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\begin{document}$$\mathcal{P}$$\end{document} be a set of n points inside a polygonal domain D\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{D}$$\end{document}. A polygonal domain with h holes (or obstacles) consists of h disjoint polygonal obstacles surrounded by a simple polygon which itself acts as an obstacle. We first study t-spanners for the set P\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{P}$$\end{document} with respect to the geodesic distance function π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document} where for any two points p and q, π(p,q)\documentclass[12pt]{minimal}
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\begin{document}$$\pi (p,q)$$\end{document} is equal to the Euclidean length of the shortest path from p to q that avoids the obstacles interiors. For a case where the polygonal domain is a simple polygon (i.e., h=0\documentclass[12pt]{minimal}
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\begin{document}$$h=0$$\end{document}), we construct a (10+ϵ\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{10}+\epsilon $$\end{document})-spanner that has O(nlog2n)\documentclass[12pt]{minimal}
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\begin{document}$$O(n \log ^2 n)$$\end{document} edges. For a case where there are h holes, our construction gives a (5+ϵ\documentclass[12pt]{minimal}
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\begin{document}$$5+\epsilon $$\end{document})-spanner with the size of O(nhlog2n)\documentclass[12pt]{minimal}
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\begin{document}$$O(n\sqrt{h}\log ^2 n)$$\end{document}. Moreover, we study t-spanners for the visibility graph of P\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{P}$$\end{document} (VG(P)\documentclass[12pt]{minimal}
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\begin{document}$$VG(\mathcal{P})$$\end{document}, for short) with respect to a hole-free polygonal domain D\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{D}$$\end{document}. The graph VG(P)\documentclass[12pt]{minimal}
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\begin{document}$$VG(\mathcal{P})$$\end{document} is not necessarily a complete graph or even connected. In this case, we propose an algorithm that constructs a (3+ϵ\documentclass[12pt]{minimal}
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\begin{document}$$3+\epsilon $$\end{document})-spanner of size O(n4/3+δ)\documentclass[12pt]{minimal}
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\begin{document}$$O(n^{4/3+\delta })$$\end{document} for some δ>0\documentclass[12pt]{minimal}
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\begin{document}$$\delta >0$$\end{document}. In addition, we show that there is a set P\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{P}$$\end{document} of n points such that any (3-ϵ)\documentclass[12pt]{minimal}
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\begin{document}$$(3-\epsilon )$$\end{document}-spanner of VG(P)\documentclass[12pt]{minimal}
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\begin{document}$$VG(\mathcal{P})$$\end{document} must contain Ω(n2)\documentclass[12pt]{minimal}
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\begin{document}$$\varOmega (n^2)$$\end{document} edges.
