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Polytree from Wikipedia, the Free Encyclopedia Contents Polytree From Wikipedia, the free encyclopedia Contents 1 Connectivity (graph theory) 1 1.1 Connected graph ............................................ 2 1.2 Definitions of components, cuts and connectivity ........................... 2 1.3 Menger’s theorem ........................................... 3 1.4 Computational aspects ......................................... 4 1.5 Examples ............................................... 4 1.6 Bounds on connectivity ........................................ 4 1.7 Other properties ............................................ 4 1.8 See also ................................................ 5 1.9 References ............................................... 5 2 Cycle (graph theory) 6 2.1 Chordless cycles ............................................ 7 2.2 Cycle space .............................................. 7 2.3 Cycle detection ............................................ 7 2.4 Covering graphs by cycles ....................................... 7 2.5 Graph classes defined by cycles .................................... 8 2.6 References ............................................... 8 3 Directed acyclic graph 9 3.1 Mathematical properties ....................................... 10 3.1.1 Reachability, transitive closure, and transitive reduction ................... 10 3.1.2 Topological ordering ..................................... 11 3.1.3 Combinatorial enumeration ................................. 11 3.1.4 Related families of graphs .................................. 11 3.2 Computational problems ....................................... 11 3.2.1 Topological sorting and recognition ............................. 11 3.2.2 Construction from cyclic graphs ............................... 12 3.2.3 Transitive closure and transitive reduction .......................... 12 3.2.4 Closure problem ....................................... 12 3.3 Applications .............................................. 12 3.3.1 Path algorithms ........................................ 12 3.3.2 Scheduling .......................................... 12 i ii CONTENTS 3.3.3 Data processing networks .................................. 13 3.3.4 Causal structures ....................................... 13 3.3.5 Genealogy and version history ................................ 13 3.3.6 Data compression ...................................... 14 3.4 References .............................................. 14 3.5 External links ............................................. 16 4 Directed graph 17 4.1 Basic terminology ........................................... 18 4.2 Indegree and outdegree ........................................ 18 4.3 Degree sequence ............................................ 19 4.4 Digraph connectivity .......................................... 19 4.5 Classes of digraphs .......................................... 20 4.6 See also ................................................ 21 4.7 Notes ................................................. 22 4.8 References ............................................... 22 5 Orientation (graph theory) 23 5.1 Oriented graphs ............................................ 23 5.2 Constrained orientations ........................................ 23 5.3 References ............................................... 24 5.4 External links ............................................. 24 6 Polytree 25 6.1 Related structures ........................................... 26 6.2 Enumeration .............................................. 26 6.3 Sumner’s conjecture .......................................... 26 6.4 Applications .............................................. 26 6.5 See also ................................................ 26 6.6 Notes ................................................. 26 6.7 References ............................................... 27 7 Tournament (graph theory) 28 7.1 Paths and cycles ............................................ 28 7.2 Transitivity .............................................. 28 7.2.1 Equivalent conditions ..................................... 28 7.2.2 Ramsey theory ........................................ 29 7.2.3 Paradoxical tournaments ................................... 30 7.2.4 Condensation ......................................... 30 7.3 Score sequences and score sets .................................... 30 7.4 See also ................................................ 31 7.5 Notes ................................................. 31 7.6 References .............................................. 31 CONTENTS iii 8 Tree (graph theory) 33 8.1 Definitions .............................................. 33 8.1.1 Plane tree ........................................... 34 8.2 Example ............................................... 34 8.3 Facts ................................................. 34 8.4 Enumeration ............................................. 35 8.4.1 Labeled trees ......................................... 35 8.4.2 Unlabeled trees ........................................ 35 8.5 Types of trees ............................................. 36 8.6 See also ................................................ 36 8.7 Notes ................................................. 36 8.8 References ............................................... 37 8.9 Further reading ............................................ 37 8.10 Text and image sources, contributors, and licenses .......................... 38 8.10.1 Text .............................................. 38 8.10.2 Images ............................................ 38 8.10.3 Content license ........................................ 39 Chapter 1 Connectivity (graph theory) This graph becomes disconnected when the right-most node in the gray area on the left is removed In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the 1 2 CHAPTER 1. CONNECTIVITY (GRAPH THEORY) This graph becomes disconnected when the dashed edge is removed. minimum number of elements (nodes or edges) that need to be removed to disconnect the remaining nodes from each other.[1] It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its robustness as a network. 1.1 Connected graph A graph is connected when there is a path between every pair of vertices. In a connected graph, there are no unreachable vertices. A graph that is not connected is disconnected. A graph with just one vertex is connected. An edgeless graph with two or more vertices is disconnected. 1.2 Definitions of components, cuts and connectivity In an undirected graph G, two vertices u and v are called connected if G contains a path from u to v. Otherwise, they are called disconnected. If the two vertices are additionally connected by a path of length 1, i.e. by a single edge, the vertices are called adjacent.A graph is said to be connected if every pair of vertices in the graph is connected. A connected component is a maximal connected subgraph of G. Each vertex belongs to exactly one connected com- ponent, as does each edge. A directed graph is called weakly connected if replacing all of its directed edges with undirected edges produces a connected (undirected) graph. It is connected if it contains a directed path from u to v or a directed path from v to u for every pair of vertices u, v. It is strongly connected or strong if it contains a directed path from u to v and a directed path from v to u for every pair of vertices u, v. The strong components are the maximal strongly connected subgraphs. A cut, vertex cut, or separating set of a connected graph G is a set of vertices whose removal renders G disconnected. The connectivity or vertex connectivity κ(G) (where G is not a complete graph) is the size of a minimal vertex cut. A graph is called k-connected or k-vertex-connected if its vertex connectivity is k or greater. More precisely, any graph G (complete or not) is said to be k-connected if it contains at least k+1 vertices, but does 1.3. MENGER’S THEOREM 3 With vertex 0 this graph is disconnected, the rest of the graph is connected. not contain a set of k − 1 vertices whose removal disconnects the graph; and κ(G) is defined as the largest k such that G is k-connected. In particular, a complete graph with n vertices, denoted K, has no vertex cuts at all, but κ(Kn) = n − 1. A vertex cut for two vertices u and v is a set of vertices whose removal from the graph disconnects u and v. The local connectivity κ(u, v) is the size of a smallest vertex cut separating u and v. Local connectivity is symmetric for undirected graphs; that is, κ(u, v) = κ(v, u). Moreover, except for complete graphs, κ(G) equals the minimum of κ(u, v) over all nonadjacent pairs of vertices u, v. 2-connectivity is also called biconnectivity and 3-connectivity is also called triconnectivity. A graph G which is con- nected but not 2-connected is sometimes called separable. Analogous concepts can be defined for edges. In the simple case in which cutting a single, specific edge would disconnect the graph, that edge is called a bridge. More generally, the edge cut of G is a group of edges whose total removal renders the graph disconnected. The edge-connectivity λ(G) is the size of a smallest edge cut, and the local edge-connectivity λ(u, v) of two vertices u, v is the size of a smallest edge cut disconnecting u from v. Again, local edge-connectivity is symmetric. A graph is called k-edge-connected if its edge connectivity is k or greater. 1.3 Menger’s theorem Main article: Menger’s theorem
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