In the design of certain kinds of electronic circuits, the following question has arisen: given a nonnegative integer k, is an electronic circuit which is treated as a graph with vertices as electronic components and ...In the design of certain kinds of electronic circuits, the following question has arisen: given a nonnegative integer k, is an electronic circuit which is treated as a graph with vertices as electronic components and edges as wires is allowed to print on a展开更多
Let k be a non-negative integer. A graph is said to be a k-bend graph if it is a planar graph in which each edge is represented by at most k+1 horizontal and vertical segments. A graph is called k-embeddable if it has...Let k be a non-negative integer. A graph is said to be a k-bend graph if it is a planar graph in which each edge is represented by at most k+1 horizontal and vertical segments. A graph is called k-embeddable if it has a planar embedding which is a k-bend graph. On the k-embeddability of a graph, [1] provided the characterizations of k-embeddability, k≤3,展开更多
In the design of certain kinds of electronic circuits the following question arises:given a non-negative integer k, what graphs admit of a plane embedding such that every edge is a broken lineformed by horizontal and ...In the design of certain kinds of electronic circuits the following question arises:given a non-negative integer k, what graphs admit of a plane embedding such that every edge is a broken lineformed by horizontal and vertical segments and having at mort k bends? Any such graph is said tobe k--rectilinear. No matter what k is, an obvious necessary condition for k-rectilinearity is that thedegree of each vertex does not exceed four.Our main result is that every planar graph H satisfying this condition is 3--rectilinear:in fact,it is 2--rectilinear with the only exception of the octahedron. We also outline a polynomial-timealgorithm which actually constructs a plane embedding of H with at most 2 bends (3 bends if H isthe octahedron) on each edge. The resulting embedding has the property that the total number ofbends does not exceed 2n, where n is the number of vertices of H.展开更多
文摘In the design of certain kinds of electronic circuits, the following question has arisen: given a nonnegative integer k, is an electronic circuit which is treated as a graph with vertices as electronic components and edges as wires is allowed to print on a
文摘Let k be a non-negative integer. A graph is said to be a k-bend graph if it is a planar graph in which each edge is represented by at most k+1 horizontal and vertical segments. A graph is called k-embeddable if it has a planar embedding which is a k-bend graph. On the k-embeddability of a graph, [1] provided the characterizations of k-embeddability, k≤3,
文摘In the design of certain kinds of electronic circuits the following question arises:given a non-negative integer k, what graphs admit of a plane embedding such that every edge is a broken lineformed by horizontal and vertical segments and having at mort k bends? Any such graph is said tobe k--rectilinear. No matter what k is, an obvious necessary condition for k-rectilinearity is that thedegree of each vertex does not exceed four.Our main result is that every planar graph H satisfying this condition is 3--rectilinear:in fact,it is 2--rectilinear with the only exception of the octahedron. We also outline a polynomial-timealgorithm which actually constructs a plane embedding of H with at most 2 bends (3 bends if H isthe octahedron) on each edge. The resulting embedding has the property that the total number ofbends does not exceed 2n, where n is the number of vertices of H.
