necessarily connected.

Assume $P_0=\left\{ (0,0), (1, \sqrt{3}), (2,0)\right\}$ which is the vertices of a equilateral triangle. Let $P_i$ be the translation of $P_0$ by value $c\times i$ horizontally where c is a sufficiently large constant positive integer. Here, we consider $c=10$ (see Fig. 4). We choose the set $P:=\bigcup _{i=0}^{k-1} P_i$.

Let $\theta$ be an angle with $\theta \ge \frac{\pi }{3}$. Consider $cY(\theta )$ on P. Note that for $\theta \ge \frac{\pi }{3}$, $cY(\theta )$ on the vertices of an equilateral triangle is exactly the complete graph of their vertices. Now, let C be a convex region such that C only contains the points $(1,\sqrt{3}),~(2,0),~ (10,0)$ and $(11,\sqrt{3})$ from P (see Fig. 4). Since in the $cY(\theta )$ the vertex (0, 0) is only connected to the vertices $(1,\sqrt{3})$ and (2, 0), clearly in $cY(\theta )\ominus C$ there is no path between (0, 0) and (12, 0). Hence $cY(\theta )\ominus C$ is not connected.

Fig. 4.

An example that $cY(\theta )$ is disconnected after convex region fault.

3 $cY(\theta )$ is Not Self-approaching

In this section, we show that there is a set P of n points in the plane such that for some angles $\theta$, Yao graph on P with parameter $\theta$ is not a self-approaching graph. Instead, the corresponding continuous Yao graph on P is a self-approaching graph. Finally, we show that there is a set of n points in the plane such that for every $\theta >0$, continuous Yao graph $cY(\theta )$ is not a self-approaching graph.

In the following, we assume that the number of cones is even. With making some modifications, similar results holds for odd number of cones.

Now, let $p=(0,0), q=(1,0), r=(r_1, r_2)$ and $s=(s_1,s_2)$ be four points in the plane. For three points x, y and z, let $\angle yxz$ be the angle between the segment xy and xz. Now, assume that $\angle qpr=\alpha _1$, $\angle pqr=\alpha _2$, $\angle prq=\alpha _3$, $\angle qps=\beta _1$, $\angle pqs=\beta _2$, $\angle psq=\beta _3$, for some angles $\alpha _i$ and $\beta _i$ with $0<\alpha _i<\pi /2$ and $0<\beta _i<\pi /2$ for $i=1,2,3$ (see Fig. 5).

We call the quadrilateral on the four points p, q, r and s a bad quadrilateral if we have the following properties:

• 1. $|pq|\cos {\alpha _1}<|pr|<|pq|.$

• 2. $|pq|\cos {\alpha _2}<|qr|<|pq|.$

• 3. $|pq|\cos {\beta _1}<|ps|<|pq|.$

• 4. $|pq|\cos {\beta _2}<|qs|<|pq|.$

Fig. 5.

A bad quadrilateral on the points p, q, r and s.

Now, we consider k cones, generated by the k-rays through origin, where the ith ray has angle $(i-1)\theta$ with the positive x-axis. The ith cones contains the points p such that the angle of line op with positive x-axis is bigger than or equal $(i-1)\theta$ and less than $i\theta$.

Let V be a set of four points $p=(0,0),~ q=(1,0),~ r=(r_1,r_2)$ and $s=(s_1,s_2)$ such that the quadrilateral on the vertices p, q, r and s is a bad quadrilateral. It is easy to see that, for every $\theta$ with $\alpha _1<\theta \le \frac{\pi }{2}$ such that $\frac{2\pi }{\theta }$ is even number and $\beta _2<\theta$, the Yao graph $Y_k$ on V where $k=\frac{2\pi }{\theta }$ does not contain the edge $\{p, q\}$ but it contains the edges $\{p, r\}, \{p, s\}, \{q, r\}$ and $\{q, s\}$. So the Yao graph $Y_k$ on the point set V is not a self-approaching graph, since every path from p to q is not a pq-path. Indeed, the following lemma shows that $cY(\theta )$ on V is a self-approaching graph.

Lemma 4

Let $\alpha _1+\beta _1\ge \frac{\pi }{2}$ or $\alpha _2+\beta _2\ge \frac{\pi }{2}$. For every $\theta$ such that

• (a) $\alpha _1<\theta \le \frac{\pi }{2}$, and

• (b) $\theta < \alpha _1+\beta _1$ or $\theta < \alpha _2+\beta _2$, and

• (c) $\frac{2\pi }{\theta }$ is even number, and

• (d) $\beta _2<\theta$,

continuous Yao graph $cY(\theta )$ on V contains the edge $\{p,q\}$. Furthermore, $cY(\theta )$ on V is a self-approaching graph.

Proof

We prove the theorem for the case that $\theta < \alpha _1+\beta _1$. Similar argument works for the case $\theta < \alpha _2+\beta _2$.

Since $\theta < \alpha _1+\beta _1$, by construction of $cY(\theta )$, clearly there is a $\theta$-cone C with apex p that only contains the point q. Hence the edge $\{p,q\}$ is in $cY(\theta )$.

On the other hand, the Yao graph on V with angle parameter $\theta$ is a subgraph of $cY(\theta )$, so the edges $\{p, r\}, \{p, s\}, \{q, r\}$ and $\{q, s\}$ are in $cY(\theta )$. So, we only need to show that there is a self-approaching path from r to s and a self-approaching path from s to r.

