Maximum Matchings and Minimum Blocking Sets in \(\varTheta _6\)-Graphs
\(\varTheta _6\)-graphs are important geometric graphs that have many applications especially in wireless sensor networks. They are equivalent to Delaunay graphs where empty equilateral triangles take the place of empty circles. We investigate lower bounds on the size of maximum matchings in these graphs. The best known lower bound is n/3, where n is the number of vertices of the graph, which comes from half-\(\varTheta _6\)-graphs that are subgraphs of \(\varTheta _6\)-graphs. Babu et al. (2014) conjectured that any \(\varTheta _6\)-graph has a (near-)perfect matching (as is true for standard Delaunay graphs). Although this conjecture remains open, we improve the lower bound to \((3n-8)/7\).
We also relate the size of maximum matchings in \(\varTheta _6\)-graphs to the minimum size of a blocking set. Every edge of a \(\varTheta _6\)-graph on point set P corresponds to an empty triangle that contains the endpoints of the edge but no other point of P. A blocking set has at least one point in each such triangle. We prove that the size of a maximum matching is at least \(\beta (n)/2\) where \(\beta (n)\) is the minimum, over all \(\varTheta _6\)-graphs with n vertices, of the minimum size of a blocking set. In the other direction, lower bounds on matchings can be used to prove bounds on \(\beta \), allowing us to show that \(\beta (n)\ge 3n/4-2\).
KeywordsTheta-six graphs Proximity graphs Maximum matching Minimum blocking set Triangular-distance Delaunay graph
This work was done by a University of Waterloo problem solving group. We thank the other participants, Alexi Turcotte and Anurag Murty Naredla, for inspiring discussions, and the anonymous reviewers for helpful comments.
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