Some Bounds on the Double Domination of Signed Generalized Petersen Graphs and Signed I-Graphs

Abstract

In a graph G, a vertex dominates itself and its neighbors. A subset D ⊆ V (G) is a double dominating set of G if D dominates every vertex of G at least twice. A signed graph Σ = (G, σ) is a graph G together with an assignment σ of positive or negative signs to all its edges. A cycle in a signed graph is positive if the product of its edge signs is positive. A signed graph is balanced if all its cycles are positive. A subset D ⊆ V (Σ) is a double dominating set of Σ if it satisfies the following conditions: (i) D is a double dominating set of G, and (ii) Σ[D : V \ D] is balanced, where Σ[D : V \ D] is the subgraph of Σ induced by the edges of Σ with one end point in D and the other end point in V \ D. The cardinality of a minimum double dominating set of Σ is the double domination number γ ×2 (Σ). In this paper, wegive bounds for the double domination number of signed cubic graphs. We also obtain some bounds on the double domination number of signed generalized Petersen graphs and signed I-graphs.