Hop-Spanners for Geometric Intersection Graphs
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A t-spanner of a graph G=(V,E) is a subgraph H=(V,E') that contains a uv-path of length at most t for every uv∈ E. It is known that every n-vertex graph admits a (2k-1)-spanner with O(n^1+1/k) edges for k≥ 1. This bound is the best possible for 1≤ k≤ 9 and is conjectured to be optimal due to Erdős' girth conjecture. We study t-spanners for t∈{2,3} for geometric intersection graphs in the plane. These spanners are also known as t-hop spanners to emphasize the use of graph-theoretic distances (as opposed to Euclidean distances between the geometric objects or their centers). We obtain the following results: (1) Every n-vertex unit disk graph (UDG) admits a 2-hop spanner with O(n) edges; improving upon the previous bound of O(nlog n). (2) The intersection graph of n axis-aligned fat rectangles admits a 2-hop spanner with O(nlog n) edges, and this bound is the best possible. (3) The intersection graph of n fat convex bodies in the plane admits a 3-hop spanner with O(nlog n) edges. (4) The intersection graph of n axis-aligned rectangles admits a 3-hop spanner with O(nlog^2 n) edges.
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