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CSE 211 Discrete Mathematics Chapter 8.4 Connectivity

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Paths in Undirected Graphs There is a path from vertex v 0 to vertex v n if there is a sequence of edges from v 0 to v n –This path is labeled as v 0,v 1,v 2,…,v n and has a length of n. The path is a circuit if the path begins and ends with the same vertex. A path is simple if it does not contain the same edge more than once.

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Graph Theoretic Foundations Paths and Cycles: Walk in a graph G is v 0, e 1, v 1, …, v l-1, e l, v l

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Graph Theoretic Foundations Paths and Cycles: IF v 0, v 1, …, v l are distinct (except possible v 0,v l ) then the walk is a path. Denoted by v 0, v 1, …, v l or e 1, e 2, …, e l Length of path is l

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Paths, Cycles, and Trails A trail is a walk with no repeated edge.

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Graph Theoretic Foundations Paths and Cycles: A path or walk is closed if v 0 = v l A closed path containing at least one edge is called a cycle.

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Paths in Undirected Graphs A path or circuit is said to pass through the vertices v 0, v 1, v 2, …, v n or traverse the edges e 1, e 2, …, e n.

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Example u 1, u 4, u 2, u 3 –Is it simple? –yes –What is the length? –3–3 –Does it have any circuits? –no u 1 u 2 u 5 u 4 u 3

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Example u 1, u 5, u 4, u 1, u 2, u 3 –Is it simple? –yes –What is the length? –5–5 –Does it have any circuits? –Yes; u 1, u 5, u 4, u 1 u 1 u 2 u 5 u 3 u 4

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Example u 1, u 2, u 5, u 4, u 3 –Is it simple? –yes –What is the length? –4–4 –Does it have any circuits? –no u 1 u 2 u 5 u 3 u 4

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Connectedness An undirected graph is called connected if there is a path between every pair of distinct vertices of the graph. There is a simple path between every pair of distinct vertices of a connected undirected graph.

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Example Are the following graphs connected? c e f a d b g e c a f b d YesNo

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Connectedness (Cont.) A graph that is not connected is the union of two or more disjoint connected subgraphs (called the connected components of the graph).

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Example What are the connected components of the following graph? b a c d e h g f

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Example What are the connected components of the following graph? b a c d e h g f {a, b, c}, {d, e}, {f, g, h}

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Cut edges and vertices If one can remove a vertex (and all incident edges) and produce a graph with more connected components, the vertex is called a cut vertex. If removal of an edge creates more connected components the edge is called a cut edge or bridge.

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Example Find the cut vertices and cut edges in the following graph. a b c d f e h g

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Example Find the cut vertices and cut edges in the following graph. a b c d f e h g Cut vertices: c and e Cut edge: (c, e)

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Graph Theoretic Foundations Connectivity: Connectivity (G) of graph G is … G is k connected if (G) k Separator or vertex-cut Cut vertex Separation pair

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Graph Theoretic Foundations Trees and Forests: Tree – connected graph without any cycle Forest – a graph without any cycle

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Connectedness in Directed Graphs A directed graph is strongly connected if there is a directed path between every pair of vertices. A directed graph is weakly connected if there is a path between every pair of vertices in the underlying undirected graph.

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Example Is the following graph strongly connected? Is it weakly connected? a b c e d This graph is strongly connected. Why? Because there is a directed path between every pair of vertices. If a directed graph is strongly connected, then it must also be weakly connected.

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Example Is the following graph strongly connected? Is it weakly connected? a b c e d This graph is not strongly connected. Why not? Because there is no directed path between a and b, a and e, etc. However, it is weakly connected. (Imagine this graph as an undirected graph.)

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Connectedness in Directed Graphs The subgraphs of a directed graph G that are strongly connected but not contained in larger strongly connected subgraphs (the maximal strongly connected subgraphs) are called the strongly connected components or strong components of G.

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Example What are the strongly connected components of the following graph? a b c e d This graph has three strongly connected components: The vertex a The vertex e The graph consisting of V = {b, c, d} and E = { (b, c), (c, d), (d, b)}

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Consider a connected graph G. The distance between vertices u and v in G, written d(u, v), is the length of the shortest path between u and v. The diameter of G, written diam(G), is the maximum distance between any two points in G. For example, in Fig. 1-8(a), d(A,F) = 2 and diam(G) = 3, whereas in Fig. 1-8(b), d(A, F) = 3 and diam(G) = 4 Distance in Trees and Graphs

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If G has a u, v-path, then the distance from u to v, written d G (u, v) or simply d(u, v), is the least length of u, v-path. The diameter (diam G) is max u,v V(G) d(u,v)

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Distance in Trees and Graphs Find the diameter, eccentricity, radius and center of the given G.

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CSE 211 Discrete Mathematics and Its Applications Chapter 8.5 Euler and Hamilton Paths

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Euler Paths and Circuits The Seven bridges of Königsberg a b c d A B C D

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Euler Paths and Circuits An Euler path is a path using every edge of the graph G exactly once. An Euler circuit is an Euler path that returns to its start. A B C D Does this graph have an Euler circuit? No.

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Necessary and Sufficient Conditions How about multigraphs? A connected multigraph has a Euler circuit iff each of its vertices has an even degree. A connected multigraph has a Euler path but not an Euler circuit iff it has exactly two vertices of odd degree.

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Example Which of the following graphs has an Euler circuit? e d a c b e d a c b e c a d b yes no no (a, e, c, d, e, b, a)

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Example Which of the following graphs has an Euler path? e d a c b e d a c b e c a d b yes noyes (a, e, c, d, e, b, a ) (a, c, d, e, b, d, a, b)

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Euler Circuit in Directed Graphs NO(a, g, c, b, g, e, d, f, a)NO

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Euler Path in Directed Graphs NO (a, g, c, b, g, e, d, f, a) (c, a, b, c, d, b)

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Hamilton Paths and Circuits A Hamilton path in a graph G is a path which visits every vertex in G exactly once. A Hamilton circuit is a Hamilton path that returns to its start.

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Hamilton Circuits Is there a circuit in this graph that passes through each vertex exactly once?

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Hamilton Circuits Yes; this is a circuit that passes through each vertex exactly once.

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Finding Hamilton Circuits Which of these three figures has a Hamilton circuit? Of, if no Hamilton circuit, a Hamilton path?

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Finding Hamilton Circuits G 1 has a Hamilton circuit: a, b, c, d, e, a G 2 does not have a Hamilton circuit, but does have a Hamilton path: a, b, c, d G 3 has neither.

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Finding Hamilton Circuits Unlike the Euler circuit problem, finding Hamilton circuits is hard. There is no simple set of necessary and sufficient conditions, and no simple algorithm.

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Properties to look for... No vertex of degree 1 If a node has degree 2, then both edges incident to it must be in any Hamilton circuit. No smaller circuits contained in any Hamilton circuit (the start/endpoint of any smaller circuit would have to be visited twice).

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A Sufficient Condition Let G be a connected simple graph with n vertices with n 3. G has a Hamilton circuit if the degree of each vertex is n/2.

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Travelling Salesman Problem A Hamilton circuit or path may be used to solve practical problems that require visiting vertices, such as: road intersections pipeline crossings communication network nodes A classic example is the Travelling Salesman Problem – finding a Hamilton circuit in a complete graph such that the total weight of its edges is minimal.

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Summary PropertyEulerHamilton Repeated visits to a given node allowed? YesNo Repeated traversals of a given edge allowed? No Omitted nodes allowed?No Omitted edges allowed?NoYes

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