
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 ...
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Optimally Guarding 2Reflex Orthogonal Polyhedra by Reflex Edge Guards
We study the problem of guarding an orthogonal polyhedron having reflex ...
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Acutely Triangulated, Stacked, and Very Ununfoldable Polyhedra
We present new examples of topologically convex edgeununfoldable polyhe...
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Faster and Enhanced InclusionMinimal Cograph Completion
We design two incremental algorithms for computing an inclusionminimal ...
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An Edge ExtrusionApproach to Generate Extruded MiuraOri and Its Double Tiling Origami Patterns
This paper proposes a family of origami tessellations called extruded Mi...
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Treelike distance colouring for planar graphs of sufficient girth
Given a multigraph G and a positive integer t, the distancet chromatic ...
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Random 2cell embeddings of multistars
By using permutation representations of maps, one obtains a bijection be...
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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 ∈ E if at least one vertex from {v,w} is on the boundary of f. For a planar graph class 𝒢 we ask for the minimal number of edges needed to guard all faces of any nvertex graph in 𝒢. We prove that ⌊ n/3 ⌋ edges are always sufficient for quadrangulations and give a construction where ⌊ (n2)/4 ⌋ edges are necessary. For 2degenerate quadrangulations we improve this to a tight upper bound of ⌊ n/4 ⌋ edges. We further prove that ⌊ 2n/7 ⌋ edges are always sufficient for stacked triangulations (that are the 3degenerate triangulations) and show that this is best possible up to a small additive constant.
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