Path graph

A path graph is a simple graph whose vertices can be ordered so that edges join exactly consecutive vertices. It is connected, has no cycles, and has two vertices of degree one when it has at least two vertices.

Graph theory: The mathematical study of graphs, structures made of vertices joined by edges. Path graphs are among the simplest structures studied in graph theory.
Adjacency matrix: A square matrix records which pairs of graph vertices are adjacent. For a path graph, its matrix has nonzero entries only beside the main diagonal, under path ordering.
Pathwidth: Pathwidth measures how closely a graph can be represented by a sequence of overlapping vertex sets. A path graph has pathwidth one when it has at least two vertices.
Cycle graph: A cycle graph connects vertices in a closed loop, with each vertex adjacent to two others. Closing the path's endpoints creates a cycle and removes its degree-one vertices.
Vertex (graph theory): A vertex is a basic object in a graph, representing an entity or position. The vertices form the ordered sequence that defines a path graph.
Graph distance: Graph distance is the length of a shortest path between two vertices. In a path graph, distance equals the number of edges between positions in its vertex order.
Dynamic programming on graphs: Dynamic programming on graphs solves problems by combining solutions for smaller subgraphs. A path's linear structure makes subproblems along successive vertices especially simple.
Star graph: A star graph has one central vertex connected to every other vertex, with no other edges. Unlike a path's maximum degree of two, a star can have a high-degree center.
Edge (graph theory): An edge is a connection between two vertices in a graph. Each edge joins one pair of consecutive vertices.
Graph traversal: A graph traversal systematically visits vertices by following edges. Following consecutive edges visits every path-graph vertex in sequence.

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