Proximal Navigation Graphs and t-spanners

Let $(X,\mathbf{d})$ be a metric space, $V\subseteq X$ a finite set, and $E \subseteq V \times V$. We call the graph $G(E,V)$ a {\em metric} graph if each edge $(u,v) \in E$ has weight $d(u,v)$. In particular edge $(u,u)$ is in the graph and have distance $0$. We call $G$ a {\em proximal navigation graph} or $PN$-graph if for each edge $(u,v) \in E$ either $u=v$ or there is a node $u_1$ such that $(u,u_1) \in E$ and $\mathbf{d}(u,v) > \mathbf{d}(u_1,v)$. In such graph it is possible to navigate greedily from an arbitrary source node to an arbitrary target node by reducing the distance between the current node and the target node in each step. The complete graph, the Delaunay triangulation and the Half Space Proximal (HSP) graph (defined below in the paper) are examples of $PN$-graphs. In this paper we study the relationship between $PN$-graphs and $t$-spanners and prove that there are $PN$-graphs that are not $t$-spanners for any $t$. On the positive side we give sufficient conditions for a $PN$-graph to be a $t$-spanner and prove that any $PN$-graph over $\mathbb{R}^n$ under the euclidean distance is a $t$-spanner.

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