The k-th Roman domination problem is polynomial on interval graphs

Let G be some simple graph and k be any positive integer. Take $$h: V(G)\rightarrow \{0,1,\ldots ,k+1\}$$ h : V ( G ) → { 0 , 1 , … , k + 1 } and $$v \in V(G)$$ v ∈ V ( G ) , let $$AN_{h}(v)$$ A N h ( v ) denote the set of vertices $$w\in N_{G}(v)$$ w ∈ N G ( v ) with $$h(w)\ge 1$$ h ( w ) ≥ 1 . Let $$AN_{h}[v] = AN_{h}(v)\cup \{v\}$$ A N h [ v ] = A N h ( v ) ∪ { v } . The function h is a [k]-Roman dominating function of G if $$h(AN_{h}[v]) \ge |AN_{h}(v)| + k$$ h ( A N h [ v ] ) ≥ | A N h ( v ) | + k holds for any $$v \in V(G)$$ v ∈ V ( G ) . The minimum weight of such a function is called the k-th Roman Domination number of G, which is denoted by $$\gamma _{kR}(G)$$ γ kR ( G ) . In 2020, Banerjee et al. presented linear time algorithms to compute the double Roman domination number on proper interval graphs and block graphs. They posed the open question that whether there is some polynomial time algorithm to solve the double Roman domination problem on interval graphs. It is argued that the interval graph is a nontrivial graph class. In this article, we design a simple dynamic polynomial time algorithm to solve the k-th Roman domination problem on interval graphs for each fixed integer $$k>1$$ k > 1 .