Improved Bounds for Guarding Plane Graphs with Edges

An edge guard set of a plane graph G is a subset Γ of edges of G such that each face of G is incident to an endpoint of an edge in Γ. Such a set is said to guardG. We improve the known upper bounds on the number of edges required to guard any n-vertex embedded planar graph G: (1) We present a simple inductive proof for a theorem of Everett and Rivera-Campo (Comput Geom Theory Appl 7:201–203, 1997) that G can be guarded with at most 2n5 edges, then extend this approach with a deeper analysis to yield an improved bound of 3n8 edges for any plane graph. (2) We prove that there exists an edge guard set of G with at most n3+α9 edges, where α is the number of quadrilateral faces in G. This improves the previous bound of n3+α by Bose et al. (Comput Geom Theory Appl 26(3):209–219, 2003). Moreover, if there is no short path between any two quadrilateral faces in G, we show that n3 edges suffice, removing the dependence on α

Rectilinear link diameter and radius in a rectilinear polygonal domain

We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of n vertices and h holes. We introduce a graph of oriented distances to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the diameter and the radius in O(min(nω, n2 + nh log h + χ^2)) time, where ω < 2.373 denotes the matrix multiplication exponent and χ ∈ Ω(n) ∩ O(n^2) is the number of edges of the graph of oriented distances. We also provide a faster algorithm for computing the diameter that runs in O(n^2 log n) time

More turán-type theorems for triangles in convex point sets

We study the following family of problems: Given a set of n points in convex position, what is the maximum number triangles one can create having these points as vertices while avoiding certain sets of forbidden configurations. As forbidden configurations we consider all 8 ways in which a pair of triangles in such a point set can interact. This leads to 256 extremal Turán-type questions. We give nearly tight (within a log n factor) bounds for 248 of these questions and show that the remaining 8 questions are all asymptotically equivalent to Stein’s longstanding tripod packing problem

Rectilinear link diameter and radius in a rectilinear polygonal domain

\u3cp\u3e We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of n vertices and h holes. We introduce a graph of oriented distances to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the diameter and the radius in O(min(n \u3csup\u3eω\u3c/sup\u3e , n \u3csup\u3e2\u3c/sup\u3e + nhlog h + χ \u3csup\u3e2\u3c/sup\u3e )) time, where ω < 2.373 denotes the matrix multiplication exponent and χ ∈ Ω(n) ∩ O(n \u3csup\u3e2\u3c/sup\u3e ) is the number of edges of the graph of oriented distances. We also provide an alternative algorithm for computing the diameter that runs in O(n \u3csup\u3e2\u3c/sup\u3e log n) time. \u3c/p\u3