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Abstract: For a positive integer k, a {k}-Roman dominating function of a graph G = (V, E) is a function f : V → {0, 1, . . . , k} satisfying f (N(v)) ≥ k for each vertex v ∈ V with f (v) = 0. Every graph G satisfies γ{Rk}(G) ≤ kγ(G), where γ{Rk}(G) denotes the minimum weight of a {k}-Roman dominating function of G and γ(G) is the domination number of G. In this work we study graphs for which the equality is reached, called {k}-Roman graphs. This extends the concept of {k}-Roman trees studied by Wang et al. in 2021 to gen- eral graphs. We prove that for every k ≥ 3, the problem of recognizing {k}-Roman graphs is NP-hard, even when restricted to split graphs. We provide partial answers to the question of which split graphs are {2}-Roman: we characterize {2}-Roman split graphs that can be decomposed with respect to the split join operation into two smaller split graphs and classify the {k}-Roman property within two specific families of split graphs that are prime with respect to the split join operation: suns and their complements.
Keywords: graph domination, {k}-Roman domination, {k}-Roman graph, split graph, split join, NP-completeness
Published in RUP: 16.12.2025; Views: 208; Downloads: 2
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