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A graph with a loop on vertex 1. In graph theory, a loop (also called a self-loop or a buckle) is an edge that connects a vertex to itself. A simple graph contains no loops. Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow the presence of loops (often in concert with allowing or disallowing ...
One may recover the surface itself by gluing a topological disk to the ribbon graph along each boundary component. The partition of the surface into vertex disks, edge disks, and face disks given by the ribbon graph and this gluing process is a different but related representation of the embedding called a band decomposition. [5] The surface ...
The brute force algorithm finds a 4-clique in this 7-vertex graph (the complement of the 7-vertex path graph) by systematically checking all C(7,4) = 35 4-vertex subgraphs for completeness. In computer science , the clique problem is the computational problem of finding cliques (subsets of vertices, all adjacent to each other, also called ...
Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring ...
Each vertex is assigned to a parallel entity. This vertex centric approach might only work well if the graph depth is very low. Graph depth in BFS is defined as the maximum distance of any vertex in the graph to the source vertex. Therefore, the vertex centric approach is well-suited for GPUs if every thread is mapped to exactly one vertex. [3]
Different total orders may lead to the same acyclic orientation, so an n-vertex graph can have fewer than n! acyclic orientations. The number of acyclic orientations is equal to |χ(−1)|, where χ is the chromatic polynomial of the given graph. [19] The yellow directed acyclic graph is the condensation of the blue directed graph.
On unit disk graphs its approximation ratio is 3. [21] The triangular prism is the smallest graph for which one of its degeneracy orderings leads to a non-optimal coloring, and the square antiprism is the smallest graph that cannot be optimally colored using any of its degeneracy orderings. [18]
Every graph has a cycle basis in which every cycle is an induced cycle. In a 3-vertex-connected graph, there always exists a basis consisting of peripheral cycles, cycles whose removal does not separate the remaining graph. [4] [5] In any graph other than one formed by adding one edge to a cycle, a peripheral cycle must be an induced cycle.