"A path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct."
A path is a sequence of edges connecting two vertices. A cycle is a path that starts and ends at the same vertex.
Graphs: A graph is a set of vertices (nodes, points) connected by edges (lines or arcs) that represent relationships or connections between the vertices.
Directed Graphs: In a directed graph, also called a digraph, each edge has a direction associated with it, meaning it goes from one vertex to another.
Simple Graphs: In a simple graph, no two edges connect the same pair of vertices, and there are no loops (edges that connect a vertex to itself).
Paths: A path is a sequence of vertices connected by edges. It can be defined as a series of two or more vertices in which each successive vertex is connected by an edge to the preceding vertex.
Cycles: A cycle is a path that starts and ends at the same vertex, without repeating any vertex (except for the starting vertex).
Connected Graphs: A graph is connected if there is a path between every pair of vertices.
Disconnected Graphs: A graph is said to be disconnected if it has two or more connected components, meaning that there are two or more sets of vertices that are not connected to each other.
Degrees of a Vertex: The degree of a vertex is the number of edges connected to that vertex.
Eulerian Paths and Cycles: In an Eulerian graph, there exists a walk that traverses every edge in the graph exactly once. Such a walk is called an Eulerian tour or cycle.
Hamiltonian Paths and Cycles: In a Hamiltonian graph, there exists a path that visits every vertex in the graph exactly once. Such a path is called a Hamiltonian path or cycle.
Trees: A tree is a special type of graph that is connected and acyclic (has no cycles).
Spanning Trees: A spanning tree of a graph is a tree that spans (includes) all the vertices of the graph.
Shortest Path: The shortest path between two vertices in a graph is the path with the minimum number of edges between them.
Isomorphism: Two graphs are isomorphic if they have the same structure, meaning that they have the same number of vertices, the same number of edges, and the same connectivity between the vertices.
Planar Graphs: A planar graph is a graph that can be drawn on a plane without any edge crossings.
Chromatic Number: The chromatic number of a graph is the smallest number of colors needed to color the vertices of the graph so that no two adjacent vertices have the same color.
Bipartite Graphs: A bipartite graph is a graph whose vertices can be divided into two disjoint sets, such that every edge connects a vertex from one set to a vertex in the other set.
Matching: A matching in a bipartite graph is a set of edges that do not share any vertices.
Network Flow: Network flow is a method of solving optimization problems in graphs, where the goal is to find the maximum flow through a network of interconnected nodes.
Graph Algorithms: There are many different algorithms that can be used to analyze and manipulate graphs, including Dijkstra's algorithm, Kruskal's algorithm, and the Floyd-Warshall algorithm. Each of these algorithms is designed to optimize a particular aspect of graph theory, such as finding the shortest path or finding the minimum spanning tree.
Walk: A sequence of vertices and edges in which no edge is repeated.
Trail: A walk in which no vertex is repeated.
Path: A trail in which no edge is repeated.
Closed walk: A walk that begins and ends at the same vertex.
Cycle: A closed walk in which no vertex, except the first and last, is repeated.
Simple path: A path in which no vertex is repeated.
Simple cycle: A cycle in which no vertex, except the first and last, is repeated.
Hamiltonian path: A path that visits each vertex exactly once.
Hamiltonian cycle: A cycle that visits each vertex exactly once.
Eulerian path: A path that visits every edge in a graph exactly once.
Eulerian cycle: A cycle that visits every edge in a graph exactly once.
Directed path: A path in which edges are directed from one vertex to another.
Directed cycle: A cycle in which edges are directed from one vertex to another.
Strongly connected graph: A graph in which there is a directed path from any vertex to any other vertex.
Weakly connected graph: A graph in which the undirected version is connected.
"The added restriction for a directed path in a directed graph is that the edges be all directed in the same direction."
"By most definitions, the vertices in a path are all distinct, and since the vertices are distinct, so are the edges."
"Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts."
"See e.g. Bondy & Murty (1976), Gibbons (1985), or Diestel (2005)."
"Korte et al. (1990) cover more advanced algorithmic topics concerning paths in graphs."
"A path in a graph can be either a finite or infinite sequence of edges."
"A directed path in a directed graph can be either a finite or infinite sequence of edges."
"No, by most definitions, the vertices in a path are all distinct."
"A directed path is sometimes called a dipath."
"Paths are fundamental concepts of graph theory."
"Paths are described in the introductory sections of most graph theory texts."
"One example of a graph theory text that covers paths is Bondy & Murty (1976)."
"Bondy & Murty (1976), Gibbons (1985), and Diestel (2005) discuss basic concepts and properties of paths in graphs."
"A path in a graph is a sequence of edges that connects a sequence of distinct vertices."
"Yes, advanced algorithmic topics concerning paths in graphs are covered by Korte et al. (1990)."
"Yes, a graph can have both paths and directed paths based on the direction of its edges."
"No, there is no limitation on the number of distinct vertices in a path."
"No, in a directed path, all edges must be directed in the same direction."
"No, a path must have at least one edge connecting the vertices."