Global Roman Domination

• Global Roman domination is a graph labeling problem where a function assigns values to vertices such that vertices with value 0 have neighbors with value 2 in both the graph and its complement.
• The topic highlights complexity differences, showing that while classical Roman domination may be solved in linear time for some graph classes, its global variant is NP-complete in structures like split and chordal bipartite graphs.
• Recent advances include efficient cotree-based algorithms for cographs and innovative distributed game-theoretic approaches to approximate global Roman domination solutions.

A global Roman dominating function (GRDF) on a graph $G=(V,E)$ is a labeling $f:V\rightarrow\{0,1,2\}$ such that for every vertex $u$ with $f(u)=0$, there exist vertices $v\in N_G(u)$ and $w\in N_{\overline{G}}(u)$ with $f(v)=f(w)=2$. The associated optimization, the Global Roman Domination (GRD) problem, seeks a GRDF $f$ minimizing the total weight $w(f)=\sum_{u\in V}f(u)$. Thus, the global Roman domination number $\gamma_{gR}(G)$ is $\min\{w(f):f\text{ is a GRDF on }G\}$. This requirement—simultaneous Roman domination for both $G$ and its complement $\overline{G}$—distinguishes the global variant from classical Roman domination, impacting both computational complexity and algorithmic approach (Reddy et al., 14 Jan 2026).

1. Formal Definitions and Theoretical Foundations

Let $G=(V,E)$ be a simple undirected graph of order $n=|V|$. A Roman dominating function (RDF) is $f:V\to\{0,1,2\}$ such that for all $u\in V$ with $f(u)=0$, some neighbor $v\in N_G(u)$ has $f(v)=2$. The Roman domination number is $\gamma_R(G)=\min\{w(f):f\text{ is an RDF on }G\}$.

In the global variant, the function $f$ must be a Roman dominating function on both $G$ and its complement $\overline{G}$. Explicitly, a global Roman dominating function (GRDF) satisfies: For every $u\in V$ with $f(u)=0$, there are $v\in N_G(u)$ and $w\in N_{\overline{G}}(u)$ with $f(v)=f(w)=2$.

The objective for GRD is:

$$\gamma_{gR}(G)=\min_{f:V\rightarrow\{0,1,2\}}\sum_{u\in V}f(u)$$

subject to $f$ being a GRDF.

2. Separation of Roman Domination and Global Roman Domination

The GRD problem exhibits substantial complexity differences from classical Roman domination. Notably, the two problems are not computationally equivalent. There exist graph classes for which one variant is efficiently solvable, while the other is NP-complete.

• Class F: On this class, Roman domination is solvable in linear time, but GRD is NP-complete. The construction involves three disjoint copies of a 3-regular graph $G$ plus special vertices $v_1,v_2,v_3$ with specific connectivity. For every $H\in F$, $\gamma_R(H)=4$ (by placing 2 on one of $v_i$ and 1 on the others), yielding tractability for RD. However, a reduction from Dominating Set shows GRD is NP-complete.
• Class G: Here, GRD is solvable in linear time while RD is NP-complete. The underlying construction leverages gadgets encoding the Exact 4-Cover (X4C) problem, with a split graph structure. For every $H\in G$, $\gamma_{gR}(H)=10\ell + 2$; the value is independent of the X4C instance, admitting an efficient algorithm. In contrast, RD inherits NP-completeness from the X4C reduction.

This separation demonstrates the need to consider the structural properties of the target graph when designing algorithms for Roman domination variants.

3. Hardness Results in Special Graph Classes

The computational intractability of GRD extends to several well-known graph classes:

• Split graphs: GRD is NP-complete via a reduction from Exact 3-Cover (X3C). The construction comprises a clique $A$, pendants $P$, and an independent set $B$, with adjacencies encoding set membership. Feasible GRDFs require a tight correspondence between label placements and the X3C solution structure.
• Chordal bipartite graphs, planar bipartite graphs of maximum degree five, and circle graphs: For each, GRD is NP-complete by reduction from Dominating Set. Each graph in these classes is extended by "tree-gadgets," ensuring that any minimal GRDF aligns with solutions to the underlying NP-complete problem, while preserving specific graph properties (planarity, bipartiteness, circle-representability).
• These reductions employ gadgets that force certain label assignments, linking the weight of a feasible GRDF to the solution cost of the base problem.

