Chordal-(2,1) graph sandwich problem with boundary conditions

In this work, we consider the graph sandwich problem for a property Π, a decision problem proposed by Golumbic, Kaplan, and Shamir as follows: given two graphs G¹ = (V, E¹) and G² = (V, E²), the question is whether there exists a graph G = (V, E) such that E¹ ⊆ E ⊆ E² and G satisfies Π. For many graph classes, this problem was settled to be NP-complete. For this reason, a different kind of approach has been proposed: instead of focusing only on property Π, we can also require special properties ${\Pi }^i$ for the input graphs Gⁱ, i = 1, 2, resulting in a problem that generalizes graph sandwich problems, called graph sandwich problems with boundary conditions. This problem is denoted by a triple (${\Pi }^1$, Π, ${\Pi }^2$)-sp. We deal with the property Π of being a chordal-(2,1)-graph when it is given that G² is a graph of a class for which there is a polynomial bound on the number of cliques, say pnc. chordal-(2,1)-sp is known to be NP-complete but, when requiring G² to be of a class in pnc, we prove that (*, chordal-(2,1), pnc)-sp is polynomially time solvable.