# Automata Computability and Formal Languages

May 1st, 2015
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Connected graph A graph G is connected if given any vertices u and v in G, there is a path from u to v (or v to u). That is, in a connected graph, we can get from any vertex to any other vertex on a path. If there’s no path between some pair of vertices then the graph is called disconnected.

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Automata, Computability and Formal LanguagesCOM273: Automata, Computability, and Formal LanguagesREVIEW NOTESConnected graph A graph G is connected if given any vertices u and v in G, there is a path from u to v (or v to u). That is, in a connected graph, we can get from any vertex to any other vertex on a path. If theres no path between some pair of vertices then the graph is called disconnected. The following shows a connected graph and a disconnected graph. Example: a b c b a ce d connected graphe d disconnected graphCycle A cycle (or a circuit) is a path of non-zero length from u to v with no repeated edges. A simple cycle is cycle from v to v, in which there are no repeated vertices (except for the initial and final vertices which are both equal to v) Example: a b d e c (a, b, d, c, b, e, c, a) is a cycle (but not simple) (a, d, c, b, e, a) is a simple cycle of length 5.Notice that a simple path of length n contains (n+1) distinct vertices, which a simple cycle of length n contains n distinct vertices.Eulerian graphs A cycle in a connected graph G that includes all the edges and all the vertices of G is called an Euler cycle (Euler is pronounced as oiler).A connected graph containing an Euler cycle is called an Eulerian graph.Finding an Euler cycle in a graph is the same as playing the following game. Draw the entire graph without lifting your pen which drawing all the edges exactly once, and visiting all the vertices, startin

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