In the mathematical subfield of graph theory a level structure of an undirected graph is a partition of the vertices into subsets that have the same distance from a given root vertex. [1]
In the mathematical subfield of graph theory a level structure of an undirected graph is a partition of the vertices into subsets that have the same distance from a given root vertex. [1]
Given a connected graph G = (V, E) with V the set of vertices and E the set of edges, and with a root vertex r, the level structure is a partition of the vertices into subsets Li called levels, consisting of the vertices at distance i from r. Equivalently, this set may be defined by setting L0 = {r}, and then, for i > 0, defining Li to be the set of vertices that are neighbors to vertices in Li − 1 but are not themselves in any earlier level. [1]
The level structure of a graph can be computed by a variant of breadth-first search: [2] : 176
algorithm level-BFS(G, r): Q ← {r} for ℓ from 0 to ∞: process(Q, ℓ) // the set Q holds all vertices at level ℓ mark all vertices in Q as discovered Q' ← {} for u in Q: for each edge (u, v): if v is not yet marked: add v to Q' if Q' is empty: return Q ← Q'In a level structure, each edge of G either has both of its endpoints within the same level, or its two endpoints are in consecutive levels. [1]
The partition of a graph into its level structure may be used as a heuristic for graph layout problems such as graph bandwidth. [1] The Cuthill–McKee algorithm is a refinement of this idea, based on an additional sorting step within each level. [3]
Level structures are also used in algorithms for sparse matrices, [4] and for constructing separators of planar graphs. [5]
In the mathematical field of graph theory, a bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets and , that is every edge connects a vertex in to one in . Vertex sets and are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.
This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges.
In the mathematical area of graph theory, a clique is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph is an induced subgraph of that is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied.
In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set of vertices such that for every two vertices in , there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in . A set is independent if and only if it is a clique in the graph's complement. The size of an independent set is the number of vertices it contains. Independent sets have also been called "internally stable sets", of which "stable set" is a shortening.
In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. In other words, a subset of the edges is a matching if each vertex appears in at most one edge of that matching. Finding a matching in a bipartite graph can be treated as a network flow problem.
In numerical linear algebra, the Cuthill–McKee algorithm (CM), named after Elizabeth Cuthill and James McKee, is an algorithm to permute a sparse matrix that has a symmetric sparsity pattern into a band matrix form with a small bandwidth. The reverse Cuthill–McKee algorithm (RCM) due to Alan George and Joseph Liu is the same algorithm but with the resulting index numbers reversed. In practice this generally results in less fill-in than the CM ordering when Gaussian elimination is applied.
In graph theory, a dominating set for a graph G is a subset D of its vertices, such that any vertex of G is either in D, or has a neighbor in D. The domination numberγ(G) is the number of vertices in a smallest dominating set for G.
In graph theory, a cut is a partition of the vertices of a graph into two disjoint subsets. Any cut determines a cut-set, the set of edges that have one endpoint in each subset of the partition. These edges are said to cross the cut. In a connected graph, each cut-set determines a unique cut, and in some cases cuts are identified with their cut-sets rather than with their vertex partitions.
In graph theory, reachability refers to the ability to get from one vertex to another within a graph. A vertex can reach a vertex if there exists a sequence of adjacent vertices which starts with and ends with .
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In graph theory, a path decomposition of a graph G is, informally, a representation of G as a "thickened" path graph, and the pathwidth of G is a number that measures how much the path was thickened to form G. More formally, a path-decomposition is a sequence of subsets of vertices of G such that the endpoints of each edge appear in one of the subsets and such that each vertex appears in a contiguous subsequence of the subsets, and the pathwidth is one less than the size of the largest set in such a decomposition. Pathwidth is also known as interval thickness, vertex separation number, or node searching number.
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In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has a vertex of degree at most k: that is, some vertex in the subgraph touches k or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of k for which it is k-degenerate. The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph.
In numerical analysis, nested dissection is a divide and conquer heuristic for the solution of sparse symmetric systems of linear equations based on graph partitioning. Nested dissection was introduced by George (1973); the name was suggested by Garrett Birkhoff.
In graph theory, a shallow minor or limited-depth minor is a restricted form of a graph minor in which the subgraphs that are contracted to form the minor have small diameter. Shallow minors were introduced by Plotkin, Rao & Smith (1994), who attributed their invention to Charles E. Leiserson and Sivan Toledo.
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In combinatorial optimization, the matroid parity problem is a problem of finding the largest independent set of paired elements in a matroid. The problem was formulated by Lawler (1976) as a common generalization of graph matching and matroid intersection. It is also known as polymatroid matching, or the matchoid problem.
In graph theory, the cutwidth of an undirected graph is the smallest integer with the following property: there is an ordering of the vertices of the graph, such that every cut obtained by partitioning the vertices into earlier and later subsets of the ordering is crossed by at most edges. That is, if the vertices are numbered , then for every , the number of edges with and is at most .