# D1 Algorithms on graphs

Cards needed for D1 AS in Further Mathematics.

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• Created by: Jonathan
• Created on: 15-01-11 20:42

Graph

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A finite number of points (vertices or nodes) connected by lines (edges or arcs).

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Path

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A finite sequence of edges such that the end vertex of one edge is the start vertex of the next.

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Cycle

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A closed path i.e the end vertex of the last edge is the start vertex of the first edge.

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Hamiltonian Cycle

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A cycle that passes through every vertex of the graph once, and only once, and returns to the start vertex.

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Eulerian Cycle

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A cycle that includes every edge of the graph exactly once (all valencies must be even).

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Subgraph

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A subset of vertices together with a subset of the edges

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Connected Graph

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All pairs of vertices on the graph are connected       ( there is a path between each of them).

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Simple Graph

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One in which there are no loops, i.e no edges with the same vertex at the end, and not more than one edge connecting any pair of vertices.

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Digraph

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A graph in which the edges are directed (they have directions associated with them).

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Tree

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A connected graph with no cycles.

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Spanning Tree

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A subgraph of a graph G that includes all the vertices of G and is also a tree

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Complete Graph

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Every vertex is connected by an edge to every other vertex. Kn denotes a complete graph with n vertices.

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Bipartite Graph

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Consists of two sets of vertices, X and Y. The edges only join vertices in X to vertices in Y, not vertices within a set. Kr,s denotes a complete bipartite graph with r vertices in X and s vertices in Y.

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Planar Graph

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No two edges meet one another, except at a vertex to which they are both incident, when the graph is drawn in a plane.

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Isomorphic Graph

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Two graphs, G1 and G2 are isomorphic if they have the same number of vertices, and the degrees of the corresponding vertices are the same.

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Network

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Each edge of the graph has a number (weight) associated with it. A network satisfies the triangle inequality if, for every triangle, no edge's weight exceeds the sum of the weights of the other two edges.

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