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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 v0 to vertex vn if there is a sequence of edges from v0 to vn This path is labeled as v0,v1,v2,…,vn 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 v0, e1, v1, …, vl-1, el, vl

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**Graph Theoretic Foundations**

Paths and Cycles: IF v0, v1, …, vl are distinct (except possible v0,vl) then the walk is a path. Denoted by v0, v1, …, vl or e1, e2, …, el 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 v0 = vl 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 v0, v1, v2, …, vn or traverse the edges e1, e2, …, en.

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**Example u1, u4, u2, u3 Is it simple? yes What is the length? 3**

Does it have any circuits? no u u2 u5 u u3

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**Example u1, u5, u4, u1, u2, u3 Is it simple? yes What is the length? 5**

Does it have any circuits? Yes; u1, u5, u4, u1 u u2 u u3 u4

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**Example u1, u2, u5, u4, u3 Is it simple? yes What is the length? 4**

Does it have any circuits? no u u2 u u3 u4

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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? Yes No c e f a d b g e c a**

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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 d**

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**Example What are the connected components of the following graph? b a**

{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.**

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? 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. a b c e d

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Example Is the following graph strongly connected? Is it weakly connected? 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.) a b c e d

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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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**Distance in Trees and Graphs**

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

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**Distance in Trees and Graphs**

If G has a u, v-path, then the distance from u to v, written dG(u, v) or simply d(u, v), is the least length of u, v-path. The diameter (diam G) is maxu,vV(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 yes no no (a, e, c, d, e, b, a)**

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 yes no yes (a, e, c, d, e, b, a ) (a, c, d, e, b, d, a, b)**

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 no yes (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**

G1 has a Hamilton circuit: a, b, c, d, e, a G2 does not have a Hamilton circuit, but does have a Hamilton path: a, b, c, d G3 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 Property Euler Hamilton**

Repeated visits to a given node allowed? Yes No Repeated traversals of a given edge allowed? Omitted nodes allowed? Omitted edges allowed?

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