GRAPH THEORY and APPLICATIONS
▫ Tree: a connected graph with no cycle (acyclic). ▫ Forest: a graph with no cycle. ▫ Paths are trees. ▫ Star: A tree consisting of one vertex adjacent to.
Chapter 10.1 Trees
Trees. Prof. Tesler. Math 184A. Winter 2017. Prof. Tesler. Ch. 10.1: Trees. Math 184A / Winter 2017. 1 / 15. Page 2. Trees. Tree in graph theory. Stick figure
Module 8: Trees and Graphs - Theme 1
Theorem 1. A tree with n nodes has n -1 edges. Proof. Every node except the root has exactly one in-coming edge. Since there
4. Trees
The following result shows the existence of spanning trees in connected graphs. Theorem 4.12 Every connected graph has at least one spanning tree. Proof Let G
GRAPHS AND TREES
Thus each edge of a directed graph can be drawn as an arrow going from the first vertex to the second vertex of the ordered pair. Graphs: Definitions and Basic
An Introduction to Combinatorics and Graph Theory
available in this pdf file. . w1 . w2 . w3 . w4 . w5 . w6 . w7 . v1 . v2 In general spanning trees are not unique
Introduction to Graph Theory
1.1s. Write down the number of vertices the number of edges
1.10. Trees and Spanning Trees. 1.10.1. Definition: Tree. ∗ ∗ ∗
3. Page 6. WUCT121. Graphs. 54. When the graph has a large number of vertices it is not easy to find all spanning trees. In fact
An introduction to chordal graphs and clique trees
Clique trees and chordal graphs have carved out a niche for themselves in recent work on sparse matrix algorithms due primarily to research questions
3.1 Characterizations and Properties of Trees 3.2 Rooted Trees
D. Page 7. GRAPH THEORY – LECTURE 4: TREES. 7. Lemma 1.10. Let v and w be two vertices in a tree T such that w is of maximum distance from v (i.e. ecc(v) =
Chapter 10.1 Trees
Trees. Tree in graph theory. Stick figure tree. Not a tree. (has cycle). Not a tree. (not connected). A tree is an undirected connected graph with no cycles
GRAPH THEORY and APPLICATIONS
? Tree: a connected graph with no cycle (acyclic). ? Forest: a graph with no cycle. ? Paths are trees. ? Star: A tree consisting of one vertex adjacent to.
GRAPHS AND TREES
Note that each directed graph has an associated ordinary. (undirected) graph which is obtained by ignoring the directions of the edges. Graphs: Definitions and
3.1 Characterizations and Properties of Trees 3.2 Rooted Trees
GRAPH THEORY – LECTURE 4: TREES. Abstract. Review from §1.5 tree = connected graph with no cycles. Def 1.1. In an undirected tree a leaf is a vertex of ...
Introduction to Graph Theory
examples of graphs connectedness
An Introduction to Combinatorics and Graph Theory
Graph theory is concerned with various types of networks new tree by introducing a new root vertex and making the children of this root the two.
GRAPH THEORY WITH APPLICATIONS
12.2 The Number of Spanning Trees. Applications. 12.3 Perfect Squares . Appendix I Hints to Starred Exercises. Appendix II Four Graphs and a Table of their
Partitions of Graphs into Trees
Maximal planar bipartite graphs have a 2-tree partition as shown by Ringel [14]. Here we give a different proof of this result with a linear time algorithm.
Chapter 6: Graph Theory
Rather than finding a minimum spanning tree that visits every vertex of a graph an Euler path or circuit can be used to find a way to visit every edge of a
Graphs and Trees
Lots of terminology surrounding graphs tons of types
Graph Theory III - MIT - Massachusetts Institute of Technology
Grow a tree one edge at a time by adding the minimum weight edge of the graph tothe tree making sure that you have a tree at each step ALG2: Select edges one at a time always choosing the minimum weight edge that does notcreate a cycle with previously selected edges
Graph Theory: Trees - IIT Kgp
GRAPH THEORY { LECTURE 4: TREES Abstract x3 1 presents some standard characterizations and properties of trees x3 2 presents several di erent types of trees x3 7 develops a counting method based on a bijection between labeled trees and numeric strings x3 8 showns how binary trees can be counted by the Catalan recursion Outline
Introduction to graph theory - University of Oxford
Trees and forests A tree (a connected acyclic graph) A forest (a graph with tree components) ©Department of Psychology University of Melbourne Bipartite graphs A bipartite graph (vertex set can be partitioned into 2 subsets and there are no edges linking vertices in the same set) A complete bipartite graph (all possible edges are present) K1
Graph Theory I - Properties of Trees
3 Trees Definition 4Given a graph G • A path in G is a sequence of edges such that each edge begins where the previous edge ends and ends where the next edge begins • A cycle in G is a path starting and ending at the same vertex G is called a tree if it contains no cycles
Lecture 12: Introduction to Graphs and Trees
Binary search tree (BST) - a tree where nodes are organized in a sorted order to make it easier to search At every node you are guaranteed: All nodes rooted at the le† child are smaller than the current node value All nodes rooted at the right child are smaller than the current node value
Searches related to trees in graph theory pdf PDF
GRAPH THEORY { LECTURE 5: SPANNING TREES 3 Choosing a Frontier Edge Def 1 3 Let T be a tree subgraph of a graph G and let S be the set of frontier edges for T The function nextEdge(GS) (usually deterministic) chooses and returns as its value the frontier edge in S that is to be added to tree T Def 1 4
What is the difference between a tree and a spanning tree?
