[Paper Review] Power domination in maximal planar graphs

Paul Dorbec, Antonio González+1 more|arXiv (Cornell University)|Jun 30, 2017

Advanced Graph Theory Research28 references3 citations

TL;DR

This paper establishes an upper bound of (n−2)/4 on the power domination number for maximal planar graphs with n ≥ 6 vertices. The authors prove that such graphs can be fully monitored using at most (n−2)/4 strategically placed measurement devices, leveraging structural properties of triangulations and propagation rules in planar embeddings, with tightness demonstrated on specific graphs like the triakis tetrahedron.

ABSTRACT

Power domination in graphs emerged from the problem of monitoring an electrical system by placing as few measurement devices in the system as possible. It corresponds to a variant of domination that includes the possibility of propagation. For measurement devices placed on a set S of vertices of a graph G, the set of monitored vertices is initially the set S together with all its neighbors. Then iteratively, whenever some monitored vertex v has a single neighbor u not yet monitored, u gets monitored. A set S is said to be a power dominating set of the graph G if all vertices of G eventually are monitored. The power domination number of a graph is the minimum size of a power dominating set. In this paper, we prove that any maximal planar graph of order n $\\ge$ 6 admits a power dominating set of size at most (n--2)/4 .

Motivation & Objective

• To determine tight upper bounds on the power domination number for maximal planar graphs.
• To analyze the structural properties of maximal planar graphs that enable efficient power domination.
• To prove that (n−2)/4 is an upper bound for the power domination number when n ≥ 6.
• To demonstrate the tightness of this bound using explicit constructions like the triakis tetrahedron.
• To provide a constructive framework for identifying small power dominating sets in triangulations using facial triangle and embedding properties.

Proposed method

• Utilizes the power domination model where monitoring propagates from initial measurements via vertices with exactly one unmonitored neighbor.
• Employs graph-theoretic techniques on maximal planar graphs (triangulations) by analyzing facial triangles and vertex degrees.
• Applies induction and case analysis on vertex degrees (≤3 or ≥4) to derive structural constraints on the graph.
• Uses embedding properties to identify facial triangles and neighbor relationships critical for propagation.
• Applies lemmas (e.g., Lemma 4.2) to rule out configurations that would violate propagation rules or embedding consistency.
• Employs Algorithm 2 to reduce induced subgraphs to canonical forms isomorphic to specific configurations in Figure 13.

Experimental results

Research questions

• RQ1 What is the maximum size of a minimum power dominating set in a maximal planar graph of order n ≥ 6?
• RQ2 Can the power domination number be bounded uniformly across all maximal planar graphs using a function of n?
• RQ3 Are there specific graph families or configurations that achieve the upper bound of (n−2)/4?
• RQ4 How do vertex degree and facial triangle structure influence the propagation of monitored vertices?
• RQ5 What structural constraints prevent smaller power dominating sets in certain maximal planar graphs?

Key findings

• The power domination number of any maximal planar graph with n ≥ 6 vertices is at most (n−2)/4.
• The bound (n−2)/4 is tight for the triakis tetrahedron, a maximal planar graph with 10 vertices and power domination number 2.
• The triakis tetrahedron achieves equality in the bound, confirming its tightness for n=10.
• For graphs with minimum vertex degree 3, the structure of facial triangles and neighbor propagation is critical in determining the power dominating set size.
• After applying Algorithm 2, all induced triangulations of a maximal planar graph are isomorphic to one of the configurations in Figure 13, enabling structural classification.
• The proof relies on case analysis based on vertex degrees and facial triangle configurations, showing that no larger dominating sets are needed than (n−2)/4.