Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: ,  define and study the new problem Simultaneous PQ-Ordering. Its input consists of a set of PQ-trees, which represent sets of circular orders of their leaves, together with a set of child-parent relations beten these PQ-trees, such that the leaves of the child form a subset of the leaves of the parent. Simultaneous PQ-Ordering asks whether orders of the leaves of each of the trees can be chosen simultaneously, that is, for every child-parent relation the order chosen for the parent is an extension of the order chosen for the child.  show that Simultaneous PQ-Ordering is NP-complete in general and  identify a family of instances that can be solved efficiently, the 2-fixed instances.  show that this result serves as a framework for several other problems that can be formulated as instances of Simultaneous PQ-Ordering.#R##N##R##N#In particular,  give linear-time algorithms for recognizing simultaneous interval graphs and extending partial interval representations. Moreover,  obtain a linear-time algorithm for Partially PQ-Constrained Planarity for biconnected graphs, which asks for a planar embedding in the presence of PQ-trees that restrict the possible orderings of edges around vertices, and a quadratic-time algorithm for Simultaneous Embedding with Fixed Edges for biconnected graphs with a connected intersection. Both results can be extended to the case where the input graphs are not necessarily biconnected but have the property that each cutvertex is contained in at most two non-trivial blocks. This includes for example the case where both graphs have maximum degree 5.
['Thomas Bläsius', 'Ignaz Rutter']
I am writing a research paper on a idea Simultaneous PQ-ordering with applications to constrained embedding problems
[{"from": "human", "value": "I am writing a research paper on a idea Simultaneous PQ-ordering with applications to constrained embedding problems"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: ,  define and study the new problem Simultaneous PQ-Ordering. Its input consists of a set of PQ-trees, which represent sets of circular orders of their leaves, together with a set of child-parent relations beten these PQ-trees, such that the leaves of the child form a subset of the leaves of the parent. Simultaneous PQ-Ordering asks whether orders of the leaves of each of the trees can be chosen simultaneously, that is, for every child-parent relation the order chosen for the parent is an extension of the order chosen for the child.  show that Simultaneous PQ-Ordering is NP-complete in general and  identify a family of instances that can be solved efficiently, the 2-fixed instances.  show that this result serves as a framework for several other problems that can be formulated as instances of Simultaneous PQ-Ordering.#R##N##R##N#In particular,  give linear-time algorithms for recognizing simultaneous interval graphs and extending partial interval representations. Moreover,  obtain a linear-time algorithm for Partially PQ-Constrained Planarity for biconnected graphs, which asks for a planar embedding in the presence of PQ-trees that restrict the possible orderings of edges around vertices, and a quadratic-time algorithm for Simultaneous Embedding with Fixed Edges for biconnected graphs with a connected intersection. Both results can be extended to the case where the input graphs are not necessarily biconnected but have the property that each cutvertex is contained in at most two non-trivial blocks. This includes for example the case where both graphs have maximum degree 5."}]