Trees and Spanning Trees •A graph having no cycles is acyclic. •A forest is an acyclic graph. •A leaf is a vertex of degree 1. •A spanning sub-graph of G is a sub-graph with vertex set V(G). •A spanning tree is a spanning sub-graph that is a tree.
What is a graph in physics?
Definition 1AgraphGis a setV(G)of points (called vertices) together with a setE(G)of edges connectingthe vertices. Though graphs are abstract objects, they are very naturally represented by diagrams, where we (usually)draw the vertices and edges in the plane.
What is a tree acyclic structure of linked nodes?
Tree- a directed, acyclic structure of linked nodesNode- an object containing a data value and links to other nodes All the blue circles Tree- a directed, acyclic structure of linked nodesEdge- directed link, representing relationships between nodes All the grey lines
How do you implement a graph search algorithm?
Goal: implement the basic graph search algorithms in timeO(m+n). This is linear time, since it takesO(m+n)time simply to read the input. Note that when we work with connected graphs, a running time ofO(m+n)is the same asO(m), sincemn 1. Breadth First Search (BFS)Depth First Search (DFS) Example… Start at the start. Look at all the neighbors.
GRAPH THEORY { LECTURE 5:
SPANNING TREES
Abstract.Several dierent problem-solving algorithms involve growing a spanning tree, one edge andone vertex at a time. All these techniques are renements and extensions of the same basic tree-growing
scheme given inx4.1.x4.2 presents depth-rst and breadth-rst search.x4.3 introduces two algorithmic computations, nding a minimum-weight spanning tree and nding a shortest path.Outline
4.1 Tree Growing
4.2 Depth-First and Breadth-First Search
4.3 Minimum Spanning Trees and Shortest Paths
12 GRAPH THEORY { LECTURE 5: SPANNING TREES
1.Tree-Growing
Def 1.1.For a given treeTin a graphG, the edges and vertices ofTare calledtree edgesandtree vertices, and the edges and vertices ofGthat are not inTare callednon-tree edgesandnon-tree vertices. Def 1.2.Afrontier edgefor a given treeTin a graph is a non-tree edge with one endpoint inT, called itstree endpoint, and one endpoint not inT, itsnon-tree endpoint.a
b c dFigure 1.1.A tree with frontier edgesa,b,c, andd. Proposition 1.1.LetTbe a tree in a graphG, and letebe a frontier edge forT. Then the subgraph ofGformed by adding edgeeto treeT is a tree.GRAPH THEORY { LECTURE 5: SPANNING TREES 3
Choosing a Frontier Edge
Def 1.3.LetTbe a tree subgraph of a graphG, and letSbe the set of frontier edges forT. The functionnextEdge(G,S)(usually deterministic) chooses and returns as its value the frontier edge inSthat is to be added to treeT. Def 1.4.After a frontier edge is added to the current tree, the procedure updateFrontier(G,S)removes fromSthose edges that are no longer frontier edges and adds toSthose that have become frontier edges. (See Fig1.2 below.)
Table 1.1.The generic tree-growing algorithm.Algorithm: Tree-Growing(G;v) Input: a connected graphG, a starting vertexv2VG, and a selection-functionnextEdge.Output: an ordered spanning treeTofGwith rootv.
Initialize treeTas vertexv.
InitializeSas the set of proper edges incident onv.WhileS6=;
Lete=nextEdge(G;S).
Letwbe the non-tree endpoint of edgee.
Add edgeeand vertexwto treeT.
updateFrontier(G;S).Return treeT.
