Let S be a set of n   points in RdRd and let t≥1t≥1 be a real number. 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||pq|, the Euclidean distance between p and q. The geometric graph G with vertex set S is called a θ-angle-constrained t-spanner if G is a t-spanner of S   and any two edges {p,q}{p,q} and {p,r}{p,r} in G make an angle of at least θ. In this paper, we show that there is a point set P   in the plane such that for every θ>π/3θ>π/3, there is no θ-angle-constrained t-spanner on P  . Moreover, we show that for θ=π/3θ=π/3 and every t<2/3, there is no θ-angle-constrained t-spanner on P. This solves an open problem posed by Paz Carmi and Michiel Smid [2].
['Davood Bakhshesh', 'Mohammad Farshi']
Angle-constrained spanners with angle at least π/3