When we consider an embedding of graphs into a plane, it would be nice if it does not intersect internally since the embedding simply shows us the structure of graphs. It is easy to embed a tree into a plane with non-self-intersections. If we embed two or more trees into a plane with non-self-intersections, what occurs? In this paper, we consider embeddings of two trees into a plane with using ...

According to the Fibonacci number which is studied by Prodinger et al., we introduce the 2-plane tree which is a planted plane tree with each of its vertices colored with one of two colors and qqppppppppppppppppp -free. The similarity of the enumeration between 2-plane trees and ternary trees leads us to build several bijections. Especially, we found a bijection between the set of 2-plane trees...

We prove new structural properties for tree-decompositions of planar graphs that we use to improve upon the runtime of tree-decomposition based dynamic programming approaches for several NP-hard planar graph problems. We give for example the fastest algorithm for Planar Dominating Set of runtime 3 · n, when we take the treewidth tw as the measure for the exponential worst case behavior. We also...

A heap ordered tree with n nodes (“size n”) is a planted plane tree together with a bijection from the nodes to the set {1, . . . , n} which is monotonically increasing when going from the root to the leaves. We consider the number of descendants of the node j in a (random) heap ordered tree of size n ≥ j. Precise expressions are derived for the probability distribution and all (factorial) mome...

Westudy some structural properties for tree-decompositions of 2-connected planar graphs that we use to improve upon the runtime of tree-decomposition based dynamic programming approaches for several NP-hard planar graph problems. E.g., we derive the fastest algorithm for Planar Dominating Set of runtime 3tw · nO(1), when we take the width tw of a given tree-decomposition as the measure for the ...

In this paper, a tree-shaped power distribution network is designed based on constructal theory for planar EBG structure power plane on PCB, in order to optimize DC performance. Planar EBG structures suppress noise, and the network provides currents to them. This network is composed of hierarchical metal paths. The geometric parameters can be optimized based on the concept of constructal theory...

A Halin graph is a plane graph constructed from a planar drawing of a tree by connecting all leaves of the tree with a cycle which passes around the boundary of the graph. The tree must have four or more vertices and no vertices of degree two. Halin graphs have many nice properties such as being Hamiltonian and remain Hamiltonian after any single vertex deletion. In 1975, Lovász and Plummer con...

In this paper, we give a finite disjunctive programming procedure to obtain the convex hull of general mixed-integer linear programs (MILP) with bounded integer variables. We propose a finitely convergent convex hull tree algorithm which constructs a linear program that has the same optimal solution as the associated MILP. In addition, we combine the standard notion of sequential cutting planes...

A straight-line drawing of a plane graph G is a planar drawing of G, where each vertex is drawn as a point and each edge is drawn as a straight line segment. Given a set S of n points in the Euclidean plane, a point-set embedding of a plane graph G with n vertices on S is a straight-line drawing of G, where each vertex of G is mapped to a distinct point of S. The problem of deciding if G admits...