## Paths, Cycles and Connectivity

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## 1.Bipartite Graphs

## 2.**Definition**: Hamiltonian Cycle

## 3.**Definition**: Eulerian Tour

## 4.**Definition**: Walks, Trails, and Paths

## 5.**Definition**: Closed Walks, Closed Trails, and Cycles

## 6.**Definition**: Connected and Disconnected Graphs, Bridges and Cutvertices

## 7.**Definition**: Weakly and Strongly Connected Digraphs

## 8.**Algorithm**: Get All Components of a Graph

## 9.**Definition**: Connected Vertices

## 10.**Definition**: Biconnected Graphs, \(k\)-Connected Graphs

## 11.**Definition**: Semi-Eulerian Tour, Open Trail

## 12.**Definition**: Semi-Hamiltonian Path

## 13.**Proposition**: Connectivity Is an Equivalence Relation - Components Are a Partition of a Graph

## 14.**Algorithm**: Get the Component Induced by Vertices Connected to a Given Vertex

## 15.**Algorithm**: Get the Cut Vertices and Biconnected Components of a Connected Graph