Improved Bounds for Guarding Plane Graphs with Edges

Abstract

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.

Guarding Quadrangulations and Stacked Triangulations with Edges

“…Considering quadrangulations was motivated by the fact that previous coloring-based approaches for general plane graphs failed on quadrangular faces. Our upper bound of n/3 as well as work from Biniaz et al [1] about quadrangular faces that are far apart from each other suggests that the difficulty is not due to the quadrangular faces themselves. Instead the currently known methods seem to be not strong enough to capture the complexity introduced by a mix of quadrangular and non-quadrangular faces.…”

“…General (not necessarily triangulated) n-vertex plane graphs might need at least n/3 edge guards, even when requiring 2-connectedness [5]. The best known upper bounds have recently been presented by Biniaz et al [1] and come in two different fashions: First, any n-vertex plane graph can be guarded by 3n/8 edge guards found in an iterative process. Second, a coloring approach yields an upper bound of n/3 + α/9 edge guards where α counts the number of quadrangular faces in G. Looking at n-vertex triangulations, Bose et al [5] give a construction for triangulations needing (4n − 8)/13 edge guards 1 .…”

“…The best known upper bounds have recently been presented by Biniaz et al [1] and come in two different fashions: First, any n-vertex plane graph can be guarded by 3n/8 edge guards found in an iterative process. Second, a coloring approach yields an upper bound of n/3 + α/9 edge guards where α counts the number of quadrangular faces in G. Looking at n-vertex triangulations, Bose et al [5] give a construction for triangulations needing (4n − 8)/13 edge guards 1 . A corresponding upper bound of n/3 edge guards was published earlier in the same year by Everett and Rivera-Campo [8].…”

“…In addition we also consider 2-degenerate quadrangulations and present an upper bound of n/4 edge guards, which is best possible. Our motivation to consider quadrangulations is that the coloring approaches developed earlier for general plane graphs [1,8] fail on quadrangular faces. As a second result in Section 3 we present a new upper bound of 2n/7 edge guards for stacked triangulations and show that it is best possible.…”

“…, sa k } is an edge guard set of size k, so Q k needs exactly k edge guards. The following definition and lemma are from Bose et al [4] and we cite it using the terminology of Biniaz et al [1]. A guard coloring of a plane graph G is a non-proper 2-coloring of its vertex set, such that each face f of G has at least one boundary vertex of each color and at least one monochromatic edge (i.e.…”