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A graph with 6 vertices and 7 edges where the vertex number 6 on the far-left is a leaf vertex or a pendant vertex. In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph ...
A vertex with degree 1 is called a leaf vertex or end vertex or a pendant vertex, and the edge incident with that vertex is called a pendant edge. In the graph on the right, {3,5} is a pendant edge. This terminology is common in the study of trees in graph theory and especially trees as data structures .
1. A leaf vertex or pendant vertex (especially in a tree) is a vertex whose degree is 1. A leaf edge or pendant edge is the edge connecting a leaf vertex to its single neighbour. 2. A leaf power of a tree is a graph whose vertices are the leaves of the tree and whose edges connect leaves whose distance in the tree is at most a given threshold.
They are the graphs that can be built up from a single vertex by a sequence of the following three operations, as shown in the illustration: Add a new pendant vertex connected by a single edge to an existing vertex of the graph. Replace any vertex of the graph by a pair of vertices, each of which has the same set of neighbors as the replaced ...
In graph theory, a universal vertex is a vertex of an undirected graph that is adjacent to all other vertices of the graph. It may also be called a dominating vertex, as it forms a one-element dominating set in the graph. A graph that contains a universal vertex may be called a cone, and its universal vertex may be called the apex of the cone. [1]
For instance, in the octahedron graph, shown in the figure, each vertex has a neighbourhood isomorphic to a cycle of four vertices, so the octahedron is locally C 4. For example: Any complete graph K n is locally K n-1. The only graphs that are locally complete are disjoint unions of complete graphs. A Turán graph T(rs,r) is locally T((r-1)s,r ...
For each vertex we store the list of adjacencies (out-edges) in order of the planarity of the graph (for example, clockwise with respect to the graph's embedding). We then initialize a counter i = n + 1 {\displaystyle i=n+1} and begin a Depth-First Traversal from s {\displaystyle s} .
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