Plane augmentation of plane graphs to meet parity constraints

A plane topological graph G=(V, E) is a graph drawn in the plane whose vertices are points in the plane and whose edges are simple curves that do not intersect, except at their endpoints. Given a plane topological graph G=(V, E) and a set CG of parity constraints, in which every vertex has assigned a parity constraint on its degree, either even or odd, we say that G is topologically augmentable to meet CG if there exists a set E' of new edges, disjoint with E, such that G'=(V, E¿E') is noncrossing and meets all parity constraints. In this paper, we prove that the problem of deciding if a plane topological graph is topologically augmentable to meet parity constraints is NP-complete, even if the set of vertices that must change their parities is V or the set of vertices with odd degree. In particular, deciding if a plane topological graph can be augmented to a Eulerian plane topological graph is NP-complete. Analogous complexity results are obtained, when the augmentation must be done by a plane topological perfect matching between the vertices not meeting their parities. We extend these hardness results to planar graphs, when the augmented graph must be planar, and to plane geometric graphs (plane topological graphs whose edges are straight-line segments). In addition, when it is required that the augmentation is made by a plane geometric perfect matching between the vertices not meeting their parities, we also prove that this augmentation problem is NP-complete for plane geometric paths. For the particular family of maximal outerplane graphs, we characterize maximal outerplane graphs that are topological augmentable to satisfy a set of parity constraints. We also provide a polynomial time algorithm that decides if a maximal outerplane graph is topologically augmentable to meet parity constraints, and if so, produces a set of edges with minimum cardinality

A survey of parameterized algorithms and the complexity of edge modification

The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio

A Polynomial-time Algorithm for Outerplanar Diameter Improvement

The Outerplanar Diameter Improvement problem asks, given a graph $G$ and an integer $D$, whether it is possible to add edges to $G$ in a way that the resulting graph is outerplanar and has diameter at most $D$. We provide a dynamic programming algorithm that solves this problem in polynomial time. Outerplanar Diameter Improvement demonstrates several structural analogues to the celebrated and challenging Planar Diameter Improvement problem, where the resulting graph should, instead, be planar. The complexity status of this latter problem is open.Comment: 24 page

Contact homology and one parameter families of Legendrian knots

We consider S^1-families of Legendrian knots in the standard contact R^3. We define the monodromy of such a loop, which is an automorphism of the Chekanov-Eliashberg contact homology of the starting (and ending) point. We prove this monodromy is a homotopy invariant of the loop. We also establish techniques to address the issue of Reidemeister moves of Lagrangian projections of Legendrian links. As an application, we exhibit a loop of right-handed Legendrian torus knots which is non-contractible in the space Leg(S^1,R^3) of Legendrian knots, although it is contractible in the space Emb(S^1,R^3) of smooth knots. For this result, we also compute the contact homology of what we call the Legendrian closure of a positive braid and construct an augmentation for each such link diagram.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper46.abs.htm

Combinatorial Optimization

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