4 GRAPH THEORY { LECTURE 5: SPANNING TREESa
b c d a b c dnextEdge(S)=c add edge c to the tree and apply update(S)Figure 1.2.Result after adding edgecto the tree; note deletion of edgedfrom the
frontier.GRAPH THEORY { LECTURE 5: SPANNING TREES 5
Discovery Order of the Vertices
Def 1.5.Thediscovery orderis a listing of the vertices of graphGin the order in which they are added (discovered) as spanning treeTis grown. Proposition 1.2.The output treeTproduced by a Tree-Growing is an ordered tree w.r.t. the discovery order of its vertices.v a b d e c f g i j h k m v h k a b d c f v a b d e c f g i j h k m v g k m j h e d nextEdge: lexicographic nextEdge: reverse lexicographic 0 1 2 3 4 5 6 7 0 1 2 3 456 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6
7Figure 1.3.Output from two instances of Tree-Growing.
Example 1.1.Both output trees of Fig 1.3 start atv. The right-gs are the left-gs redrawn to display the spanning trees as ordered trees. Notice that the left-to-right order of the children of each vertex is consistent with the discovery order, as asserted by Prop 1.2.6 GRAPH THEORY { LECTURE 5: SPANNING TREES
Two Kinds of Non-Tree Edges
The non-tree edges that appear in both output trees in Figure 1.3 above, or any other output tree, fall into two categories. Def 1.6.Two vertices in a rooted tree arerelatedif one is a descendant of the other. Def 1.7.For a given output tree grown by Tree-Growing, askip-edgeis a non-tree edge whose endpoints are related; across-edgeis a non-tree edge whose endpoints are not related.Example 1.1, continued:For the output
tree at the right, which is reproduced from the top half of Figure 1.3, there are three skip-edges (04, 05, 26) and two cross-edges (56, 67). For the other output tree, there are four skip-edges and one cross-edge.v h k a b d c f 0 1 2 3 4 5 6 7GRAPH THEORY { LECTURE 5: SPANNING TREES 7
DFS and BFS as Tree-Growing
Preview ofx4.2: Thedepth-rstandbreadth-rstsearches use opposite versions ofnextEdge. depth-rst:nextEdgeselects a frontier edge whose tree endpoint was most recently discovered. breadth-rst:nextEdgeselects a frontier edge whose tree endpoint was discovered earliest (least recently).Resolving Ties for Next-Frontier-Edge
Typically, ties are resolved by somedefault prioritythat is likely to be part of the implementation of the data structures involved. We will often rely on the somewhat articial lexicographic (alphabetical) order of the edge names and/or vertex names to resolve ties in choosing the next frontier edge.8 GRAPH THEORY { LECTURE 5: SPANNING TREES
Tree-Growing in a Non-Connected Graph
Review fromx2.3Thecomponent of a vertexvin a graphG, denotedCG(v), is the subgraph ofGinduced on the set of vertices that are reachable fromv. Proposition 1.3.Let treeTbe the output of Tree-Growing (Algorithm1.1) on a graphG(not necessarily connected), starting at a vertex
v2VG. Then treeTspans the componentCG(v). Proof.The result follows by induction, using Prop 1.1. big Corollary 1.4.A graphGis connected i the output tree produced fromTree-Growing inGis a spanning tree ofG.
GRAPH THEORY { LECTURE 5: SPANNING TREES 9
Tree-Growing in a Digraph
Def 1.8.Afrontier arcfor a rooted treeTin a digraph is an arc whose tail is inTand whose head is not inT. In contrast with undirected graphs, the number of vertices in the output tree for a digraph depends on the choice of a starting vertex, as the next example shows. Example 1.4.In Fig 1.4, the number of vertices in the output tree ranges between 1 and 5, depending on the starting vertex.u x yz vFigure 1.4.Output tree depends on the starting vertex.10 GRAPH THEORY { LECTURE 5: SPANNING TREES
Def 1.9.For tree-growing in digraphs, the non-tree edges (arcs) fall into three categories: Aback-arcis directed from a vertex to one of its ancestors. Aforward-arcis directed from a vertex to one of its descendants. Across-arcis directed from a vertex to another vertex that is unre- lated. Def 1.10.There are two kinds of cross-arcs: aleft-to-right cross-arc is directed from smaller discovery number to larger one; aright-to-left cross-arcis the opposite.GRAPH THEORY { LECTURE 5: SPANNING TREES 11
Forest-Growing
Def 1.11.Afull spanning forestof a graphGis a spanning forest consisting of a collection of trees, such that each tree is a spanning tree of a dierent component ofG. Table 1.2.Forest-growing and component-counting.Algorithm: Forest-Growingquotesdbs_dbs17.pdfusesText_23[PDF] tremolo harmonica lessons for beginners pdf
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