Apex graphIn graph theory, a branch of mathematics, an apex graph is a graph that can be made planar by the removal of a single vertex. The deleted vertex is called an apex of the graph. It is an apex, not the apex because an apex graph may have more than one apex; for example, in the minimal nonplanar graphs K_5 or K_3,3, every vertex is an apex. The apex graphs include graphs that are themselves planar, in which case again every vertex is an apex. The null graph is also counted as an apex graph even though it has no vertex to remove.
K-treeIn 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. The k-trees are exactly the maximal graphs with a treewidth of k ("maximal" means that no more edges can be added without increasing their treewidth).
Wagner graphIn the mathematical field of graph theory, the Wagner graph is a 3-regular graph with 8 vertices and 12 edges. It is the 8-vertex Möbius ladder graph. As a Möbius ladder, the Wagner graph is nonplanar but has crossing number one, making it an apex graph. It can be embedded without crossings on a torus or projective plane, so it is also a toroidal graph. It has girth 4, diameter 2, radius 2, chromatic number 3, chromatic index 3 and is both 3-vertex-connected and 3-edge-connected.
Tree-depthIn 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.
Caterpillar treeIn graph theory, a caterpillar or caterpillar tree is a tree in which all the vertices are within distance 1 of a central path. Caterpillars were first studied in a series of papers by Harary and Schwenk. The name was suggested by Arthur Hobbs. As colorfully write, "A caterpillar is a tree which metamorphoses into a path when its cocoon of endpoints is removed." The following characterizations all describe the caterpillar trees: They are the trees for which removing the leaves and incident edges produces a path graph.
Degeneracy (graph theory)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.
Apollonian networkIn combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing construction.
Haven (graph theory)In graph theory, a haven is a certain type of function on sets of vertices in an undirected graph. If a haven exists, it can be used by an evader to win a pursuit–evasion game on the graph, by consulting the function at each step of the game to determine a safe set of vertices to move into. Havens were first introduced by as a tool for characterizing the treewidth of graphs. Their other applications include proving the existence of small separators on minor-closed families of graphs, and characterizing the ends and clique minors of infinite graphs.
Circle graphIn graph theory, a circle graph is the intersection graph of a chord diagram. That is, it is an undirected graph whose vertices can be associated with a finite system of chords of a circle such that two vertices are adjacent if and only if the corresponding chords cross each other. gives an O(n2)-time algorithm that tests whether a given n-vertex undirected graph is a circle graph and, if it is, constructs a set of chords that represents it. A number of other problems that are NP-complete on general graphs have polynomial time algorithms when restricted to circle graphs.
Planar graphIn 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.
