DeepAI AI Chat
Log In Sign Up

Improved Bounds for Guarding Plane Graphs with Edges

by   Ahmad Biniaz, et al.
Carleton University
Université Libre de Bruxelles

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 guard G. 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 (1997) that G can be guarded with at most 2n/5 edges, then extend this approach with a deeper analysis to yield an improved bound of 3n/8 edges for any plane graph. 2- We prove that there exists an edge guard set of G with at most n/3+α/9 edges, where α is the number of quadrilateral faces in G. This improves the previous bound of n/3 + α by Bose, Kirkpatrick, and Li (2003). Moreover, if there is no short path between any two quadrilateral faces in G, we show that n/3 edges suffice, removing the dependence on α.


page 1

page 2

page 3

page 4


Guarding Quadrangulations and Stacked Triangulations with Edges

Let G = (V,E) be a plane graph. A face f of G is guarded by an edge vw ∈...

Saturated k-Plane Drawings with Few Edges

A drawing of a graph is k-plane if no edge is crossed more than k times....

Bounds on Ramsey Games via Alterations

This note contains a refined alteration approach for constructing H-free...

Improved Bounds for Covering Paths and Trees in the Plane

A covering path for a planar point set is a path drawn in the plane with...

A Tight Extremal Bound on the Lovász Cactus Number in Planar Graphs

A cactus graph is a graph in which any two cycles are edge-disjoint. We ...

Temporal Cliques admit Sparse Spanners

Let G=(G,λ) be a labeled graph on n vertices with λ:E_G→N a locally inj...

Sharp bounds for the chromatic number of random Kneser graphs

Given positive integers n> 2k, a Kneser graph KG_n,k is a graph whose ve...