Discover published sets by community

A graph is an ordered pair $G = (V, E)$ comprising a set $V$ of vertices or nodes together with a set $E$ of edges or arcs, which are 2-element subsets of $V$.

A subgraph is a graph $S = (V', E')$ where $V' \subseteq V$, $E' \subseteq E$, and every edge in $E'$ connects two vertices in $V'$.

An adjacency list is a collection of unordered lists used to represent a finite graph. Each list describes the set of neighbors of a vertex in the graph.

A weighted graph is a graph in which each edge is assigned a weight or cost, which is often used to represent the cost of traveling between the vertices connected by the edge.

A bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets $U$ and $V$ such that every edge connects a vertex in $U$ to one in $V$.

A planar graph is a graph that can be drawn on a plane without any edges crossing each other.

A Hamiltonian path is a path in an undirected or directed graph that visits each vertex exactly once.

A tree is an undirected graph in which any two vertices are connected by exactly one path. In other words, a connected graph without cycles.

A cycle in a graph is a path that starts and ends at the same vertex and includes at least one other vertex, with no vertices repeated.

A complete graph is a simple graph in which every pair of distinct vertices is connected by a unique edge.

The chromatic number of a graph is the smallest number of colors needed to color the vertices of a graph so that no two adjacent vertices share the same color.

A directed graph or digraph is a graph in which the edges have a direction, indicated by an arrowhead, and the edges are called arcs.

A graph is connected if there is a path between every pair of vertices.

A forest is a disjoint union of trees, or equivalently, an undirected graph without cycles.

A vertex cover of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. The size of the smallest possible vertex cover is called the vertex cover number.

Euler's formula states that for a connected planar graph with $V$ vertices, $E$ edges, and $F$ faces (including the outer one), the formula $V - E + F = 2$ holds.

An adjacency matrix is a square matrix used to represent a finite graph where each element of the matrix indicates whether pairs of vertices are adjacent or not in the graph.

A path in a graph is a finite or infinite sequence of edges which connect a sequence of vertices which, by most definitions, are all distinct from one another.

A simple graph is a graph that does not contain any loops or multiple edges. The edges in a simple graph are implied to be undirected.

An incidence matrix is a matrix that shows the relationship between edges and vertices in a graph where rows represent vertices and columns represent edges, indicating which vertex is incident to which edge.

An Eulerian path is a path in a graph which visits every edge exactly once.

The degree of a vertex in a graph is the number of edges that are incident to the vertex, with loops counted twice.

A Hamiltonian cycle is a cycle in an undirected or directed graph that visits each vertex exactly once and returns to the starting vertex.

An Eulerian cycle is a cycle in a graph that visits every edge exactly once and returns to the starting vertex.