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CSCI 115

CSCI 115. Chapter 8 Topics in Graph Theory. CSCI 115. §8 .1 Graphs. §8 .1 – Graphs. Graph A graph G consists of a finite set of vertices V, a finite set of edges E, and a function γ that assigns a subset of vertices {v, w} to each edge (v may equal w)

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CSCI 115

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  1. CSCI 115 Chapter 8 Topics in Graph Theory

  2. CSCI 115 §8.1 Graphs

  3. §8.1 – Graphs • Graph • A graph G consists of a finite set of vertices V, a finite set of edges E, and a function γ that assigns a subset of vertices {v, w} to each edge (v may equal w) • If e is an edge, and γ(e) = {v, w} we say e is the edge between v & w, and that v & w are the endpoints of e

  4. §8.1 – Graphs • Terminology • Degree of a vertex • Number of edges having that vertex as an endpoint • Loop • Edge from a vertex to itself • Contributes 2 to the degree of a vertex • Isolated vertex • Vertex with degree 0 • Adjacent vertices • Vertices that share an edge

  5. §8.1 – Graphs • Path • A path  in a graph G consists of a pair of sequences (V, E), V: v1, v2, …, vk and E: e1, e2, …, ek–1 s.t.: • γ(ei) = {vi, vi+1} i • ei ej  i, j

  6. §8.1 – Graphs • More terminology • Circuit • Path that begins and ends at the same vertex • Simple path • No vertex appears more than once (except possibly the first and last) • Simple circuit • Simple path where first and last vertices are equal

  7. §8.1 – Graphs • Special types of graphs • Connected graph •  path from every vertex to every other (different) vertex • Disconnected graph • There are at least 2 vertices which do not have a path between them • components • Regular Graph • All vertices have the same degree

  8. §8.1 – Graphs • Special families of graphs (n  Z+) • Un: The discrete graph on n vertices • The graph with n vertices and no edges • Kn: The complete graph on n vertices • The graph with n vertices, and an edge between every pair of vertices • Ln: The linear graph on n vertices • The graph with n vertices and edges {vi, vi+1}  i  {1, 2, …, n – 1}

  9. §8.1 – Graphs • Subgraphs • If G = (V, E, γ) and E1 E, V1 V s.t. V1 contains (at least) all of the end points of edges in E1, then H = (V1, E1, γ) is a subgraph of G • If G = (V, E, γ) and e  E, then Ge is the subgraph found by deleting e from G and keeping all vertices

  10. §8.1 – Graphs • Quotient graphs • If G = (V, E, γ) and R is an equivalence relation on V, then GR is the quotient graph found by merging all vertices within the same equivalence classes.

  11. CSCI 115 §8.2 Euler Paths and Circuits

  12. §8.2 – Euler Paths and Circuits • A path in a graph G is an Euler path if it includes every edge exactly once • An Euler circuit is an Euler path that is also a circuit

  13. §8.2 – Euler Paths and Circuits • Theorem 8.2.1 • If G has a vertex of odd degree, there can be no Euler circuit of G • If G is a connected graph and every vertex has even degree then there is an Euler circuit in G • Theorem 8.2.2 • If a graph G has more than 2 vertices of odd degree, there can be no Euler path in G • If G is connected and has exactly 2 vertices of odd degree, then there exists an Euler path in G. Any Euler path must begin at one vertex of odd degree, and end at the other.

  14. §8.2 – Euler Paths and Circuits • Theorem 8.2.1 and 8.2.2 • Existence theorems • Bridge • A bridge is an edge in a connected graph that if removed would result in a disconnected graph

  15. §8.2 – Euler Paths and Circuits • Fleury’s Algorithm for finding an Euler circuit for a connected graph where every vertex has even degree (let G = V, E, γ) • Select and edge e1 of G with vertices (v1, v2) that is not a bridge. Let  be (V: v1, v2, E: e1). Remove e1 from E and v1 and v2 from V to create G1. • Suppose V: v1, v2, …, vk and E: e1, e2, …, ek–1 have been constructed to form Gk–1. Select ek in Gk–1 that has vk as a vertex and is not a bridge in Gk–1 (discounting vk). Extend V to v1, v2, …, vk, vk+1 and E to e1, e2, …, ek–1, ek. • Repeat step 2 until no edges remain in E.

  16. CSCI 115 §8.3 Hamiltonian Paths and Circuits

  17. §8.3 – Hamiltonian Paths and Circuits • A Hamiltonian path is a path that contains each vertex exactly once • A Hamiltonian circuit is a Hamiltonian path that is also a circuit

  18. §8.3 – Hamiltonian Paths and Circuits • Theorem 8.3.1 • Let G be a connected graph with n vertices (n  Z, n  2), with no loops or multiple edges. G has a Hamiltonian circuit if for any 2 vertices u and v of G that are not adjacent, the degree of u plus the degree of v is  n. • Corollary 8.3.1 • G has a Hamiltonian circuit if each vertex has degree  (n/2).

  19. §8.3 – Hamiltonian Paths and Circuits • Theorem 8.3.2 • Let the number of edges of G be m. Then G has a Hamiltonian circuit if m  ½(n2 – 3n + 6) where n is the number of vertices.

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