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

2025-08-15

AI Summary

This paper addresses the vertex bipartition problem for planar graphs: partitioning the vertex set into two subsets $V_1$ and $V_2$, each of which is simultaneously a dominating set (i.e., every vertex intersects its closed neighborhood with $V_i$) and a face-hitting set (i.e., every face is incident to at least one vertex in $V_i$). We present a fully constructive linear-time algorithm ($O(n)$) that avoids reliance on the Four-Color Theorem. Our approach leverages decomposition into 2-connected components, ear decomposition, and perfect matchings in 3-regular planar graphs, while reusing known linear-time subroutines. This is the first solution that is simultaneously constructive, runs in linear time, and requires no advanced graph-theoretic tools—such as the Four-Color Theorem—thereby significantly improving the efficiency of solving this dual-cover partition problem on large-scale planar graphs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Activity and Plan Recognition
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• Finding vertex partitions into two dominating face-hitting sets
• Providing constructive proof for planar graph decomposition
• Developing linear-time algorithm for face-hitting dominating sets

Innovation

• Constructive proof using ear decomposition
• Linear-time algorithm via perfect matching
• Avoids 4-color theorem with simpler components