Graph Class	RD Complexity	GRD Complexity
Class F (special gadget)	Linear time (P)	NP-complete
Class G (special gadget)	NP-complete	Linear time (P)
Split graphs	?	NP-complete
Chordal bipartite	?	NP-complete
Planar bipartite $\Delta\leq 5$	?	NP-complete
Circle graphs	?	NP-complete

A plausible implication is that algorithm design for coverage-based graph labeling must assess both the primal and complement structure of input classes.

4. Algorithmic Approaches: Linear-Time Solution for Cographs

A cograph, defined by the absence of induced $P_4$ subgraphs, admits a modular decomposition via a cotree. This enables efficient computation for both RD and GRD.

• Cotree-centric Algorithm: Internal nodes of cotree $T_G$ are labeled $\bigsqcup$ (disjoint union) or $\oplus$ (join). Leaves correspond to single vertices.
• For leaves: $\gamma_R(\{v\})=1$, $\gamma_{gR}(\{v\})=2$.
• For internal nodes:
  • If $G=G_1\bigsqcup G_2$ and $|G_1|,|G_2|\ge2$: $$\gamma_{gR}(G)=\gamma_R(G)=\gamma_R(G_1)+\gamma_R(G_2)$$.
  • If $G=G_1\oplus G_2$ with $|G_1|,|G_2|\ge 2$: $\gamma_{gR}(G)=\gamma_R(G_1)+\gamma_R(G_2)$.
• Special cases (universal/isolated vertices) are handled in constant time by closed-form rules.
• The overall running time is $O(n+m)$.

This efficient computation aligns with the additive decomposability of Roman domination numbers on cographs, and no additional complexity is incurred by the global requirement (Reddy et al., 14 Jan 2026).

5. Pertinent Reductions and Gadget Constructions

Reduction techniques establish the NP-completeness of GRD on various classes by encoding hard combinatorial problems:

• Exact 3-Cover and Exact 4-Cover: Used as sources for reductions to split and special gadget graphs.
• Tree-Gadgets: Attached to vertices to force label-2 assignments, ensuring that every feasible GRDF reflects a solution to the base problem.
• Planarity and Chordality Preservation: These reductions guarantee that global Roman domination hardness is retained under graph property constraints, e.g., preservation under gadget attachment.

Key lemmas include:

• The weight lower bound on gadgets, ensuring each attachment incurs minimal cost under any RDF or GRDF.
• Additive behavior under join/union decompositions for cographs, facilitating linear-time computation via modular decomposition.
• Separation results via construction, indicating non-equivalence between RD and GRD even within highly restricted graph families.

6. Connections to Distributed and Game-Theoretic Algorithms

Recent work recasts Roman domination as a vertex-based potential game (Chen et al., 2023):

• Each vertex is treated as a player choosing strategies $c_i\in\{0,1,2\}$, interacting locally.
• The game defines utilities penalizing both troop deployment and exposure of neighbors.
• Nash equilibria of the resulting game correspond to strong minimal Roman dominating functions and are Pareto-optimal.
• Distributed algorithms, notably the game-based synchronous algorithm (GSA), converge to NE in $O(n)$ rounds; an enhanced version (EGSA) employs coalitional moves for better approximate solutions in $O(n^2)$ rounds.
• Empirical results indicate scalability and competitive accuracy against centralized methods.

This suggests potential for distributed heuristics applied to the general GRD framework, particularly where centralized computation is infeasible due to size or dynamics.

7. Conclusion and Directions

The global Roman domination problem integrates combinatorial graph labeling with dual constraints from both primal and complement graphs. Rigorous complexity-theoretic separations highlight new boundaries in algorithmic graph theory, while efficient algorithms for cographs indicate the impact of graph structure on tractability. NP-completeness on split, chordal bipartite, planar bipartite, and circle graphs connects GRD complexity to classic covering and packing hardness. Concurrently, distributed/game-theoretic algorithms offer scalable approaches for networked environments. These findings collectively advance understanding and establish new directions in domination-type graph optimization (Reddy et al., 14 Jan 2026, Chen et al., 2023).