[Paper Review] Domination Cover Pebbling: Structural Results

Nathaniel G. Watson, Carl Yerger|ArXiv.org|Sep 23, 2005

Advanced Graph Theory Research11 references3 citations

TL;DR

This paper establishes structural bounds for the domination cover pebbling number ψ(G) in graphs of bounded diameter, proving ψ(G) ≤ n−1 for diameter-2 graphs and ψ(G) ≤ 2^{d−2}(n−2)+1 for diameter d. It also derives a lower bound of 3 for the ratio λ(G)/ψ(G) in diameter-2 graphs and introduces subversion domination cover pebbling, analyzing the parameter Ωω(G) for undominated connected components of size at most ω.

ABSTRACT

This paper continues the results of "Domination Cover Pebbling: Graph Families." An almost sharp bound for the domination cover pebbling (DCP) number for graphs G with specified diameter has been computed. For graphs of diameter two, a bound for the ratio between the cover pebbling number of G and the DCP number of G has been computed. A variant of domination cover pebbling, called subversion DCP is introducted, and preliminary results are discussed.

Motivation & Objective

• . The paper aims to establish tight structural bounds for the domination cover pebbling number ψ(G) in graphs of bounded diameter.
• It investigates the relationship between ψ(G) and the cover pebbling number λ(G), particularly their ratio in diameter-2 graphs.
• The authors introduce and analyze a variant called subversion domination cover pebbling, defined via the parameter Ωω(G), which measures the minimum number of pebbles needed to ensure all but at most ω vertices are dominated.
• The study extends prior work on domination cover pebbling by exploring how the presence of undominated connected components affects pebbling requirements.

Proposed method

• . The proof for ψ(G) ≤ n−1 in diameter-2 graphs uses case analysis based on vertex sets S1 (vertices with >1 pebble), S2 (empty vertices adjacent to S1), and S3 (vertices not dominated by S1 or S2).
• The pairing number P(c′) = ∑v∈G max{0, ⌈(c′(v)−1)/2⌉} is used to estimate the number of pebbling moves available from a configuration.
• For diameter-d graphs, the bound ψ(G) ≤ 2^{d−2}(n−2)+1 is derived via inductive reasoning and distance-based vertex partitioning.
• The ratio λ(G)/ψ(G) ≥3 for diameter-2 graphs is established by analyzing worst-case configurations and leveraging the fact that λ(G) accounts for full vertex coverage, while ψ(G) only requires domination.
• The subversion DCP model introduces Ωω(G), the minimal pebble count such that after pebbling, all but at most ω vertices are dominated, with undominated vertices forming a connected component.
• The proof for Ωω(G) ≤ n−1−ω in diameter-2 graphs uses a reduction argument: removing ω vertices from T2 and applying the diameter-2 argument to the remaining graph G′, leveraging preserved distance properties.

Experimental results

Research questions

• RQ1 . What is the tightest possible upper bound for the domination cover pebbling number ψ(G) in graphs of diameter 2?
• RQ2 . How does the cover pebbling number λ(G) compare to ψ(G) in diameter-2 graphs, and what is the minimal possible ratio λ(G)/ψ(G)?
• RQ3 . Can a generalized variant of domination cover pebbling, called subversion DCP, be meaningfully defined and bounded, particularly in terms of undominated connected components of size ω?
• RQ4 . What structural properties of graphs with higher diameter (e.g., d=3) allow for analogous bounds on Ωω(G), and how can such bounds be constructed?

Key findings

• . For all graphs G of order n with diameter 2, the domination cover pebbling number satisfies ψ(G) ≤ n−1, and this bound is sharp, as demonstrated by the star graph with n−1 pebbles on all but one leaf.
• . The ratio between the cover pebbling number λ(G) and the domination cover pebbling number ψ(G) is at least 3 for all graphs of diameter 2.
• . For graphs of diameter 2, the parameter Ωω(G), which bounds the minimal pebble count to ensure all but at most ω vertices are dominated, satisfies Ωω(G) ≤ n−1−ω.
• . A construction of a graph Hn with a star structure and ω additional edges among outer vertices shows that Ωω(Hn) > n−2−ω, proving the bound is sharp.
• . The paper conjectures that for diameter-3 graphs, Ωω(G) ≤ ⌊3/2(n−2−ω) + 1⌋, with a construction demonstrating that this bound is tight up to the floor function.
• . The analysis shows that even after removing ω vertices from a diameter-2 graph, the remaining graph G′ retains the property that every vertex in T′1 is within distance 2 of any vertex in G′, preserving the applicability of the diameter-2 argument.