[Paper Review] A Note on Roman \{2\}-domination problem in graphs

Hangdi Chen, Changhong Lü|arXiv (Cornell University)|Apr 25, 2018

Advanced Graph Theory Research16 references3 citations

TL;DR

This paper establishes the NP-completeness of the Roman {2}-domination problem in split graphs and presents a linear-time algorithm for computing the independent Roman {2}-domination number in trees, resolving an open problem. It further provides a linear-time algorithm for Roman {2}-domination in block graphs using dynamic programming on block-cutpoint trees with state classification based on vertex types and function values.

ABSTRACT

For a graph $G=(V,E)$, a Roman $\{2\}$-dominating function (R2DF)$f:V ightarrow \{0,1,2\}$ has the property that for every vertex $v\in V$ with $f(v)=0$, either there exists a neighbor $u\in N(v)$, with $f(u)=2$, or at least two neighbors $x,y\in N(v)$ having $f(x)=f(y)=1$. The weight of a R2DF is the sum $f(V)=\sum_{v\in V}{f(v)}$, and the minimum weight of a R2DF is the Roman $\{2\}$-domination number $γ_{\{R2\}}(G)$. A R2DF is independent if the set of vertices having positive function values is an independent set. The independent Roman $\{2\}$-domination number $i_{\{R2\}}(G)$ is the minimum weight of an independent Roman $\{2\}$-dominating function on $G$. In this paper, we show that the decision problem associated with $γ_{\{R2\}}(G)$ is NP-complete even when restricted to split graphs. We design a linear time algorithm for computing the value of $i_{\{R2\}}(T)$ for any tree $T$. This answers an open problem raised by Rahmouni and Chellali [Independent Roman $\{2\}$-domination in graphs, Discrete Applied Mathematics 236 (2018), 408-414]. Chellali, Haynes, Hedetniemi and McRae \cite{chellali2016roman} have showed that Roman $\{2\}$-domination number can be computed for the class of trees in linear time. As a generalization, we present a linear time algorithm for solving the Roman $\{2\}$-domination problem in block graphs.

Motivation & Objective

• To prove that the decision problem for Roman {2}-domination is NP-complete even when restricted to split graphs.
• To resolve an open problem posed by Rahmouni and Chellali regarding the existence of a linear-time algorithm for computing the independent Roman {2}-domination number in trees.
• To extend efficient computation of Roman {2}-domination to block graphs, a broader class than trees.
• To develop a dynamic programming framework on block-cutpoint trees for computing the Roman {2}-domination number in linear time.

Proposed method

• Reduction from the Vertex Cover problem to prove NP-completeness of Roman {2}-domination in split graphs.
• Design of a dynamic programming algorithm on the block-cutpoint tree of a block graph, using six state types to classify vertex function assignments.
• Classification of vertices into cut-vertices and block-vertices of types 0, 1, and 2, each with specific initial state vectors.
• State transitions defined via recursive combinations of subtrees, with min-operations over valid function value combinations to preserve domination constraints.
• Use of a tree ordering and bottom-up traversal of the block-cutpoint tree to compute optimal function values efficiently.
• Final output derived as the minimum of the first three states at the root, representing valid Roman {2}-dominating functions.

Experimental results

Research questions

• RQ1 Is the Roman {2}-domination decision problem NP-complete for split graphs?
• RQ2 Can the independent Roman {2}-domination number be computed in linear time for any tree?
• RQ3 Does a linear-time algorithm exist for computing the Roman {2}-domination number in block graphs?
• RQ4 What is the structure of valid Roman {2}-dominating functions in block graphs, and how can they be efficiently enumerated via dynamic programming?

Key findings

• The Roman {2}-domination decision problem is NP-complete for split graphs, as proven via a polynomial-time reduction from Vertex Cover.
• A linear-time algorithm is successfully designed and proven correct for computing the independent Roman {2}-domination number in any tree, resolving an open problem.
• The algorithm for block graphs runs in O(n) time, where n is the number of vertices, and is based on dynamic programming over the block-cutpoint tree.
• The state classification scheme (B1 to B6 and A1 to A6) enables correct propagation of domination constraints through tree decomposition.
• The final result is obtained as the minimum among the first three states at the root, ensuring the minimal weight of a valid Roman {2}-dominating function.
• The algorithm correctly handles all block types (0, 1, 2) and cut-vertices by initializing appropriate state vectors and combining them via min-operations with defined transition rules.