[Paper Review] On the Strong Roman Domination Number of Graphs

M. P. Álvarez-Ruiz, Ismael G. Yero+3 more|arXiv (Cornell University)|Feb 13, 2015

Advanced Graph Theory Research3 citations

TL;DR

This paper introduces the strong Roman domination number γStR(G), a new graph invariant modeling defense against multiple simultaneous attacks, where each undefended vertex must be protected by a neighbor with two legions. The authors prove the problem is NP-complete, establish tight bounds—particularly γStR(T) ≤ 6n/7 for trees—and fully characterize realizability of values between ⌈(n+1)/2⌉ and ⌊6n/7⌋ for trees.

ABSTRACT

Based on the history that the Emperor Constantine decreed that any undefended place (with no legions) of the Roman Empire must be protected by a "stronger" neighbor place (having two legions), a graph theoretical model called Roman domination in graphs was described. A Roman dominating function for a graph $G=(V,E)$, is a function $f:V ightarrow \{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ has at least a neighbor $w$ in $G$ for which $f(w)=2$. The Roman domination number of a graph is the minimum weight, $\sum_{v\in V}f(v)$, of a Roman dominating function. In this paper we initiate the study of a new parameter related to Roman domination, which we call strong Roman domination number and denote it by $γ_{StR}(G)$. We approach the problem of a Roman domination-type defensive strategy under multiple simultaneous attacks and begin with the study of several mathematical properties of this invariant. In particular, we first show that the decision problem regarding the computation of the strong Roman domination number is NP-complete, even when restricted to bipartite graphs. We obtain several bounds on such a parameter and give some realizability results for it. Moreover, we prove that for any tree $T$ of order $n\ge 3$, $γ_{StR}(T)\le 6n/7$ and characterize all extremal trees.

Motivation & Objective

• To formalize a new graph-theoretic model for defending against multiple simultaneous attacks, inspired by the historical Roman Empire's defense strategy.
• To define and investigate the strong Roman domination number γStR(G), a variant of Roman domination under multiple attack scenarios.
• To establish computational complexity, providing a proof that determining γStR(G) is NP-complete, even for bipartite graphs.
• To derive tight upper and lower bounds for γStR(G) in terms of graph parameters such as order, diameter, and girth.
• To characterize all trees achieving the upper bound γStR(T) ≤ 6n/7 and prove realizability of all integer values of γStR(T) within the range [⌈(n+1)/2⌉, ⌊6n/7⌋]

Proposed method

• Define a strong Roman dominating function (SRDF) as a function f: V → {0,1,2} such that every vertex with f(v)=0 has a neighbor w with f(w)=2.
• Introduce the strong Roman domination number γStR(G) as the minimum weight ∑f(v) over all SRDFs on G.
• Prove NP-completeness of the decision problem for γStR(G) via reduction techniques, even when restricted to bipartite graphs.
• Derive general bounds for γStR(G) using structural graph parameters such as order n, diameter Diam(G), and girth g(G).
• Construct a family of graphs Gₙ(q,j,l) parameterized by integers q, j, l to realize any desired value p of γStR(G) within the specified range.
• Use algebraic manipulation of systems of Diophantine equations to ensure integer solutions for parameters q, j, l given n and p, ensuring graph realizability

Experimental results

Research questions

• RQ1 What is the computational complexity of determining the strong Roman domination number γStR(G) for a given graph?
• RQ2 What are the tightest possible upper and lower bounds for γStR(G) in terms of graph order and other structural parameters?
• RQ3 What is the maximum possible value of γStR(T) for a tree T of order n, and which trees achieve this bound?
• RQ4 For any integer p between ⌈(n+1)/2⌉ and ⌊6n/7⌋, does there exist a tree T of order n such that γStR(T) = p?
• RQ5 How can a graph be systematically constructed to realize a given strong Roman domination number p for a specified order n?

Key findings

• The decision problem for computing the strong Roman domination number γStR(G) is NP-complete, even when restricted to bipartite graphs.
• For any tree T of order n ≥ 3, the strong Roman domination number satisfies γStR(T) ≤ 6n/7, and this bound is tight.
• All extremal trees achieving the upper bound γStR(T) = 6n/7 are completely characterized in terms of their structural parameters.
• For any integers n ≥ 3 and p satisfying ⌈(n+1)/2⌉ ≤ p ≤ ⌊6n/7⌋, there exists a tree T of order n such that γStR(T) = p, proving full realizability of values in this range.
• The construction of such graphs relies on solving a system of Diophantine equations to determine parameters q, j, l such that Gₙ(q,j,l) realizes the desired γStR value.
• The solution process ensures that for any valid p, integer values of q and l are always obtained, confirming the existence of such graphs for all feasible p