Since $\alpha _1+\beta _1\ge \frac{\pi }{2}$ or $\alpha _2+\beta _2\ge \frac{\pi }{2}$, the path $r\rightarrow q\rightarrow s$ or $r\rightarrow p\rightarrow s$ is an rs-path and in the reverse direction is an sr-path. So there is an xy-path between all ordered pairs $\{x,y\}$ for $x,y\in V$. Hence, $cY(\theta )$ is a self-approaching graph on V. This completes the proof.    $\square$

Next, we will give another point set such that the Yao graph on the point set is not self-approaching graph, but the continuous Yao graph is a self-approaching graph.

Let P be a set of n points as follows:

$$\begin{aligned} P=V\cup \{x_1, x_2, \ldots , x_{n-4}\}, \end{aligned}$$

(1)

with $x_j=(c+j, 0)$ for $1\le j\le n-4$ where c is a sufficiently large constant positive integer. Here, we consider $c=19$ (see Fig. 6).

Fig. 6.

The set P with $n=8$.

Now, using Lemma 4 we can easily find the following result.

Lemma 5

Let $\alpha _1+\beta _1\ge \frac{\pi }{2}$ or $\alpha _2+\beta _2\ge \frac{\pi }{2}$. For every $\theta$ such that

• (a) $\alpha _1<\theta \le \frac{\pi }{2}$, and

• (b) $\theta < \alpha _1+\beta _1$ or $\theta < \alpha _2+\beta _2$, and

• (c) $\frac{2\pi }{\theta }$ is even number, and

• (d) $\beta _2<\theta$,

continuous Yao graph $cY(\theta )$ on P is self-approaching and $Y_k$ is not a self-approaching for $k=\frac{2\pi }{\theta }$.

Note that there are some point set that satisfy the conditions mentioned in above lemmas. For example, let $V=\{p, q, r, s\}$ with $p=(0,0), q=(1,0), r=(0.7839, 0.4422), s=(0.2161, -0.4422)$, and suppose that $k=4$ and $\theta =\frac{\pi }{2}$. We can easily verify that the quadrilateral on the points of V is a bad quadrilateral. Moreover we have

$$\begin{aligned} \alpha _1=29.4271^\circ , \alpha _2=63.9535^\circ , \alpha _3=86.6198^\circ , \end{aligned}$$

$$\begin{aligned} \beta _1=29.4271^\circ , \beta _2=63.9535^\circ , \beta _3=86.6198^\circ . \end{aligned}$$

Now, suppose that $P=V\cup \{x_1, x_2, \ldots , x_{n-4}\}$ where $x_i=(19+i, 0)$. This point set satisfies Lemma 5.

At the first view, it seems that the continuous Yao graph is a self-approaching graph, but this is not true in general. Next, we give a point set such that for each $\theta$, the continuous Yao graph $cY(\theta )$ on the point set is not a self-approaching graph.

Theorem 3

There is a set P of n points such that for every $\theta >0$, $cY(\theta )$ on P is not a self-approaching graph.

Proof

We prove the theorem for $0<\theta \le \frac{2\pi }{3}$. Since for every $\alpha _1$ and $\alpha _2$ with $\alpha _1\le \alpha _2$ we have $cY(\alpha _2)\subseteq cY(\alpha _1)$, the theorem holds for every $\theta >0$.

Consider two points $p=(0,0)$ and $q=(1,0)$. Let C be a circle centered at the midpoint of the segment pq, with radius $\frac{1}{2}$. Let $D_p$ and $D_q$ be circles respectively centered at p and q with radius one. Let $\ell$ be the perpendicular bisector of segment pq (see Fig. 7).

Consider x and y as points outside C and inside $D_p\cap D_q$ such that $\angle xpq<\frac{\theta }{2}$ and $\angle ypq<\frac{\theta }{2}$. Let $x'$ and $y'$ be the symmetries of x and y with respect to line $\ell$, respectively.

Fig. 7.

Illustrating of the proof of Theorem 3.

Now let $V=\{p,q, x, y, x', y'\}$, and consider $cY(\theta )$ on V. Since $\angle xpy=\angle x'qy'<\theta$, $cY(\theta )$ does not contain the edge $\{p,q\}$. So, by our selection of points $x,y,x'$ and $y'$, there is no self-approach path between p and q. Hence, $cY(\theta )$ on V is not self-approaching.

One can extend this point set to a point set with arbitrary number of points such that $cY(\theta )$ is not a self-approaching graph of the point set. To this end, one can add any point to V which is sufficiently far from the points in the original set V, because adding points which are far can not help in making a self-approaching path from p to q.    $\square$

4 Concluding Remarks

We proved that for $0<\theta <\pi /3$ and $t\ge \frac{1}{1-2\sin (\theta /2)}$, the continuous Yao graph $cY(\theta )$ is a $\mathcal {C}$-fault-tolerant geometric t-spanner. Furthermore, we showed that for every $\theta \le \pi$ and every half-plane h, $cY(\theta )\ominus h$ is connected. Finally, we showed that for every $\theta >0$, $cY(\theta )$ is not necessarily a self-approaching graph. The question whether $cY (\theta )$ for $\frac{\pi }{3} \le \theta \le \pi$ is a $\mathcal {C}$-fault-tolerant geometric spanner with constant stretch factor remains open.