Computing the security number of gap based graphs of symmetric numerical semigroups

A simple undirected graph G is called a gap-based graph if there exists a numerical semigroup ξ such that V(G)=N∖ξ and E(G)={xy:x−y or y−x∈ξ}. In this paper, we characterize a class of symmetric numerical semigroups satisfying 2m<F<3m in terms of the gap sets, where m is the multiplicity and F is the Frobenius number. Also, we compute the secure set and the security number of its gap-based graphs.