Graph Terminology, Section 5.1

(definitions pilfered -- with slight modifications, perhaps -- from www.cs.usask.ca/resources/tutorials/csconcepts/graphs/tutorial/definitions.html -- now a dead link).

• graph: consists of a set of nodes (or vertices), a set of arcs (or edges), and a function that associates a pair of vertices with each edge

• vertex: a point or node in a graph
• adjacent vertices: two distinct vertices are said be adjacent if there is an edge connecting them
• isolated vertex: a vertex of degree zero (no edges going in or out of it)
• degree of a vertex: the number of edges incident to a given vertex
• reachability: describes a relationship between two vertices in which one vertex is reachable from the other via a path

• edge (or arc): a connection made between two vertices in a graph
• parallel: where there is more than a single edge connecting two given vertices
• loop (or sling): where an edge joins a vertex to itself

• path: a sequence of edges that begins at an initial vertex and ends at a terminal vertex
• length: the number of edges appearing in the sequence of a path
• cycle (or circuit): a path that begins and terminates in the same vertex

• digraph (or directed graph): a graph in which every edge is directed
• labeled graph: a graph in which each vertex is labeled
• weight: a number assigned to an edge
• weighted graph: a connected graph in which a positive real number has been assigned to each edge
• loop-free graph: a graph which is free of loops (well what did you expect?!)
• simple graph: a graph that contains no loops or parallel edges
• complete graph: a simple graph in which every pair of distinct vertices are adjacent
• connected graph: a graph in which for any given vertex in the graph, all the other vertices are reachable from it
• acyclic graph: a graph that contains no cycles
• subgraph: if every edge of graph A is also an edge of graph B, then graph A is a subgraph of graph B

• adjacent edges: two distinct edges are said be adjacent if they have at least one vertex in common
• distance: the length of a minimum path length from one specific vertex to another vertex.
• elementary path: a path in which all the vertices are distinct
• elementary cycle: a cycle through a graph that does not visit any vertex more than once
• incident: an edge that joins two specified vertices is said to be incident to them
• indegree: the number of edges terminating in a given node in a directed graph
• minimum path length: the path of minimum length between two vertices that are reachable from one another
• mixed graph: a graph that consists of both directed and undirected edges
• multigraph: a graph with multiple edges between the same vertices
• null graph (or totally-disconnected graph): a graph that contains only isolated vertices (a graph with no edges)
• outdegree: the number of edges originating from a given vertex in a directed graph
• simple path: a path in a directed graph in which all the edges are distinct
• strongly-connected: describes the connectedness of a simple directed graph in which, for any two vertices, both vertices are reachable from the other
• terminal vertex (or endpoint): a vertex of degree one
• total degree: the sum of the indegree and outdegree of a vertex
• transport network: a connected, directed graph (with no self loops) that has only one vertex of indegree zero (the source), and one vertex of outdegree zero (the sink). Each edge is associated to a number which represents the limit to the rate of transportation of the product (called the capacity).
• tree: a connected graph in which there is only one path connecting each pair of vertices
• undirected graph: a graph in which every edge is undirected
• unilaterally-connected: describes the connectedness of a simple directed graph in which, for any two vertices, at least one vertex is reachable from the other of the pair
• weakly-connected: describes the connectedness of a simple directed graph in which the direction of the edges is disregarded