In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.
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 graph theory, an undirected graph H is called a minor of the graph G if H can be formed from G by deleting edges, vertices and by contracting edges.
In theoretical computer science, the subgraph isomorphism problem is a computational task in which two graphs G and H are given as input, and one must determine whether G contains a subgraph that is isomorphic to H. Subgraph isomorphism is a generalization of both the maximum clique problem and the problem of testing whether a graph contains a Hamiltonian cycle, and is therefore NP-complete. However certain other cases of subgraph isomorphism may be solved in polynomial time.
The arboricity of an undirected graph is the minimum number of forests into which its edges can be partitioned. Equivalently it is the minimum number of spanning forests needed to cover all the edges of the graph. The Nash-Williams theorem provides necessary and sufficient conditions for when a graph is k-arboric.
In graph theory, a book embedding is a generalization of planar embedding of a graph to embeddings into a book, a collection of half-planes all having the same line as their boundary. Usually, the vertices of the graph are required to lie on this boundary line, called the spine, and the edges are required to stay within a single half-plane. The book thickness of a graph is the smallest possible number of half-planes for any book embedding of the graph. Book thickness is also called pagenumber, stacknumber or fixed outerthickness. Book embeddings have also been used to define several other graph invariants including the pagewidth and book crossing number.
In graph theory, a pseudoforest is an undirected graph in which every connected component has at most one cycle. That is, it is a system of vertices and edges connecting pairs of vertices, such that no two cycles of consecutive edges share any vertex with each other, nor can any two cycles be connected to each other by a path of consecutive edges. A pseudotree is a connected pseudoforest.
In graph theory, the Laman graphs are a family of sparse graphs describing the minimally rigid systems of rods and joints in the plane. Formally, a Laman graph is a graph on n vertices such that, for all k, every k-vertex subgraph has at most 2k − 3 edges, and such that the whole graph has exactly 2n − 3 edges. Laman graphs are named after Gerard Laman, of the University of Amsterdam, who in 1970 used them to characterize rigid planar structures. This characterization, however, had already been discovered in 1927 by Hilda Geiringer.
In the mathematical field of graph theory, a star coloring of a graph G is a (proper) vertex coloring in which every path on four vertices uses at least three distinct colors. Equivalently, in a star coloring, the induced subgraphs formed by the vertices of any two colors has connected components that are star graphs. Star coloring has been introduced by Grünbaum (1973). The star chromatic number of G is the fewest colors needed to star color G.
In graph theory, the tree-depth of a connected undirected graph is a numerical invariant of , the minimum height of a Trémaux tree for a supergraph of . This invariant and its close relatives have gone under many different names in the literature, including vertex ranking number, ordered chromatic number, and minimum elimination tree height; it is also closely related to the cycle rank of directed graphs and the star height of regular languages. Intuitively, where the treewidth of a graph measures how far it is from being a tree, this parameter measures how far a graph is from being a star.
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 graph theory, a k-tree is an undirected graph formed by starting with a (k + 1)-vertex complete graph and then repeatedly adding vertices in such a way that each added vertex v has exactly k neighbors U such that, together, the k + 1 vertices formed by v and U form a clique.
In graph theory, an acyclic orientation of an undirected graph is an assignment of a direction to each edge that does not form any directed cycle and therefore makes it into a directed acyclic graph. Every graph has an acyclic orientation.
In the mathematics of structural rigidity, a rigidity matroid is a matroid that describes the number of degrees of freedom of an undirected graph with rigid edges of fixed lengths, embedded into Euclidean space. In a rigidity matroid for a graph with n vertices in d-dimensional space, a set of edges that defines a subgraph with k degrees of freedom has matroid rank dn − k. A set of edges is independent if and only if, for every edge in the set, removing the edge would increase the number of degrees of freedom of the remaining subgraph.
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.
In the mathematical field of graph theory, the queue number of a graph is a graph invariant defined analogously to stack number using first-in first-out (queue) orderings in place of last-in first-out (stack) orderings.
In graph theory, a family of graphs is said to have bounded expansion if all of its shallow minors are sparse graphs. Many natural families of sparse graphs have bounded expansion. A closely related but stronger property, polynomial expansion, is equivalent to the existence of separator theorems for these families. Families with these properties have efficient algorithms for problems including the subgraph isomorphism problem and model checking for the first order theory of graphs.
In the mathematical fields of graph theory and finite model theory, the logic of graphs deals with formal specifications of graph properties using sentences of mathematical logic. There are several variations in the types of logical operation that can be used in these sentences. The first-order logic of graphs concerns sentences in which the variables and predicates concern individual vertices and edges of a graph, while monadic second-order graph logic allows quantification over sets of vertices or edges. Logics based on least fixed point operators allow more general predicates over tuples of vertices, but these predicates can only be constructed through fixed-point operators, restricting their power.
A sparsity matroid is a mathematical structure that captures how densely a multigraph is populated with edges. To unpack this a little, sparsity is a measure of density of a graph that bounds the number of edges in any subgraph. The property of having a particular matroid as its density measure is invariant under graph isomorphisms and so it is a graph invariant.
The twin-width of an undirected graph is a natural number associated with the graph, used to study the parameterized complexity of graph algorithms. Intuitively, it measures how similar the graph is to a cograph, a type of graph that can be reduced to a single vertex by repeatedly merging together twins, vertices that have the same neighbors. The twin-width is defined from a sequence of repeated mergers where the vertices are not required to be twins, but have nearly equal sets of neighbors.
