Good spanning tree

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Conditions of good spanning tree Good spanning tree conditions.svg
Conditions of good spanning tree

In the mathematical field of graph theory, a good spanning tree [1] of an embedded planar graph is a rooted spanning tree of whose non-tree edges satisfy the following conditions.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

Graph theory study of graphs, which are mathematical structures used to model pairwise relations between objects

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices which are connected by edges. A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges, then called arrows, link two vertices asymmetrically; see Graph for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics.

In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.

Contents

Formal definition [1] [2]

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sets of edges GST conditions.svg
An illustration for and sets of edges

Let be a plane graph. Let be a rooted spanning tree of . Let be the path in from the root to a vertex . The path divides the children of , , except , into two groups; the left group and the right group . A child of is in group and denoted by if the edge appears before the edge in clockwise ordering of the edges incident to when the ordering is started from the edge . Similarly, a child of is in the group and denoted by if the edge appears after the edge in clockwise order of the edges incident to when the ordering is started from the edge . The tree is called a good spanning tree [1] of if every vertex of satisfies the following two conditions with respect to .

Applications

In monotone drawing of graphs, [2] in 2-visibility representation of graphs. [1]

Finding good spanning tree

Every planar graph has an embedding such that contains a good spanning tree. A good spanning tree and a suitable embedding can be found from in linear-time. [1] Not all embeddings of contain a good spanning tree.

See also

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References

  1. 1 2 3 4 5 Hossain, Md. Iqbal; Rahman, Md. Saidur (23 November 2015). "Good spanning trees in graph drawing". Theoretical Computer Science. 607: 149–165. doi:10.1016/j.tcs.2015.09.004.
  2. 1 2 Hossain, Md Iqbal; Rahman, Md Saidur (28 June 2014). "Monotone Grid Drawings of Planar Graphs". Frontiers in Algorithmics. Springer, Cham: 105–116. arXiv: 1310.6084 Lock-green.svg. doi:10.1007/978-3-319-08016-1_10.