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Face-hitting dominating sets in planar graphs: Alternative proof and linear-time algorithm

Therese Biedl

Abstract

In a recent paper, Francis, Illickan, Jose and Rajendraprasad showed that every $n$-vertex plane graph $G$ has (under some natural restrictions) a vertex-partition into two sets $V_1$ and $V_2$ such that each $V_i$ is \emph{dominating} (every vertex of $G$ contains a vertex of $V_i$ in its closed neighbourhood) and \emph{face-hitting} (every face of $G$ is incident to a vertex of $V_i$). Their proof works by considering a supergraph $G'$ of $G$ that has certain properties, and among all such graphs, taking one that has the fewest edges. As such, their proof is not algorithmic. Their proof also relies on the 4-color theorem, for which a quadratic-time algorithm exists, but it would not be easy to implement. In this paper, we give a new proof that every $n$-vertex plane graph $G$ has (under the same restrictions) a vertex-partition into two dominating face-hitting sets. Our proof is constructive, and requires nothing more complicated than splitting a graph into 2-connected components, finding an ear decomposition, and computing a perfect matching in a 3-regular plane graph. For all these problems, linear-time algorithms are known and so we can find the vertex-partition in linear time.

Face-hitting dominating sets in planar graphs: Alternative proof and linear-time algorithm

Abstract

In a recent paper, Francis, Illickan, Jose and Rajendraprasad showed that every -vertex plane graph has (under some natural restrictions) a vertex-partition into two sets and such that each is \emph{dominating} (every vertex of contains a vertex of in its closed neighbourhood) and \emph{face-hitting} (every face of is incident to a vertex of ). Their proof works by considering a supergraph of that has certain properties, and among all such graphs, taking one that has the fewest edges. As such, their proof is not algorithmic. Their proof also relies on the 4-color theorem, for which a quadratic-time algorithm exists, but it would not be easy to implement. In this paper, we give a new proof that every -vertex plane graph has (under the same restrictions) a vertex-partition into two dominating face-hitting sets. Our proof is constructive, and requires nothing more complicated than splitting a graph into 2-connected components, finding an ear decomposition, and computing a perfect matching in a 3-regular plane graph. For all these problems, linear-time algorithms are known and so we can find the vertex-partition in linear time.

Paper Structure

This paper contains 12 sections, 13 theorems, 1 equation, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bipartite Edge Covers
  4. Triangulated graphs
  5. 2-connected Plane Graphs
  6. Putting It All Together
  7. Hitting faces of smaller sizes
  8. Making Everyone Happy
  9. Some bad examples
  10. Triangulating a single face
  11. Triangulating all faces
  12. Further Thoughts

Key Result

Theorem 1

There exists a linear-time algorithm for the following task: Given a plane graph $G=(V,E)$ without isolated vertices or faces incident to at most two vertices, partition $V$ into two vertex sets $V_1,V_2$ that both are dominating as well as face-hitting.

Figures (7)

  • Figure 1: A triangulated planar graph $G$, not simple, with its dual graph (red dotted) and a perfect matching in it (thick dashed). We also show the resulting bipartite graph $H$, where each colour-part is a face-hitting dominating set of $G$.
  • Figure 2: Graphs where any face-hitting dominating set $S$ must be large. (a) With faces of degree 1, we may need $|S|=n$. (b) With loops, we may need $|S|\geq n-2$ even if all faces have degree 2 or more. (c) With bigons, we may need $|S|\geq \tfrac{3}{4}n$ even if there are no loops.
  • Figure 3: Graphs where triangulations have few happy angles (shaded green). (a) A graph with cutvertices. (b) Demanding a simple triangulation. (c) We cannot make two consecutive angles happy.
  • Figure 4: Making a prescribed set of angles (indicated by green circles) happy.
  • Figure 5: Finding an ear decomposition of a plane 2-connected graph via a topological order in the bipolar orientation of the dual graph.
  • ...and 2 more figures

Theorems & Definitions (24)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • Corollary 6
  • proof
  • Corollary 7
  • ...and 14 more