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In this paper we deal with the signed Roman domination and signed total Romandomination problems. For each problem we propose two integer linear programming(ILP) formulations, the constraint programming (CP) formulation and variableneighborhood search (VNS) method. We present proofs for the correctness of theILP formulations and a polyhedral study in which we show that the polyhedronsof the two model relaxations are equivalent.  VNS uses specifically designed penalty function that allows the appearance ofslightly infeasible solutions. The acceptance of these solutions directs theoverall search process to the promising areas in the long run.  All proposed approaches are tested on the large number of instances.Experimental results indicate that all of them reach optimal solutions for themost of small and middle scale instances. Both ILP models have proven to bemore successful than the other two methods.

Solving the signed Roman domination and signed total Roman domination problems with exact and heuristic methods